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https://www.jiskha.com/search?query=the+activity+of+a+sample+of+carbon+has+one-eighth+of+the+initial+activity.+carbon-14+has+a+half-life+of+5730+yr.+how+old+is+the+sample+in+yr%3F	1,534,430,970,000,000,000	text/html	crawl-data/CC-MAIN-2018-34/segments/1534221211000.35/warc/CC-MAIN-20180816132758-20180816152758-00083.warc.gz	909,541,915	12,822	# the activity of a sample of carbon has one-eighth of the initial activity. carbon-14 has a half-life of 5730 yr. how old is the sample in yr? 42,855 results 1. ## college physics part 2 the activity of a sample of carbon has one-eighth of the initial activity. carbon-14 has a half-life of 5730 yr. how old is the sample in yr? 2. ## ChemB The activity of carbon 14 in living tissue is 15.3 disintegrations per minute per gram of carbon. The limit for reliable determination of age is based on carbon 14 is .10 disintegration per min per gram of carbon. Calculate the max age of a sample that can 3. ## science The half-life of carbon 14 is 5730 years. if a 1 g sample of old carbon is 1/8 as radioactive as 1 g of a current sample, then the age of the old sample is about ___________years. 22,900 17,200 11,500 716 4. ## geology suppose you have a sample of clamshell at a paleoindian site and you measure the 14c activity of aa 100-gram sample of carbon as 320 disintegretions per scond (or 3.2 disingtegrations per gram of carbon per second). a. what was the activity(in 5. ## Physicr A 10 kg sample from a living plant has actiuity 30 000 Bq due to Carbon-14. A 500g sample of the same type of plant but dead has activity 1 000 Bq. Calculate the age of the dead sample given that the half life of Carbon-14 is 5568 years 6. ## nuclear Calculate the age of a plant sample that shows about one-eighth of the carbon 14 of a living sample. The half-life of carbon-14 is about 5760 a. Help please... 7. ## nuclear physics Calculate the age of a plant sample that shows about one-eighth of the carbon 14 of a living sample. The half-life of carbon 14 is about 5760a How do you do this? 8. ## physics A sample has a activity of 0.0065 Bq per gram of carbon. (a) Find the age of the sample, assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq. (b) Evidence suggests that the value of 0.23 Bq might have 9. ## physics 2 A sample has a activity of 0.0065 Bq per gram of carbon. (a) Find the age of the sample, assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq. (b) Evidence suggests that the value of 0.23 Bq might have Beryllium-7 has a half-life of 53 D. A sample was observed for one minute and there were 26,880 decays. a.) what is the activity level of the sample? b.) what will the activity level of the sample be after 265 d? c.) after how many days will be activity 11. ## Chemistry Carbon-14 has a half-life of 5730 y. How much of a 144 g sample of a carbon-14 will remain after 1.719 (times) 10 4y? 12. ## Chemistry Carbon-14 has a half-life of 5730 y. How much of a 144 g sample of a carbon-14 will remain after 1.719 (times) 10 4y? 13. ## Physics When any radioactive dating method is used, experimental error in the measurement of the sample's activity leads to error in the estimated age. In an application of the radiocarbon dating technique to certain fossils, an activity of 0.15 Bq per gram of 14. ## chemistry A wooden object from a prehistoric site has a carbon-14 activity of 10 counts per min.(cpm) compared to 40 cpm for new wood. If carbon-14 has a half life of 5730 years what is the age of the wood? Please get me started. Thank you 15. ## chem Can someone please help me with the following questions? 1:Suppose you were given an ancient wooden box. If you analyze the box for carbon-14 activity and find that it is 50% of that of a new piece of wood of the same size, how old is the wood in the box? 16. ## chemistry The half-life for the radioactive decay of calcium-47 is 4.5 days.If a sample has an activity of 4.00uCi after 13.5 days, what was the initial activity of the sample? 17. ## college geology help please I was the same person from yesterday. I need help on a question. I don't type very good so I have one of my host family do it for me. so please help us. Suppose you have a sample of clamshell at a Paleoindian site and you measure the 14C activity of a 18. ## college Geol 101 Suppose you have a sample of clamshell at a Paleoindian site and you measure the 14C activity of a 100-gram sample of carbon as 320 disintegrations per minute (or 3.2 disintegrations per gram of carbon per minute). a.What was the activity (in 19. ## physics The half-life of carbon-14 is 5730 years. How much of a 50 g sample of carbon-14 is present after 11,460 years? 20. ## calculus Carbon-14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a half-life (the time it takes for half the amount of a sample to decay) of approximately 5730 years. Using 21. ## Physics The isotope carbon-14, 146C, is radioactive and has a half-life of 5730 years. If you start with a sample of 1000 carbon-14 nuclei, how many will still be around in 22920 years? I got 62.5 atoms, am I correct?? Thank you!! The activity of a sample of Ba-137m has decreased by 75% from its initial activity after 5.10 minutes. What is the half-life of Ba-137m? What fraction of activity remains after 6.25 minutes? (A=Aoe (-kt); k = .693/t1/2) 23. ## Physics University students, studying the activity of a particular radioactive isotope which had a half-life of 12.0 hours. If the original activity of the sample was 448 kBq, what would the activity be 3.00 days later? 24. ## science The half-life of carbon-14 is 5730 years. If we start with 10 grams of carbon-14, after 5730 years, we will have _____ of carbon-14 left. 25. ## science The half-life of carbon-14 is 5730 years. If we start with 10 grams of carbon-14, after 5730 years, we will have _____ of carbon-14 left. 26. ## half-life ok so i got... the half life of carbon-14 is 5730 years, the relation c+(1/2)^ N/5730 is used to calc. the concentration, c, in parts per trillion remaining n years after death determine the carbon concentration in a 11460 year old bone. please help :) 27. ## half-life ok so i got... the half life of carbon-14 is 5730 years, the relation c+(1/2)^ N/5730 is used to calc. the concentration, c, in parts per trillion remaining n years after death determine the carbon concentration in a 11460 year old bone. please help :) 28. ## chemistry A 1.00-g sample of carbon from a modern source gave 15.3 disintegrations per minute. A sample of carbon from an “old” source gave 920 disintegrations per hour. What is the age of the “old” sample of carbon? The half-life of carbon-14 is 5.73×103 29. ## chemistry A 1.00-g sample of carbon from a modern source gave 15.3 disintegrations per minute. A sample of carbon from an “old” source gave 920 disintegrations per hour. What is the age of the “old” sample of carbon? The half-life of carbon-14 is 5.73×103 30. ## Nuclear Physics An investigator receives Co-60 (5.27 year half life) for use in a research study. Unfortunately the Co-60 is contaminated with Cs-137 (30.0 year half life). The initial Co-60 activity is 400 times the initial Cs-137 activity. How long after the initial 31. ## physics Suppose 32000 radioactive nuclei are in a sample. About how many remain after two days if the half-life is 22 hrs? What is the initial activity of the sample in decays per minute? 32. ## physics How do I do this ? Calculate the age of a plant sample that shows about one-eigth of the carbon 14 of a living sample. The half-life of carbon-14 is about 5760a I don't have the notes on how to do this one... Help 33. ## physics After 1.84 days, the activity of a sample of an unknown type radioactive material has decreased to 85.8% of the initial activity. What is the half-life of this material? (in days) 34. ## Chemistry The cloth shroud from around a mummy is found to have a carbon-14 activity of 8.1 disintegrations per minute per gram of carbon as compared with living organisms that undergo 15.2 disintegrations per minute per gram of carbon. From the half-life for 35. ## Algebra An artifact was found and tested for its carbon-14 content. If 74% of the orginal carbon-14 was still present, what is its probable age (to the nearest 100 years)? (Carbon-14 has a half life of 5730 years). 36. ## physics before 1900 the activity per mass of atmospheric carbon due to the presence of carbon-14 averaged about 0.255bq/g of carbon 1. what fraction of carbon is carbon-14 37. ## physics before 1900 the activity per mass of atmospheric carbon due to the presence of carbon-14 averaged about 0.255bq/g of carbon 1. what fraction of carbon is carbon-14 38. ## Pre Calculus The burial cloth of an Egyptian mummy is estimated to contain 56% of the carbon-14 it contained originally. How long ago was the mummy buried? (the half-life of carbon-14 is 5730). Please round the answer to the nearest tenth. I have figured that: m(t) = 39. ## physics A wooden artifact is found in an ancient tomb. Its carbon-14 activity is measured to be 60.0% of that in a fresh sample of wood from the same region. Assuming the same amount of carbon-14 was initially present in the wood from which the artifact was made, 40. ## Math Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react w/ nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide. As long as it is alive, carbon-14 is maintained in the 41. ## Math The half-life of carbon-14, which is used in dating archaeological finds, is 5730 yr. Assume that 100% of the carbon-14 is present at time 0 yr, or x=0. Write the equation that expresses the percentage of carbon-14 remaining as a function of time. 42. ## Calculus An artifact was found with 63.8% of Carbon 14, how old is the mummy, assuming the half-life of Carbon 14 is 5730 years. 43. ## Chemistry A piece of wood found in an ancient city has a carbon-14 to carbon-12 ratio that is one-eighth the carbon-14 to carbon-12 ratio of a tree growing nearby. How old is the piece of wood? (The half-life of carbon-14 is 5,715 years.) 44. ## Physics Carbon-15 has a half-life of 5730 years. As observed from Earth, what would the half-life of carbon-15 be if it traveled through space at 25% of the speed of light, relative to Earth? 45. ## Calculus The amount of carbon-14 still present in a sample after t years is given by the function where C0 is the initial amount. Estimate the age of a sample of wood discovered by an archeologist if the carbon level in the sample is only 18% of its original 46. ## Chem Living organisms give a Carbon-14 decay rate of 15.3 counts/min for each gram of carbon in the sample. A sample of bristle cone pine wood has a decay rate of 3.20 counts/min per gram of carbon. Calculate the age of the wood from the radiochemical evidence. 47. ## Biology The half-life of Carbon-14 is 5730 years. What is the age of a fossil containing 1/16 the amount of Carbon-14 of living organisms? Explain your calculation. 48. ## Physics (Inside the atom) A sample of radioactive isotope I is to be used for medical diagnosis of the kidneys. The isotope has a half-life of 8.0 days, and the sample is required to have an activity of 8 x 10^5 per second at the time it is given to the patient. Calculate the mass 49. ## chemistry The half-life of oxygen-15 is 124 s. If a sample of oxygen-15 has an activity of 4800 Bq, how many minutes will elapse before it has an activity of 600 Bq? 50. ## chem the half life of oxygen-15 is 124 s. if a sample of oxygen-15 has an activity of 4000 Bq, how many minutes will elapse before it reaches an activity of 500Bq 51. ## ChemB measurements of carbon 14 taken from linen wrappings of the book of isaiah from the dead sea scrolls indicate that the scrolls contain 79.5% of the carbon 14 expected in living tissue. how old are these scrolls if the half life of carbon 14 is 5730 years? 52. ## chem A sample of radon has an activity of 80,000 Bq. If the half-life of radon is 15 h, how long before the sample's activity is 5,000 Bq? By comparing the amount of carbon-14 to amount of carbon-12, one can determine approx how long ago the organism died. The half-life of carbon-14 is 5730 years. Assume the initial quantity of carbon-14 is 600 milligrams. the equation is A (t)=600*(.5)^(t) 54. ## Physics (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample 55. ## Physics (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample 56. ## Chemistry Radium-223 has a half-life of 11.4 days.Approximately how long would it take for the activity of a sample of 223Ra to decrease to 2.00 % of its initial value? 57. ## Chemistry Radium-223 has a half-life of 11.4 days. Approximately how long would it take for the activity of a sample of 223Ra to decrease to 2.00 % of its initial value? 58. ## Intermediate Maths (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample 59. ## Final Maths (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample 60. ## Chemistry What is the activity of a 18.9 μCi sample of carbon-14 in becquerels? 61. ## math: pre-calculus You have 5 grams of carbon-14; whose half-life is 5730 years. a)Write the rule of the function that gives the amount of carbon-14 remaining after x years. b)How much carbon-14 will be left after 4,000 years? c)When will there be just 1 gram left? 62. ## PreCalculus The half life of Carbon-14 is 5730 years. a. find the rule of the function that gives the amount remaining from an initial quantity of 100 milligrams of carbon14 after t years. b. the burial cloth of an egyptian mummy i sestimated to contain 59% of the 63. ## Chemistry A sample of sodium-24 with an activity of 12 mCi is used to study the rate of blow flow in the circulatory system. If sodium -24 has a half-life of 15 hours, what is the activity of the sodium after 2.5 d ? 64. ## Chemistry Numerical The half life period of a radioactive element is 27.96 days. Calculate the time taken by a given sample to reduce to 1/8th of its initial activity 65. ## chemistry a sample of sodium-24 with an activity of 12 mCi is used to study the rate of blood flow in the circulatory system. If sodium-24 has a half-life of 15 h, what is the activity of the sodium after 2.5 days. please show work. 66. ## math only 5% of the carbon-14 in a small wooden bowl remains , how old is the bowl? Hint, half-life for carbon –14 us 5730 years 67. ## math Carbon-14 has a half-life of 5730 years. How long will it take 17 grams of carbon-14 to be reduced to 10 grams? Round to the nearest integer. 68. ## Math A plant fossil lost 40 % of its carbon-14. How old is the fossil? (Half-life of carbon-14 is 5730 years.) 69. ## physics If an archaeologist finds an ancient fire pit containing partially consumed firewood and the 14C content of the wood is only 10% of an equal carbon sample from present day tree, what is the age of ancient site. 14 C has half life of 5730 years 70. ## physical science a scientist found a fossilized bison. using Carbon-14 dating, she found that the fossil was about 45,840 years old. Carbon-14 has a half-life of 5730 years. how many half-lives have passed? 71. ## chemistry The amount of organic carbon in a sample can be obtained by IR analysis of the carbon dioxide (CO2) released following complete combustion of the sample in oxygen. A reference sample of an organic compound is usually used to calibrate the instrument 72. ## Precalculus The burial cloth of an Egyptian mummy is estimated to contain 57% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.) 73. ## Precalculus The burial cloth of an Egyptian mummy is estimated to contain 57% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.) 74. ## calculus A wooden artifact recovered from a tomb contains 29% of the carbon-14 that is present in living trees. The half life of carbon-14 is 5730 years. How long ago was the artifact made? 75. ## Physics A radioactive isotope of Bismuth undergoes Beta Decay with a half-life of 22 days. How long (in days) will it take the Activity (A) of the sample to decrease to one-ninth of its original value? Assume that the original activity is A0. 76. ## Chemistry One last question...even though no one seems to be willing to help me out without being rude. The blood volume in a cancer patient was measured by injecting 5.0 mL of Na2SO4(aq) labeled with 35S (t1/2 = 87.4 d). The activity of the sample was 300 µCi. 77. ## chemistry The radioactive decay of carbon-14 is first-order and the half-life is 5800 years. While a plant or animal is living, it has a constant proportion of carbon-14 (relative to carbon-12) in its composition. When the organism dies, the proportion of carbon-14 78. ## Chemistry A sample of charcoal is found to have only 1/8th of Carbon-14. Half-life of Carbon-14 is 5730yrs, what is the age of the charcoal? 79. ## Chemistry Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq? 80. ## Chemistry Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq? 81. ## Chemistry Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq? 82. ## Chemistry Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq? 83. ## math A wooden artifact from an ancient tomb contains 70% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.) 84. ## Biology The half-life of carbon-14 is 5,700 years. If a sample originally had 26 g of carbon-14, how much would it contain after 22,800 years? 85. ## chem A typical dose of a radioactive sample is 27.0 mCi. How long does it take for the activity to reduce to 0.100 mCi? The half life of the sample is 211,000 y. 86. ## MATH 123 A wooden artifact from an ancient tomb contains 55% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.) 87. ## Intro to Astronomy The half-life of carbon 14, which is commonly used to date organic materials, is 5700 years. What is the minimum age of sample in which less than 1% of the organic carbon 14 is left? 88. ## Chemistry The strontium-90 isotope decays by the reaction below. It has a half-life of 28 yr. If the initial activity of this isotope was 168dpm, what would the activity be after 28.0 yr? 89. ## Chemistry Radium-223 has a half-life of 11.4 days.Approximately how long would it take for the activity of a sample of Radium-223 to decrease to 1.00% of its initial value? 90. ## Calculus A fragment of bone is discovered to contain 20% of the usual carbon-14 concentration. Estimate the age of the bone to the nearest hundred years, given that Carbon-1 is radioactive with half-life of 5730 years and the rate of decay is given by the following 91. ## College Algebra This exercise uses the radioactive decay model. A wooden artifact from an ancient tomb contains 80% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the 92. ## chem Radioactive substances decay by first-order kinetics. How many years would be required for a sample containing strontium-90 to decrease to 58.32% of its initial activity? The half-life of strontium-90 is 2.88e1 years. 93. ## Physics 30 The activity of a sample of a radioactive isotope is 1800Bq. If this isotope has a half-life of 16 days, what is the activity after 16 days?, 24 days?, and 60 days? 1. A granite rock is thought to be about 2 billion years old. why is it not possible to determine the age of the Rock using carbon-14 dating? 2. A hair sample has 80% of its original carbon-14 present. what is the age of the sample? 3. A bone fragment has 95. ## Chem If you have a 12.01g sample carbon. What would be the average mass of a carbon atoms if it is 1.994 x 10 to the 23 power g. How many carbon atoms are in the sample? 96. ## math 1.The remains of an old campﬁre are unearthed and it is found that there is only 80% as much radioactive carbon-14 in the charcoal samples from the campﬁre as there is in modern living trees. If the half-life of carbon- 14 is 5730 years, how 97. ## Chemistry An artifact has a carbon-14 to carbon-12 ratio that is one-fourth the carbon-14 to carbon-12 ratio of a similar modern object. How old is the artifact? (The half-life of carbon-14 is 5,715 years.) 98. ## Chemistry Q1. The blood volume in a cancer patient was measured by injecting 5.0 mL of Na2SO4(aq) labeled with 35S (t1/2 = 87.4 d). The activity of the sample was 300 µCi. After 22 min, 12.9 mL of blood was withdrawn from the man and the activity of that sample was 99. ## chemistry You have a 12.01 g sample of carbon. The average mass of a carbon atoms is 1.994 x 1023 g. How many carbon atoms are in the sample? 100. ## chemistry Assayed for LDH activity were 5 microliters of a sample that was diluted 6 to 1. The activity in the reaction vessel, which has a volume of 3 mililiters, is 0.30 U. What is the ΔA/min observed? What is the relative activity of the original sample?	6,016	22,391	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.578125	4	CC-MAIN-2018-34	latest	en	0.921395	# the activity of a sample of carbon has one-eighth of the initial activity. carbon-14 has a half-life of 5730 yr. how old is the sample in yr?. 42,855 results. 1. ## college physics part 2. the activity of a sample of carbon has one-eighth of the initial activity. carbon-14 has a half-life of 5730 yr. how old is the sample in yr?. 2. ## ChemB. The activity of carbon 14 in living tissue is 15.3 disintegrations per minute per gram of carbon. The limit for reliable determination of age is based on carbon 14 is .10 disintegration per min per gram of carbon. Calculate the max age of a sample that can. 3. ## science. The half-life of carbon 14 is 5730 years. if a 1 g sample of old carbon is 1/8 as radioactive as 1 g of a current sample, then the age of the old sample is about ___________years. 22,900 17,200 11,500 716. 4. ## geology. suppose you have a sample of clamshell at a paleoindian site and you measure the 14c activity of aa 100-gram sample of carbon as 320 disintegretions per scond (or 3.2 disingtegrations per gram of carbon per second). a. what was the activity(in. 5. ## Physicr. A 10 kg sample from a living plant has actiuity 30 000 Bq due to Carbon-14. A 500g sample of the same type of plant but dead has activity 1 000 Bq. Calculate the age of the dead sample given that the half life of Carbon-14 is 5568 years. 6. ## nuclear. Calculate the age of a plant sample that shows about one-eighth of the carbon 14 of a living sample. The half-life of carbon-14 is about 5760 a. Help please.... 7. ## nuclear physics. Calculate the age of a plant sample that shows about one-eighth of the carbon 14 of a living sample. The half-life of carbon 14 is about 5760a How do you do this?. 8. ## physics. A sample has a activity of 0.0065 Bq per gram of carbon. (a) Find the age of the sample, assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq. (b) Evidence suggests that the value of 0.23 Bq might have. 9. ## physics 2. A sample has a activity of 0.0065 Bq per gram of carbon. (a) Find the age of the sample, assuming that the activity per gram of carbon in a living organism has been constant at a value of 0.23 Bq. (b) Evidence suggests that the value of 0.23 Bq might have. Beryllium-7 has a half-life of 53 D. A sample was observed for one minute and there were 26,880 decays. a.) what is the activity level of the sample? b.) what will the activity level of the sample be after 265 d? c.) after how many days will be activity. 11. ## Chemistry. Carbon-14 has a half-life of 5730 y. How much of a 144 g sample of a carbon-14 will remain after 1.719 (times) 10 4y?. 12. ## Chemistry. Carbon-14 has a half-life of 5730 y. How much of a 144 g sample of a carbon-14 will remain after 1.719 (times) 10 4y?. 13. ## Physics. When any radioactive dating method is used, experimental error in the measurement of the sample's activity leads to error in the estimated age. In an application of the radiocarbon dating technique to certain fossils, an activity of 0.15 Bq per gram of. 14. ## chemistry. A wooden object from a prehistoric site has a carbon-14 activity of 10 counts per min.(cpm) compared to 40 cpm for new wood. If carbon-14 has a half life of 5730 years what is the age of the wood? Please get me started. Thank you. 15. ## chem. Can someone please help me with the following questions? 1:Suppose you were given an ancient wooden box. If you analyze the box for carbon-14 activity and find that it is 50% of that of a new piece of wood of the same size, how old is the wood in the box?. 16. ## chemistry. The half-life for the radioactive decay of calcium-47 is 4.5 days.If a sample has an activity of 4.00uCi after 13.5 days, what was the initial activity of the sample?. 17. ## college geology help please. I was the same person from yesterday. I need help on a question. I don't type very good so I have one of my host family do it for me. so please help us. Suppose you have a sample of clamshell at a Paleoindian site and you measure the 14C activity of a. 18. ## college Geol 101. Suppose you have a sample of clamshell at a Paleoindian site and you measure the 14C activity of a 100-gram sample of carbon as 320 disintegrations per minute (or 3.2 disintegrations per gram of carbon per minute). a.What was the activity (in. 19. ## physics. The half-life of carbon-14 is 5730 years. How much of a 50 g sample of carbon-14 is present after 11,460 years?. 20. ## calculus. Carbon-14 is a radioactive substance produced in the Earth's atmosphere and then absorbed by plants and animals on the surface of the earth. It has a half-life (the time it takes for half the amount of a sample to decay) of approximately 5730 years. Using. 21. ## Physics. The isotope carbon-14, 146C, is radioactive and has a half-life of 5730 years. If you start with a sample of 1000 carbon-14 nuclei, how many will still be around in 22920 years? I got 62.5 atoms, am I correct?? Thank you!!. The activity of a sample of Ba-137m has decreased by 75% from its initial activity after 5.10 minutes. What is the half-life of Ba-137m? What fraction of activity remains after 6.25 minutes? (A=Aoe (-kt); k = .693/t1/2). 23. ## Physics. University students, studying the activity of a particular radioactive isotope which had a half-life of 12.0 hours. If the original activity of the sample was 448 kBq, what would the activity be 3.00 days later?. 24. ## science. The half-life of carbon-14 is 5730 years. If we start with 10 grams of carbon-14, after 5730 years, we will have _____ of carbon-14 left.. 25. ## science. The half-life of carbon-14 is 5730 years. If we start with 10 grams of carbon-14, after 5730 years, we will have _____ of carbon-14 left.. 26. ## half-life. ok so i got... the half life of carbon-14 is 5730 years, the relation c+(1/2)^ N/5730 is used to calc. the concentration, c, in parts per trillion remaining n years after death determine the carbon concentration in a 11460 year old bone. please help :). 27. ## half-life. ok so i got... the half life of carbon-14 is 5730 years, the relation c+(1/2)^ N/5730 is used to calc. the concentration, c, in parts per trillion remaining n years after death determine the carbon concentration in a 11460 year old bone. please help :). 28. ## chemistry. A 1.00-g sample of carbon from a modern source gave 15.3 disintegrations per minute. A sample of carbon from an “old” source gave 920 disintegrations per hour. What is the age of the “old” sample of carbon? The half-life of carbon-14 is 5.73×103. 29. ## chemistry. A 1.00-g sample of carbon from a modern source gave 15.3 disintegrations per minute. A sample of carbon from an “old” source gave 920 disintegrations per hour. What is the age of the “old” sample of carbon? The half-life of carbon-14 is 5.73×103. 30. ## Nuclear Physics. An investigator receives Co-60 (5.27 year half life) for use in a research study. Unfortunately the Co-60 is contaminated with Cs-137 (30.0 year half life). The initial Co-60 activity is 400 times the initial Cs-137 activity. How long after the initial. 31. ## physics. Suppose 32000 radioactive nuclei are in a sample. About how many remain after two days if the half-life is 22 hrs? What is the initial activity of the sample in decays per minute?. 32. ## physics. How do I do this ? Calculate the age of a plant sample that shows about one-eigth of the carbon 14 of a living sample. The half-life of carbon-14 is about 5760a I don't have the notes on how to do this one... Help. 33. ## physics. After 1.84 days, the activity of a sample of an unknown type radioactive material has decreased to 85.8% of the initial activity. What is the half-life of this material? (in days). 34. ## Chemistry. The cloth shroud from around a mummy is found to have a carbon-14 activity of 8.1 disintegrations per minute per gram of carbon as compared with living organisms that undergo 15.2 disintegrations per minute per gram of carbon. From the half-life for. 35. ## Algebra. An artifact was found and tested for its carbon-14 content. If 74% of the orginal carbon-14 was still present, what is its probable age (to the nearest 100 years)? (Carbon-14 has a half life of 5730 years).. 36. ## physics. before 1900 the activity per mass of atmospheric carbon due to the presence of carbon-14 averaged about 0.255bq/g of carbon 1. what fraction of carbon is carbon-14. 37. ## physics. before 1900 the activity per mass of atmospheric carbon due to the presence of carbon-14 averaged about 0.255bq/g of carbon 1. what fraction of carbon is carbon-14. 38. ## Pre Calculus. The burial cloth of an Egyptian mummy is estimated to contain 56% of the carbon-14 it contained originally. How long ago was the mummy buried? (the half-life of carbon-14 is 5730). Please round the answer to the nearest tenth. I have figured that: m(t) =. 39. ## physics. A wooden artifact is found in an ancient tomb. Its carbon-14 activity is measured to be 60.0% of that in a fresh sample of wood from the same region. Assuming the same amount of carbon-14 was initially present in the wood from which the artifact was made,. 40. ## Math. Cosmic ray bombardment of the atmosphere produces neutrons, which in turn react w/ nitrogen to produce radioactive carbon-14. Radioactive carbon-14 enters all living tissue through carbon dioxide. As long as it is alive, carbon-14 is maintained in the. 41. ## Math. The half-life of carbon-14, which is used in dating archaeological finds, is 5730 yr. Assume that 100% of the carbon-14 is present at time 0 yr, or x=0. Write the equation that expresses the percentage of carbon-14 remaining as a function of time.. 42. ## Calculus. An artifact was found with 63.8% of Carbon 14, how old is the mummy, assuming the half-life of Carbon 14 is 5730 years.. 43. ## Chemistry. A piece of wood found in an ancient city has a carbon-14 to carbon-12 ratio that is one-eighth the carbon-14 to carbon-12 ratio of a tree growing nearby. How old is the piece of wood? (The half-life of carbon-14 is 5,715 years.). 44. ## Physics. Carbon-15 has a half-life of 5730 years. As observed from Earth, what would the half-life of carbon-15 be if it traveled through space at 25% of the speed of light, relative to Earth?. 45. ## Calculus. The amount of carbon-14 still present in a sample after t years is given by the function where C0 is the initial amount. Estimate the age of a sample of wood discovered by an archeologist if the carbon level in the sample is only 18% of its original. 46. ## Chem. Living organisms give a Carbon-14 decay rate of 15.3 counts/min for each gram of carbon in the sample. A sample of bristle cone pine wood has a decay rate of 3.20 counts/min per gram of carbon. Calculate the age of the wood from the radiochemical evidence.. 47. ## Biology. The half-life of Carbon-14 is 5730 years. What is the age of a fossil containing 1/16 the amount of Carbon-14 of living organisms? Explain your calculation.. 48. ## Physics (Inside the atom). A sample of radioactive isotope I is to be used for medical diagnosis of the kidneys. The isotope has a half-life of 8.0 days, and the sample is required to have an activity of 8 x 10^5 per second at the time it is given to the patient. Calculate the mass. 49. ## chemistry. The half-life of oxygen-15 is 124 s. If a sample of oxygen-15 has an activity of 4800 Bq, how many minutes will elapse before it has an activity of 600 Bq?. 50. ## chem. the half life of oxygen-15 is 124 s. if a sample of oxygen-15 has an activity of 4000 Bq, how many minutes will elapse before it reaches an activity of 500Bq.	51. ## ChemB. measurements of carbon 14 taken from linen wrappings of the book of isaiah from the dead sea scrolls indicate that the scrolls contain 79.5% of the carbon 14 expected in living tissue. how old are these scrolls if the half life of carbon 14 is 5730 years?. 52. ## chem. A sample of radon has an activity of 80,000 Bq. If the half-life of radon is 15 h, how long before the sample's activity is 5,000 Bq?. By comparing the amount of carbon-14 to amount of carbon-12, one can determine approx how long ago the organism died. The half-life of carbon-14 is 5730 years. Assume the initial quantity of carbon-14 is 600 milligrams. the equation is A (t)=600*(.5)^(t). 54. ## Physics. (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample. 55. ## Physics. (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample. 56. ## Chemistry. Radium-223 has a half-life of 11.4 days.Approximately how long would it take for the activity of a sample of 223Ra to decrease to 2.00 % of its initial value?. 57. ## Chemistry. Radium-223 has a half-life of 11.4 days. Approximately how long would it take for the activity of a sample of 223Ra to decrease to 2.00 % of its initial value?. 58. ## Intermediate Maths. (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample. 59. ## Final Maths. (a)-A fossilised tree was tested and contains 10 grams of Carbon-14. Given that there was 12 grams of Carbon-14 present when it died, determine the age of the fossil? Half life of Carbon-14 is 5700 years. (b)How much Carbon-14 will be present in the sample. 60. ## Chemistry. What is the activity of a 18.9 μCi sample of carbon-14 in becquerels?. 61. ## math: pre-calculus. You have 5 grams of carbon-14; whose half-life is 5730 years. a)Write the rule of the function that gives the amount of carbon-14 remaining after x years. b)How much carbon-14 will be left after 4,000 years? c)When will there be just 1 gram left?. 62. ## PreCalculus. The half life of Carbon-14 is 5730 years. a. find the rule of the function that gives the amount remaining from an initial quantity of 100 milligrams of carbon14 after t years. b. the burial cloth of an egyptian mummy i sestimated to contain 59% of the. 63. ## Chemistry. A sample of sodium-24 with an activity of 12 mCi is used to study the rate of blow flow in the circulatory system. If sodium -24 has a half-life of 15 hours, what is the activity of the sodium after 2.5 d ?. 64. ## Chemistry Numerical. The half life period of a radioactive element is 27.96 days. Calculate the time taken by a given sample to reduce to 1/8th of its initial activity. 65. ## chemistry. a sample of sodium-24 with an activity of 12 mCi is used to study the rate of blood flow in the circulatory system. If sodium-24 has a half-life of 15 h, what is the activity of the sodium after 2.5 days. please show work.. 66. ## math. only 5% of the carbon-14 in a small wooden bowl remains , how old is the bowl? Hint, half-life for carbon –14 us 5730 years. 67. ## math. Carbon-14 has a half-life of 5730 years. How long will it take 17 grams of carbon-14 to be reduced to 10 grams? Round to the nearest integer.. 68. ## Math. A plant fossil lost 40 % of its carbon-14. How old is the fossil? (Half-life of carbon-14 is 5730 years.). 69. ## physics. If an archaeologist finds an ancient fire pit containing partially consumed firewood and the 14C content of the wood is only 10% of an equal carbon sample from present day tree, what is the age of ancient site. 14 C has half life of 5730 years. 70. ## physical science. a scientist found a fossilized bison. using Carbon-14 dating, she found that the fossil was about 45,840 years old. Carbon-14 has a half-life of 5730 years. how many half-lives have passed?. 71. ## chemistry. The amount of organic carbon in a sample can be obtained by IR analysis of the carbon dioxide (CO2) released following complete combustion of the sample in oxygen. A reference sample of an organic compound is usually used to calibrate the instrument. 72. ## Precalculus. The burial cloth of an Egyptian mummy is estimated to contain 57% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.). 73. ## Precalculus. The burial cloth of an Egyptian mummy is estimated to contain 57% of the carbon-14 it contained originally. How long ago was the mummy buried? (The half-life of carbon-14 is 5730 years.). 74. ## calculus. A wooden artifact recovered from a tomb contains 29% of the carbon-14 that is present in living trees. The half life of carbon-14 is 5730 years. How long ago was the artifact made?. 75. ## Physics. A radioactive isotope of Bismuth undergoes Beta Decay with a half-life of 22 days. How long (in days) will it take the Activity (A) of the sample to decrease to one-ninth of its original value? Assume that the original activity is A0.. 76. ## Chemistry. One last question...even though no one seems to be willing to help me out without being rude. The blood volume in a cancer patient was measured by injecting 5.0 mL of Na2SO4(aq) labeled with 35S (t1/2 = 87.4 d). The activity of the sample was 300 µCi.. 77. ## chemistry. The radioactive decay of carbon-14 is first-order and the half-life is 5800 years. While a plant or animal is living, it has a constant proportion of carbon-14 (relative to carbon-12) in its composition. When the organism dies, the proportion of carbon-14. 78. ## Chemistry. A sample of charcoal is found to have only 1/8th of Carbon-14. Half-life of Carbon-14 is 5730yrs, what is the age of the charcoal?. 79. ## Chemistry. Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq?. 80. ## Chemistry. Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq?. 81. ## Chemistry. Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq?. 82. ## Chemistry. Cobalt-60, which undergoes beta decay, has a half-life of 5.26 yr. How many beta particles are emitted in 190s by a 3.10mg sample of Co-60? What is the activity of the sample in Bq?. 83. ## math. A wooden artifact from an ancient tomb contains 70% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.). 84. ## Biology. The half-life of carbon-14 is 5,700 years. If a sample originally had 26 g of carbon-14, how much would it contain after 22,800 years?. 85. ## chem. A typical dose of a radioactive sample is 27.0 mCi. How long does it take for the activity to reduce to 0.100 mCi? The half life of the sample is 211,000 y.. 86. ## MATH 123. A wooden artifact from an ancient tomb contains 55% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the nearest whole number.). 87. ## Intro to Astronomy. The half-life of carbon 14, which is commonly used to date organic materials, is 5700 years. What is the minimum age of sample in which less than 1% of the organic carbon 14 is left?. 88. ## Chemistry. The strontium-90 isotope decays by the reaction below. It has a half-life of 28 yr. If the initial activity of this isotope was 168dpm, what would the activity be after 28.0 yr?. 89. ## Chemistry. Radium-223 has a half-life of 11.4 days.Approximately how long would it take for the activity of a sample of Radium-223 to decrease to 1.00% of its initial value?. 90. ## Calculus. A fragment of bone is discovered to contain 20% of the usual carbon-14 concentration. Estimate the age of the bone to the nearest hundred years, given that Carbon-1 is radioactive with half-life of 5730 years and the rate of decay is given by the following. 91. ## College Algebra. This exercise uses the radioactive decay model. A wooden artifact from an ancient tomb contains 80% of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon-14 is 5730 years. Round your answer to the. 92. ## chem. Radioactive substances decay by first-order kinetics. How many years would be required for a sample containing strontium-90 to decrease to 58.32% of its initial activity? The half-life of strontium-90 is 2.88e1 years.. 93. ## Physics 30. The activity of a sample of a radioactive isotope is 1800Bq. If this isotope has a half-life of 16 days, what is the activity after 16 days?, 24 days?, and 60 days?. 1. A granite rock is thought to be about 2 billion years old. why is it not possible to determine the age of the Rock using carbon-14 dating? 2. A hair sample has 80% of its original carbon-14 present. what is the age of the sample? 3. A bone fragment has. 95. ## Chem. If you have a 12.01g sample carbon. What would be the average mass of a carbon atoms if it is 1.994 x 10 to the 23 power g. How many carbon atoms are in the sample?. 96. ## math. 1.The remains of an old campﬁre are unearthed and it is found that there is only 80% as much radioactive carbon-14 in the charcoal samples from the campﬁre as there is in modern living trees. If the half-life of carbon- 14 is 5730 years, how. 97. ## Chemistry. An artifact has a carbon-14 to carbon-12 ratio that is one-fourth the carbon-14 to carbon-12 ratio of a similar modern object. How old is the artifact? (The half-life of carbon-14 is 5,715 years.). 98. ## Chemistry. Q1. The blood volume in a cancer patient was measured by injecting 5.0 mL of Na2SO4(aq) labeled with 35S (t1/2 = 87.4 d). The activity of the sample was 300 µCi. After 22 min, 12.9 mL of blood was withdrawn from the man and the activity of that sample was. 99. ## chemistry. You have a 12.01 g sample of carbon. The average mass of a carbon atoms is 1.994 x 1023 g. How many carbon atoms are in the sample?. 100. ## chemistry. Assayed for LDH activity were 5 microliters of a sample that was diluted 6 to 1. The activity in the reaction vessel, which has a volume of 3 mililiters, is 0.30 U. What is the ΔA/min observed? What is the relative activity of the original sample?.
https://gateoverflow.in/220957/iit-m-video-questions	1,576,133,642,000,000,000	text/html	crawl-data/CC-MAIN-2019-51/segments/1575540537212.96/warc/CC-MAIN-20191212051311-20191212075311-00055.warc.gz	380,315,592	18,309	81 views P(x,y,z), xy=z, Universe is interger; write in logic form If xy=x for all y, then x =0. Thank you \| 81 views +1 vote It's a simple "If----then" statement. Hence, the formulation would be of type $A \rightarrow B$ Here, $A$ is "xy=x for all y"...Or rephrasing, "For all y, xy = x" Hence, $A$ = $\forall y (P(x,y,x))$ And $B$ is "$x = 0$" So, "If xy=x for all y, then x =0" $\equiv$ $\forall y (P(x,y,x))$ $\rightarrow$ ($x = 0$) You can check the above Propositional expression is Always True. by Boss (26.4k points) 0 '=' operator is used here too 0 '=' operator is used here too I have also used...see.. ∀y(P(x,y,x)) → (x=0) Since, Universe is integer, You could use $x = 0$ if need be. 0 Sir, what about ${\forall y (P(x,y,x)) \to P(x,1,0)}$? Can it be a solution? $\forall y\exists x\left (\left (P\left ( x ,y,z\right )=x\right )\rightarrow\left ( xy=x \right )\Lambda \left ( x=0 \right ) \right )$ by Veteran (117k points) edited by 0 What "=" stands for ?? And What will be the English interpretation of this expression? 0 = is just for the interpretation of statement May be a bracket need to clear the statement "If xy=x for all y, then x =0." 0 P(x,y,z)=x What does it mean? 0 ultimately we need to get x as result and we are operating on function P(x,y,z) i.e. P(x,y,z)=x and for that conditions are (xy=x) and (x=0) I think it is resolution method http://nptel.ac.in/courses/106106140/39 1	487	1,440	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.03125	4	CC-MAIN-2019-51	longest	en	0.806885	81 views. P(x,y,z), xy=z, Universe is interger;. write in logic form. If xy=x for all y, then x =0.. Thank you. \| 81 views. +1 vote. It's a simple "If----then" statement.. Hence, the formulation would be of type $A \rightarrow B$. Here, $A$ is "xy=x for all y"...Or rephrasing, "For all y, xy = x". Hence, $A$ = $\forall y (P(x,y,x))$. And $B$ is "$x = 0$". So, "If xy=x for all y, then x =0" $\equiv$ $\forall y (P(x,y,x))$ $\rightarrow$ ($x = 0$). You can check the above Propositional expression is Always True.. by Boss (26.4k points). 0. '=' operator is used here too. 0. '=' operator is used here too. I have also used...see.. ∀y(P(x,y,x)) → (x=0). Since, Universe is integer, You could use $x = 0$ if need be.	0. Sir, what about ${\forall y (P(x,y,x)) \to P(x,1,0)}$? Can it be a solution?. $\forall y\exists x\left (\left (P\left ( x ,y,z\right )=x\right )\rightarrow\left ( xy=x \right )\Lambda \left ( x=0 \right ) \right )$. by Veteran (117k points). edited by. 0. What "=" stands for ??. And What will be the English interpretation of this expression?. 0. = is just for the interpretation of statement. May be a bracket need to clear the statement "If xy=x for all y, then x =0.". 0. P(x,y,z)=x. What does it mean?. 0. ultimately we need to get x as result and we are operating on function P(x,y,z) i.e. P(x,y,z)=x. and for that conditions are (xy=x) and (x=0). I think it is resolution method http://nptel.ac.in/courses/106106140/39. 1.
https://eftradioonline.com/mph-to-degrees-per-second-2022.html	1,660,339,132,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882571758.42/warc/CC-MAIN-20220812200804-20220812230804-00081.warc.gz	225,399,126	18,297	# Mph To Degrees Per Second 2022 Mph To Degrees Per Second 2022. For 250057 mph the best unit of measurement is metres per second, and the amount is 111785.48128 mps. Miles per hour to speed. Once you know what 1 mph is in feet per second, you can. You can also go to the universal conversion page. A distance of one international mile or 1 760 international yards or exactly 1609.344 meters travelled in one hour or exactly 3 600 seconds. ### 2 Miles Per Second = 7200 Miles Per Hour: If not, is it better to express the speed of rotation in degrees or rpms?” using linear speed, like miles per hour, to describe a rotating. To switch the unit simply find the one you want on the page and click it. For example, here's how to convert 5 miles per hour to feet per second using the formula above. ### To Get The Distance Travelled During One Minute, The Previous Value Must Be Multiplied By 60. Meters per second is a unit of speed or velocity in the metric system. You can also go to the universal conversion page. In that case, m/sec = (degrees/sec *. ### 2500 Miles Per Second = 8999998.71 Miles Per Hour: Miles per hour or mile/second. Then click the convert me button. In this case, all you need to know is that 1 mph is equal to 1.4666679325038 ftps. ### The Speed In Feet Per Second Is Equal To The Miles Per Hour Multiplied By 1.466667. We all use different units of measurement every day. We assume you are converting between mile/hour and mile/second. Once you know what 1 mph is in feet per second, you can. ### 1 Miles Per Second = 3600 Miles Per Hour: What is meters per second (m/s)? 1 degree per second is comparative to 0.00277777777777778 hertz. 1,000 degrees per second to radians per second = 17.4533.	446	1,731	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.515625	4	CC-MAIN-2022-33	latest	en	0.840815	# Mph To Degrees Per Second 2022. Mph To Degrees Per Second 2022. For 250057 mph the best unit of measurement is metres per second, and the amount is 111785.48128 mps. Miles per hour to speed.. Once you know what 1 mph is in feet per second, you can. You can also go to the universal conversion page. A distance of one international mile or 1 760 international yards or exactly 1609.344 meters travelled in one hour or exactly 3 600 seconds.. ### 2 Miles Per Second = 7200 Miles Per Hour:. If not, is it better to express the speed of rotation in degrees or rpms?” using linear speed, like miles per hour, to describe a rotating. To switch the unit simply find the one you want on the page and click it. For example, here's how to convert 5 miles per hour to feet per second using the formula above.. ### To Get The Distance Travelled During One Minute, The Previous Value Must Be Multiplied By 60.. Meters per second is a unit of speed or velocity in the metric system. You can also go to the universal conversion page.	In that case, m/sec = (degrees/sec *.. ### 2500 Miles Per Second = 8999998.71 Miles Per Hour:. Miles per hour or mile/second. Then click the convert me button. In this case, all you need to know is that 1 mph is equal to 1.4666679325038 ftps.. ### The Speed In Feet Per Second Is Equal To The Miles Per Hour Multiplied By 1.466667.. We all use different units of measurement every day. We assume you are converting between mile/hour and mile/second. Once you know what 1 mph is in feet per second, you can.. ### 1 Miles Per Second = 3600 Miles Per Hour:. What is meters per second (m/s)? 1 degree per second is comparative to 0.00277777777777778 hertz. 1,000 degrees per second to radians per second = 17.4533.
http://mathhelpforum.com/pre-calculus/132596-x-iy.html	1,480,736,483,000,000,000	text/html	crawl-data/CC-MAIN-2016-50/segments/1480698540804.14/warc/CC-MAIN-20161202170900-00367-ip-10-31-129-80.ec2.internal.warc.gz	177,726,939	8,798	1. ## x + iy just a quick question on expressing $(1 + i) ^3$ in the form x + iy the answer i came to was: $2 + 4i + i^2 + i^2 = 2 + 4i -1 -1 = 0 + 4i$ however im not sure if this is correct... 2. Originally Posted by murielx just a quick question on expressing $(1 + i) ^3$ in the form x + iy the answer i came to was: $2 + 4i + i^2 + i^2 = 2 + 4i -1 -1 = 0 + 4i$ however im not sure if this is correct... I am not sure how you expanded the above but it can be treated like a polynomial in i. Using the binomial theorem or by the distributive law you get $(1+i)^3=\sum_{n=0}^{3}\binom{3}{n}1^{3-n}i^{n}=1+3i+3i^2+i^3=1+3i-3-i=-2+2i$	253	639	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.765625	4	CC-MAIN-2016-50	longest	en	0.84213	1. ## x + iy. just a quick question on expressing $(1 + i) ^3$ in the form x + iy. the answer i came to was:. $2 + 4i + i^2 + i^2. = 2 + 4i -1 -1. = 0 + 4i$. however im not sure if this is correct.... 2. Originally Posted by murielx.	just a quick question on expressing $(1 + i) ^3$ in the form x + iy. the answer i came to was:. $2 + 4i + i^2 + i^2. = 2 + 4i -1 -1. = 0 + 4i$. however im not sure if this is correct.... I am not sure how you expanded the above but it can be treated like a polynomial in i.. Using the binomial theorem or by the distributive law you get. $(1+i)^3=\sum_{n=0}^{3}\binom{3}{n}1^{3-n}i^{n}=1+3i+3i^2+i^3=1+3i-3-i=-2+2i$.
https://edurev.in/studytube/Class-12-Mathematics-CBSE-Sample-Question-Paper-Te/1e4c95ba-724c-4822-8449-85038c149f86_t	1,722,944,553,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640484318.27/warc/CC-MAIN-20240806095414-20240806125414-00797.warc.gz	171,727,324	72,286	Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1 # Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1 \| Mathematics (Maths) Class 12 - JEE PDF Download Table of contents Class-XII Time: 90 Minutes Max. Marks: 40 Section - A Section - B Section - C ## Max. Marks: 40 General Instructions : 1. This question paper contains three sections – A, B and C. Each part is compulsory. 2. Section - A has 20 MCQs, attempt any 16 out of 20. 3. Section - B has 20 MCQs, attempt any 16 out of 20. 4. Section - C has 10 MCQs, attempt any 8 out of 10. 5. There is no negative marking. 6. All questions carry equal marks ## Section - A Q.1: 1. What is the principal value branch of sec–1 x ? (a) (–1, 1) (b) [–1, 1] (c) (d) [0, π] The sec function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat. For sec function this interval is considered as Thus when we take the inverse of the function the domain becomes range and the range becomes domain. Hence the principal value branch is the range of sec–1x that is Q.2: What is the derivative of the function y = xtanx ? (a) xtanx(xsecx + sec x logx) (b) (c) xtanx (2x sec x + tanx log x)) (d) y = xtan x log y = tanx logx Differentiating both side w.r.t. x. Q.3: Matrix is a square matrix if (a) m > n (b) m < n (c) m = 1 (d) m = n Given matrix is said to be square matrix if number of rows are equal to number of columns. Therefore, is a square matrix only if m = n. Q.4: Calculate the determinant of the given matrix (a) 1/2 (b) -1/2 (c) 3/2 (d) None of the above Q.5: Absolute maximum of the function 2x + 5 in [5, 10] ? (a) 5 (b) 10 (c) 20 (d) 25 Given, f '(x) = 2x + 5 Therefore, f '(x) = 2 > 0 Since, f '(x) > 0 in the maximum value is at upper and point f(10) = 2 × 10 + 5 = 25 Q.6: if then the value of x is: (a) –6 (b) –36 (c) 6 (d) 36 For any matrices A and B of suitable orders, we have (a) (A')' = A (b) (A + B)' = A' + B' (c) (kA)' = kA' (where k is any constant) (d) (A B)' = B'A' Q.7: Given set A = {a, b, c}. An identity relation in set A is: (a) R= {(a, b), (a, c)} (b) R= {(a, a), (b, b), (c, c)} (c) R= {(a, a), (b, b), (c, c), (a, c)} (d) R= {(c, a), (b, a), (a, a)} Identity relation is function that always returns the same value that was used as its argument. That is, f(x) = x for all elements in set A. Q.8: For a square matrix A = [aij] the quantity calculated for any element aij in A as the product of (-1)i+j and determinant of the square sub-matrix of order (n-1) obtained by leaving the ith row and jth column of A is known as (a) Cofactor (b) Minor (c) Coefficient (d) Elements The cofactor of an element aij in A is calculated as the product of ( -1)i+j and determinant of the square sub-matrix of order (n-1) obtained by leaving the ith row and jth column of A. Q.9: What is the absolute minimum of the function \|x – 3\| in the interval [4, 5] ? (a) 2 (b) 4 (c) 6 (d) 8 Since the given function is increasing continuously in the given interval, maximum value is at the extreme end point. Q.10: What is the general interval for sine function to become a bijective function? (a) (b) (c) (d) The sine function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat. Q.11: Let R be relation from R to R the set of real numbers defined by R = {(x, y): x, y ∈ R and x – y + √3 is an irrational number}. Then, R is: (a) Reflexive (b) Transitive (c) Symmetric (d) An equivalence relation For reflexive, let (x, x) ∈ R ⇒ x - x + √3 = √3 which is an irrational number. Hence, it is reflexive. For symmetric, let f(x, y) ∈ R ⇒ x - y + √3 which is an irrational number. This means y - x + √3 is an irrational number. So, f(y, x) ∈ R. Hence, it is symmetric. For transitive, let f(x, y)∈ R ⇒ x - y + √3 and f(y, z) ∈ R ⇒ y - z + √3. Now adding these equations, we will get x - z + √3 ⇒ (x, z)∈ R. Hence, it is transitive. Therefore, it is an equivalence relation. Q.12: if x = at4, y = at3 then dy/dx will be (a) 3/4t (b) 3/4t2 (c) 3/4 (d) 3t/4 Q.13: Every Identity matrix is a: (a) Zero matrix (b) Row matrix (c) Scalar matrix (d) Column matrix A scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix. Q.14: If y = sin x log x then the value of dy/dx is (a) sin x log x – 1 (b) (c) (d) y = sinx logx Q.15: For Matrix (adj A)' is equal to (a) (b) (c) (d) Q.16: Which of the following line perpendicular to the tangent to curve y = x2 – 5 at x=1. (a) 2y+x - 35 = 0 (b) 2x−3y+35 = 0 (c) 4x+7y+35 = 0 (d) 3x+7y+21= 0 Slope of tangent = dy/dx = 2x at x = 1, dy/dx = 2 slope of the perpendicular is - 1/slope = - 1/2 Required equation is 2y + x – 35 =0 Q.17: Calculate the value of x such that the matrix is singular. (a) –1, 2 (b) 2, 3 (c) 1 (d) No such value exist + 1[1 – x + 1] = 0 ⇒ (x – 1)(x2 + 1 – 2x – 1) – x + 2 + 2 – x = 0 ⇒ (x – 1)(x2 – 2x) – 2(x – 2) = 0 ⇒ (x – 2)(x2 – x – 2) = 0 Q.18: If then dy/dx is equal to (a) (b) (c) (d) Given that, Differentiate with respect to x, we have Q.19: Maximize Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0. (a) the value of z is minimum at every point on line x – y = –1 (b) there is no feasible region with these constraints. (c) the value of z is minimum at every point on line –x + y = 1 (d) None The region determined by the constraints, is as follows. There is no feasible region and thus, Z has x – y ≤ –1, –x + y ≤ 0, x, y≥ 0 Q.20: Which of the following is true for the given function? (a) Continuous at x = 0 (b) Not continuous at 0 (c) differentiable at 0 (d) None of the above Given, ## Section - B Q.21: Let A = {a, b, c} and B = {1, 2, 3} and f: A→ B is defined by f = {(a, 2), (b, 1), (c, 3)}. Is the function oneone and onto. (a) both one-one and onto (b) only one-one (c) only onto (d) neither of them All the elements in the domain has a unique value in the range. Also the codomain of the function is equal to its range. Q.22: Find the dy/dx of yx+xy = 0 ? (a) (b) (c) 0 (d) None of these yx+xy = 0 or exlogy + eylogx = 0 Differentiating both sides w.r.t. x. Q.23: Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4x + 6y be the objective function. The minimum value of F occurs at (a) (0, 2) only (b) (3, 0) only (c) the mid-point of the line segment joining the points (0, 2) and (3, 0) only (d) any point on the line segment joining the points (0, 2) and (3, 0) Hence, minimum value of F occurs at any points on the line segment joining the points (0, 2) and (3, 0). Q.24: Consider the curve y = x2/4. The Slope of the line parallel to tangent to the curve at x = 1 is (a) 1/4 (b) 1/3 (c) - 1/2 (d) 1/2 y = x2/4 Slope of the curve Parallel lines have same slopes ∴ Slope of tangent = 1/2 Q.25: if then the value of x is (a) 3 (b) ±3 (c) ±6 (d) 6 The process described in the reason statement is the correct procedure to solve the given question. For the given determinant, 2x2 – 40 = 32 2x2 = 72 x = ±6 Q.26: Given a function f (x)= 2x3 −21x2 +60x+48, it has local maximum at x = (a) 2 (b) 3 (c) 5 (d) 4 f(x) = 2x3 – 21x2 + 60x + 48 f'(x) = 6x2 – 42x + 60 f'(x) = 0 ⇒ 6x2 – 42x + 60 = 0 6 (x2 – 7x + 10) = 0 6 (x – 2) (x – 5) = 0 x = 2, 5 f''(x) = 12x – 42 f''(2) = –18 < 0 f''(5) = 60 – 42 = 18 > 0 ∴ f(x) is maximum at x = 2. Q.27: The principal value of (a) π/4 (b) π/6 (c) -π/4 (d) π/3 The principal value of means that we need to find an angle in the principal branch of the function where the sine function is equal to - 1/√2. Hence the required value is -π/4. Q.28: Suppose P and Q are two different matrices of order 3 × n and n × p, then the order of the matrix P × Q is ? (a) 3 × p (b) p × 3 (c) n × n (d) 3 × 3 Q.29: Let f(x) = \|sin x\|, then (a) f is everywhere differentiable (b) f is everywhere continuous but not differentiable at x = n π, n ∈ Z. (c) f is everywhere continuous but not differentiable at (d) none of these Given that, f(x) = \|sin x\| The functions \|x\| and sin x are continuous function for all real value of x. Thus, the function f(x) = \|sin x\| is continuous function everywhere. Now, \|x\| is non-differentiable function at x = 0. Since f(x) = \|sin x\| is non-differentiable function at sin x = 0 Thus, f is everywhere continuous but not differentiable at x = n π, n ∈ Z. Q.30: If function f : R → R defined as f(x) = x2 then f(x) is (a) onto (b) one-one and onto (c) one-one (d) None of these f(x) is a one-one function if f(x1) = f(x2) ⇒ x1= x2 Let f(x1) = f(x2) for some x1, x2 ∈ R ⇒(x1)2=(x2)2 ⇒ x1 = ±x2 Hence f(x) is one-one. Q.31: Which of these intervals, the function f (x)= √2 cos x+x−35 is monotonic? (a) (b) (c) (d) A function is Monotonic if its first derivative’s sign doesn’t change in the given interval. Q.32: If then x equals (a) 0 (b) -2 (c) -1 (d) 2 Q.33: Objective function: Maximise Z = 1000x + 600y Constraints: x + y ≥ 200 y ≥ 20, x ≥ 0 4x Z is maximum at point (a) (20, 80) (b) (20, 180) (c) (0, 0) (d) (40, 160) The corner points are A(20, 180), B(40, 160), C(20, 80) Evaluating the objective function Z = 1,000x + 600y at A, B and C At A(20, 180), Z = 1,000 × 20 + 600 × 180 = 20,000 + 1,08,000 = ₹1,28,000 At B(40, 160), Z = 1,000 × 40 + 600 × 160 = 40,000 + 96,000 = ₹1,36,000 (max.) At C(20, 80), Z = 1000 × 20 + 600 × 80 = 20,000 + 48,000 = ₹68,000 or Z is maximum, when x = 40, y = 160. Q.34: A particle moves along the curve x2 = 2y. The point at which, ordinate increases at the same rate as the abscissa is ________ (a) (1,2) (b) (1/2, 1) (c) (1/2, 1/2) (d) (1, 1/2) Q.35: If then AB + XY equals (a) [28] (b) [24] (c) 28 (d) 24 Given, A = [2 -3 4] , = [6 – 6 + 8] + [2 + 6 + 12] = [8] + [20] = [28] Q.36: If function Then the domain of the function is: (a) (b) R (c) R - {1} (d) R - {5} 10xy – 2y = 2x + 5 10xy – 2x = 5 + 2y 2x(5y – 1) = 5 + 2y Q.37: The maximum number of equivalence relations on the set A = {1, 2, 3} are (a) 1 (b) 2 (c) 3 (d) 5 Given that, A = {1, 2, 3} Now, number of equivalence relations are as follows: R1 = {(1, 1), (2, 2), (3, 3)} R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} R3 = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)} R4 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)} R5 = {(1, 2, 3) ⇔ A × A = A2} ∴ Maximum number of equivalence relations on the set A = {1, 2, 3} = 5 Q.38: If A is any square matrix of order 3 × 3 such that \|A\| = 3, then the value of \|adj A\| is? (a) 3 (b) 1/3 (c) 9 (d) 27 \|A\| = 3, n = 3 \|adj A\| = \|A\|2 = 32 = 9 Q.39: The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is: (a) x + 5y = 2 (b) x – 5y = 2 (c) 5x – y = 2 (d) 5x + y = 2 Given that the equation of curve is y(1 + x2) = 2 – x ...(i) On differentiating with respect to x, we get Since, the given curve passes through -axis, i.e y = 0 ∴ 0(1 + x2) = 2 - x [by using eq. (i) ] ⇒ x = 2 So the curve passes through the point (2, 0). ∴ Equation of tangent to the curve passsing through (2, 0) is Q.40: A = [aij]m×n is a square matrix, if (a) m < n (b) m > n (c) m = n (d) None of these It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns. Therefore, A = [aij]m x n is a square matrix, if m = n. ## Section - C Q.41: For an objective function Z = ax + by, where a, b > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is: (a) b − 3a = 0 (b) a = 3b (c) a + 2b = 0 (d) 2a − b = 0 As Z is maximum at (30, 30) and (0, 40) ⇒ 30a + b = 40b ⇒ b – 3a = 0 Q.42: For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x? (a) 1 2 (b) 1 (c) 2 (d) 3 y = mx + 1 ...(i) and y2 = 4x ...(ii) Substituting (i) in (ii) : (mx + 1)2 = 4x ⇒ m2x2 + (2m – 4)x + 1 = 0 ...(iii) As line is tangent to the curve ⇒ line touches the curve at only one point ⇒ (2m – 4)2 – 4m2 = 0 ⇒ m = 1 Q.43: The maximum value of [x(x+1)+1]1/3, 0 ≤ x ≤ 1 is: (a) 0 (b) 1/2 (c) 1 (d) Let f(x) = [x(x – 1) + 1]1/3,0 ≤ x ≤ 1 Q.44: In a linear programming problem, the constraints on the decision variables x and y are x − 3y ≥ 0, y ≥ 0, 0 ≤ x ≤ 3. The feasible region (a) is not in the first quadrant (b) is bounded in the first quadrant (c) is unbounded in the first quadrant (d) does not exist Feasible region is bounded in the first quadrant Q.45: Let where 0 ≤ α ≤ 2π, then: (A) \|A\|= 0 (B) \|A\| ∈ (2, ∞) (C) \|A\| ∈ (2, 4) (D) \|A\| ∈ [2, 4] \|A\| = 2 + 2sin2θ As –1 ≤ sin θ ≤ 1, ∀ 0 ≤ θ ≤ 2π ⇒ 2 ≤ 2 + 2sin2θ ≤ 4 ⇒ \|A\|∈ [2, 4] Questions 46-50 are based on a Case-Study Case-Study The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹48 per hour at speed 16 km per hour and the fixed charges to run the train amount to ₹1200 per hour. Assume the speed of the train as v km/h. Based on the given information, answer the following questions. Q.46: Given that the fuel cost per hour is k times the square of the speed the train generates in km/h, the value of k is: (a) 16/3 (b) 1/3 (c) 3 (d) 3/16 Fuel cost = k(speed)2 ⇒ 48 = k.162 ⇒ k = 3/16 Q.47: If the train has travelled a distance of 500km, then the total cost of running the train is given by function: (a) (b) (c) (d) Total cost of running train Distance covered = 500 km Q.48: The most economical speed to run the train is: (a) 18km/h (b) 5km/h (c) 80km/h (d) 40km/h Q.49: The fuel cost for the train to travel 500 km at the most economical speed is: (a) ₹3750 (b) 750 (c) 7500 (d) 75000 Fuel cost for running 500 km Q.50: The total cost of the train to travel 500km at the most economical speed is: (a) 3750 (b) 75000 (c) 7500 (d) 15000 Total cost for running 500 km The document Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1 \| Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12. All you need of JEE at this link: JEE ## Mathematics (Maths) Class 12 204 videos\|288 docs\|139 tests ## Mathematics (Maths) Class 12 204 videos\|288 docs\|139 tests ### Up next Explore Courses for JEE exam Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests. 10M+ students study on EduRev Related Searches , , , , , , , , , , , , , , , , , , , , , ;	5,594	14,633	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.4375	4	CC-MAIN-2024-33	latest	en	0.783867	Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1. # Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1 \| Mathematics (Maths) Class 12 - JEE PDF Download. Table of contents Class-XII Time: 90 Minutes Max. Marks: 40 Section - A Section - B Section - C. ## Max. Marks: 40. General Instructions :. 1. This question paper contains three sections – A, B and C. Each part is compulsory.. 2. Section - A has 20 MCQs, attempt any 16 out of 20.. 3. Section - B has 20 MCQs, attempt any 16 out of 20.. 4. Section - C has 10 MCQs, attempt any 8 out of 10.. 5. There is no negative marking.. 6. All questions carry equal marks. ## Section - A. Q.1: 1. What is the principal value branch of sec–1 x ?. (a) (–1, 1). (b) [–1, 1]. (c). (d) [0, π]. The sec function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat. For sec function this interval is considered as. Thus when we take the inverse of the function the domain becomes range and the range becomes domain. Hence the principal value branch is the range of sec–1x that is. Q.2: What is the derivative of the function y = xtanx ?. (a) xtanx(xsecx + sec x logx). (b). (c) xtanx (2x sec x + tanx log x)). (d). y = xtan x. log y = tanx logx. Differentiating both side w.r.t. x.. Q.3: Matrix is a square matrix if. (a) m > n. (b) m < n. (c) m = 1. (d) m = n. Given matrix is said to be square matrix if number of rows are equal to number of columns. Therefore, is a square matrix only if m = n.. Q.4: Calculate the determinant of the given matrix. (a) 1/2. (b) -1/2. (c) 3/2. (d) None of the above. Q.5: Absolute maximum of the function 2x + 5 in [5, 10] ?. (a) 5. (b) 10. (c) 20. (d) 25. Given, f '(x) = 2x + 5. Therefore, f '(x) = 2 > 0. Since, f '(x) > 0 in the maximum value is at upper and point f(10) = 2 × 10 + 5 = 25. Q.6: if then the value of x is:. (a) –6. (b) –36. (c) 6. (d) 36. For any matrices A and B of suitable orders, we have. (a) (A')' = A. (b) (A + B)' = A' + B'. (c) (kA)' = kA' (where k is any constant). (d) (A B)' = B'A'. Q.7: Given set A = {a, b, c}. An identity relation in set A is:. (a) R= {(a, b), (a, c)}. (b) R= {(a, a), (b, b), (c, c)}. (c) R= {(a, a), (b, b), (c, c), (a, c)}. (d) R= {(c, a), (b, a), (a, a)}. Identity relation is function that always returns the same value that was used as its argument. That is, f(x) = x for all elements in set A.. Q.8: For a square matrix A = [aij] the quantity calculated for any element aij in A as the product of (-1)i+j and determinant of the square sub-matrix of order (n-1) obtained by leaving the ith row and jth column of A is known as. (a) Cofactor. (b) Minor. (c) Coefficient. (d) Elements. The cofactor of an element aij in A is calculated as the product of ( -1)i+j and determinant of the square sub-matrix of order (n-1) obtained by leaving the ith row and jth column of A.. Q.9: What is the absolute minimum of the function \|x – 3\| in the interval [4, 5] ?. (a) 2. (b) 4. (c) 6. (d) 8. Since the given function is increasing continuously in the given interval, maximum value is at the extreme end point.. Q.10: What is the general interval for sine function to become a bijective function?. (a). (b). (c). (d). The sine function is periodic so to calculate its inverse function we need to make the function bijective. For that we have to consider an interval in which all values of the function exist and do not repeat.. Q.11: Let R be relation from R to R the set of real numbers defined by R = {(x, y): x, y ∈ R and x – y + √3 is an irrational number}. Then, R is:. (a) Reflexive. (b) Transitive. (c) Symmetric. (d) An equivalence relation. For reflexive, let (x, x) ∈ R. ⇒ x - x + √3 = √3 which is an irrational number. Hence, it is reflexive.. For symmetric, let f(x, y) ∈ R. ⇒ x - y + √3 which is an irrational number.. This means y - x + √3 is an irrational number. So, f(y, x) ∈ R. Hence, it is symmetric.. For transitive, let f(x, y)∈ R. ⇒ x - y + √3 and f(y, z) ∈ R. ⇒ y - z + √3. Now adding these equations, we will get x - z + √3. ⇒ (x, z)∈ R. Hence, it is transitive. Therefore, it is an equivalence relation.. Q.12: if x = at4, y = at3 then dy/dx will be. (a) 3/4t. (b) 3/4t2. (c) 3/4. (d) 3t/4. Q.13: Every Identity matrix is a:. (a) Zero matrix. (b) Row matrix. (c) Scalar matrix. (d) Column matrix. A scalar matrix is an identity matrix when k = 1. But every identity matrix is clearly a scalar matrix.. Q.14: If y = sin x log x then the value of dy/dx is. (a) sin x log x – 1. (b). (c). (d). y = sinx logx. Q.15: For Matrix (adj A)' is equal to. (a). (b). (c). (d). Q.16: Which of the following line perpendicular to the tangent to curve y = x2 – 5 at x=1.. (a) 2y+x - 35 = 0. (b) 2x−3y+35 = 0. (c) 4x+7y+35 = 0. (d) 3x+7y+21= 0. Slope of tangent = dy/dx = 2x at x = 1, dy/dx = 2. slope of the perpendicular is - 1/slope. = - 1/2. Required equation is 2y + x – 35 =0. Q.17: Calculate the value of x such that the matrix is singular.. (a) –1, 2. (b) 2, 3. (c) 1. (d) No such value exist. + 1[1 – x + 1] = 0. ⇒ (x – 1)(x2 + 1 – 2x – 1) – x + 2 + 2 – x = 0. ⇒ (x – 1)(x2 – 2x) – 2(x – 2) = 0. ⇒ (x – 2)(x2 – x – 2) = 0. Q.18: If then dy/dx is equal to. (a). (b). (c). (d). Given that,. Differentiate with respect to x, we have. Q.19: Maximize Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0.. (a) the value of z is minimum at every point on line x – y = –1. (b) there is no feasible region with these constraints.. (c) the value of z is minimum at every point on line –x + y = 1. (d) None. The region determined by the constraints, is as follows. There is no feasible region and thus, Z has. x – y ≤ –1, –x + y ≤ 0, x, y≥ 0. Q.20: Which of the following is true for the given function?. (a) Continuous at x = 0. (b) Not continuous at 0. (c) differentiable at 0. (d) None of the above. Given,. ## Section - B. Q.21: Let A = {a, b, c} and B = {1, 2, 3} and f: A→ B is defined by f = {(a, 2), (b, 1), (c, 3)}. Is the function oneone and onto.. (a) both one-one and onto. (b) only one-one. (c) only onto. (d) neither of them. All the elements in the domain has a unique value in the range. Also the codomain of the function is equal to its range.. Q.22: Find the dy/dx of yx+xy = 0 ?. (a). (b). (c) 0. (d) None of these. yx+xy = 0. or exlogy + eylogx = 0. Differentiating both sides w.r.t. x.. Q.23: Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F = 4x + 6y be the objective function.. The minimum value of F occurs at. (a) (0, 2) only. (b) (3, 0) only. (c) the mid-point of the line segment joining the points (0, 2) and (3, 0) only. (d) any point on the line segment joining the points (0, 2) and (3, 0). Hence, minimum value of F occurs at any points on the line segment joining the points (0, 2) and (3, 0).. Q.24: Consider the curve y = x2/4. The Slope of the line parallel to tangent to the curve at x = 1 is. (a) 1/4. (b) 1/3. (c) - 1/2. (d) 1/2. y = x2/4. Slope of the curve. Parallel lines have same slopes. ∴ Slope of tangent = 1/2. Q.25: if then the value of x is. (a) 3. (b) ±3. (c) ±6. (d) 6. The process described in the reason statement is the correct procedure to solve the given question. For the given determinant,. 2x2 – 40 = 32. 2x2 = 72. x = ±6. Q.26: Given a function f (x)= 2x3 −21x2 +60x+48, it has local maximum at x =. (a) 2. (b) 3. (c) 5. (d) 4. f(x) = 2x3 – 21x2 + 60x + 48. f'(x) = 6x2 – 42x + 60. f'(x) = 0. ⇒ 6x2 – 42x + 60 = 0. 6 (x2 – 7x + 10) = 0. 6 (x – 2) (x – 5) = 0. x = 2, 5. f''(x) = 12x – 42. f''(2) = –18 < 0. f''(5) = 60 – 42. = 18 > 0. ∴ f(x) is maximum at x = 2.. Q.27: The principal value of. (a) π/4. (b) π/6.	(c) -π/4. (d) π/3. The principal value of means that we need to find an angle in the principal branch of the function where the sine function is equal to - 1/√2. Hence the required value is -π/4.. Q.28: Suppose P and Q are two different matrices of order 3 × n and n × p, then the order of the matrix P × Q is ?. (a) 3 × p. (b) p × 3. (c) n × n. (d) 3 × 3. Q.29: Let f(x) = \|sin x\|, then. (a) f is everywhere differentiable. (b) f is everywhere continuous but not differentiable at x = n π, n ∈ Z.. (c) f is everywhere continuous but not differentiable at. (d) none of these. Given that, f(x) = \|sin x\|. The functions \|x\| and sin x are continuous function for all real value of x.. Thus, the function f(x) = \|sin x\| is continuous function everywhere.. Now, \|x\| is non-differentiable function at x = 0.. Since f(x) = \|sin x\| is non-differentiable function at sin x = 0. Thus, f is everywhere continuous but not differentiable at x = n π, n ∈ Z.. Q.30: If function f : R → R defined as f(x) = x2 then f(x) is. (a) onto. (b) one-one and onto. (c) one-one. (d) None of these. f(x) is a one-one function. if f(x1) = f(x2) ⇒ x1= x2. Let f(x1) = f(x2) for some x1, x2 ∈ R. ⇒(x1)2=(x2)2. ⇒ x1 = ±x2. Hence f(x) is one-one.. Q.31: Which of these intervals, the function f (x)= √2 cos x+x−35 is monotonic?. (a). (b). (c). (d). A function is Monotonic if its first derivative’s sign doesn’t change in the given interval.. Q.32: If then x equals. (a) 0. (b) -2. (c) -1. (d) 2. Q.33: Objective function:. Maximise Z = 1000x + 600y. Constraints:. x + y ≥ 200. y ≥ 20, x ≥ 0. 4x. Z is maximum at point. (a) (20, 80). (b) (20, 180). (c) (0, 0). (d) (40, 160). The corner points are A(20, 180), B(40, 160), C(20, 80). Evaluating the objective function Z = 1,000x + 600y at A, B and C. At A(20, 180),. Z = 1,000 × 20 + 600 × 180 = 20,000 + 1,08,000 = ₹1,28,000. At B(40, 160), Z = 1,000 × 40 + 600 × 160 = 40,000 + 96,000 = ₹1,36,000 (max.). At C(20, 80), Z = 1000 × 20 + 600 × 80 = 20,000 + 48,000 = ₹68,000 or. Z is maximum, when x = 40, y = 160.. Q.34: A particle moves along the curve x2 = 2y. The point at which, ordinate increases at the same rate as the abscissa is ________. (a) (1,2). (b) (1/2, 1). (c) (1/2, 1/2). (d) (1, 1/2). Q.35: If then AB + XY equals. (a) [28]. (b) [24]. (c) 28. (d) 24. Given, A = [2 -3 4] ,. = [6 – 6 + 8] + [2 + 6 + 12]. = [8] + [20]. = [28]. Q.36: If function Then the domain of the function is:. (a). (b) R. (c) R - {1}. (d) R - {5}. 10xy – 2y = 2x + 5. 10xy – 2x = 5 + 2y. 2x(5y – 1) = 5 + 2y. Q.37: The maximum number of equivalence relations on the set A = {1, 2, 3} are. (a) 1. (b) 2. (c) 3. (d) 5. Given that, A = {1, 2, 3}. Now, number of equivalence relations are as follows:. R1 = {(1, 1), (2, 2), (3, 3)}. R2 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}. R3 = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}. R4 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}. R5 = {(1, 2, 3) ⇔ A × A = A2}. ∴ Maximum number of equivalence relations on the set A = {1, 2, 3} = 5. Q.38: If A is any square matrix of order 3 × 3 such that \|A\| = 3, then the value of \|adj A\| is?. (a) 3. (b) 1/3. (c) 9. (d) 27. \|A\| = 3, n = 3. \|adj A\| = \|A\|2 = 32 = 9. Q.39: The equation of tangent to the curve y(1 + x2) = 2 – x, where it crosses x-axis is:. (a) x + 5y = 2. (b) x – 5y = 2. (c) 5x – y = 2. (d) 5x + y = 2. Given that the equation of curve is. y(1 + x2) = 2 – x ...(i). On differentiating with respect to x, we get. Since, the given curve passes through -axis,. i.e y = 0. ∴ 0(1 + x2) = 2 - x [by using eq. (i) ]. ⇒ x = 2. So the curve passes through the point (2, 0).. ∴ Equation of tangent to the curve passsing through (2, 0) is. Q.40: A = [aij]m×n is a square matrix, if. (a) m < n. (b) m > n. (c) m = n. (d) None of these. It is known that a given matrix is said to be a square matrix if the number of rows is equal to the number of columns.. Therefore,. A = [aij]m x n is a square matrix, if m = n.. ## Section - C. Q.41: For an objective function Z = ax + by, where a, b > 0; the corner points of the feasible region determined by a set of constraints (linear inequalities) are (0, 20), (10, 10), (30, 30) and (0, 40). The condition on a and b such that the maximum Z occurs at both the points (30, 30) and (0, 40) is:. (a) b − 3a = 0. (b) a = 3b. (c) a + 2b = 0. (d) 2a − b = 0. As Z is maximum at (30, 30) and (0, 40). ⇒ 30a + b = 40b. ⇒ b – 3a = 0. Q.42: For which value of m is the line y = mx + 1 a tangent to the curve y2 = 4x?. (a) 1 2. (b) 1. (c) 2. (d) 3. y = mx + 1 ...(i). and y2 = 4x ...(ii). Substituting (i) in (ii) :. (mx + 1)2 = 4x. ⇒ m2x2 + (2m – 4)x + 1 = 0 ...(iii). As line is tangent to the curve. ⇒ line touches the curve at only one point. ⇒ (2m – 4)2 – 4m2 = 0. ⇒ m = 1. Q.43: The maximum value of [x(x+1)+1]1/3, 0 ≤ x ≤ 1 is:. (a) 0. (b) 1/2. (c) 1. (d). Let f(x) = [x(x – 1) + 1]1/3,0 ≤ x ≤ 1. Q.44: In a linear programming problem, the constraints on the decision variables x and y are x − 3y ≥ 0, y ≥ 0, 0 ≤ x ≤ 3. The feasible region. (a) is not in the first quadrant. (b) is bounded in the first quadrant. (c) is unbounded in the first quadrant. (d) does not exist. Feasible region is bounded in the first quadrant. Q.45: Let where 0 ≤ α ≤ 2π, then:. (A) \|A\|= 0. (B) \|A\| ∈ (2, ∞). (C) \|A\| ∈ (2, 4). (D) \|A\| ∈ [2, 4]. \|A\| = 2 + 2sin2θ. As –1 ≤ sin θ ≤ 1, ∀ 0 ≤ θ ≤ 2π. ⇒ 2 ≤ 2 + 2sin2θ ≤ 4. ⇒ \|A\|∈ [2, 4]. Questions 46-50 are based on a Case-Study. Case-Study. The fuel cost per hour for running a train is proportional to the square of the speed it generates in km per hour. If the fuel costs ₹48 per hour at speed 16 km per hour and the fixed charges to run the train amount to ₹1200 per hour. Assume the speed of the train as v km/h. Based on the given information, answer the following questions.. Q.46: Given that the fuel cost per hour is k times the square of the speed the train generates in km/h, the value of k is:. (a) 16/3. (b) 1/3. (c) 3. (d) 3/16. Fuel cost = k(speed)2. ⇒ 48 = k.162. ⇒ k = 3/16. Q.47: If the train has travelled a distance of 500km, then the total cost of running the train is given by function:. (a). (b). (c). (d). Total cost of running train. Distance covered = 500 km. Q.48: The most economical speed to run the train is:. (a) 18km/h. (b) 5km/h. (c) 80km/h. (d) 40km/h. Q.49: The fuel cost for the train to travel 500 km at the most economical speed is: (a) ₹3750. (b) 750. (c) 7500. (d) 75000. Fuel cost for running 500 km. Q.50: The total cost of the train to travel 500km at the most economical speed is:. (a) 3750. (b) 75000. (c) 7500. (d) 15000. Total cost for running 500 km. The document Class 12 Mathematics: CBSE Sample Question Paper- Term I (2021-22)- 1 \| Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.. All you need of JEE at this link: JEE. ## Mathematics (Maths) Class 12. 204 videos\|288 docs\|139 tests. ## Mathematics (Maths) Class 12. 204 videos\|288 docs\|139 tests. ### Up next. Explore Courses for JEE exam. Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.. 10M+ students study on EduRev. Related Searches. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ;.
https://www.jiskha.com/questions/642516/almost-everyone-at-the-bright-corporation-reads-a-daily-newspaper-on-a-regular-basis-in	1,638,423,966,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964361169.72/warc/CC-MAIN-20211202054457-20211202084457-00468.warc.gz	878,495,449	5,563	# math Almost everyone at the Bright Corporation reads a daily newspaper on a regular basis. In fact, altogether te company employs 2,940 people,of whom70% read The Daily News,45% read The Courier and 60% read The inquirer. One fourth of the employees reads The Daily News and The Courier, and 30% read The Inquirer and The Courier, and 35% read The Daily News and The Inquirer. One tenth of the people read all three newspapers on a regular basis. a. how many people read only The Daily news? b. How many people read only The Courier? c. How many people read on The Inquirer? d. How many people don't read any newspaper? 1. 👍 2. 👎 3. 👁 1. Draw your 3-circle Venn diagram. Label the circles D,C,I DCI = .1 so DC only = .15 (.25-.10) CI only = .20 (.30-.10) DI only = .25 (.35-.10) D only = .20 (.70 - .15 - .10 - .25) C only = .00 (.45 - .15 - .10 - .20) I only = .05 (.60 - .25 - .10 - .20) Add up all the percentages and it is clear that only 90% of the employees read any papers. That makes .92940 = 2646 readers. So, a. D only = .22940 = 588 b. C only = 0 c. I only = .052940 = 147 d. None = .102940 = 294 1. 👍 2. 👎 2. 95% of the employees read any newspaper, therefore 5% don't read and newspaper. D should be 147 1. 👍 2. 👎 ## Similar Questions 1. ### social studies (check my answers) What is the definition of civic-mindedness? paying attention to the needs of one's community treating others with respect following the newspaper on a daily basis following and showing respect for the law 2. ### English 1. My father reads the newspaper every day. [Does this sentence mean that he reads the same newspaper every day{gain and again}? Or was 'the newspaper' in generic use?] 2. My father reads a newspaper every day. [What about this 3. ### Math in a survey of 290 newspaper readers, 181 of them read the Daily Times, 142 Read the Guardian 117 read punch and each reads at least one of the three papers if 75 read the Daily Times, and the Guardian, 60 read the Daily Times and 4. ### English What distinguishes a fact from an opinion? -A fact can be proven true. -A fact can be changed over time. -A fact can be supported with evidence. -A fact can be supported with examples. 1. ### Art Art can serve many purposes. here, an artist created bowls and plates that are used every day. what other artistic creations are used on a daily basis? clothing**** sculptures paintings drawings The **** is what i think my 2. ### Mathematics In a surveyor of 200 newspaper readers,181 of them read daily times,142 read the guardian,117 read the punch and each reads at least one of the three papers. If 75 read the daily times and the guardian,60 read the daily times and 3. ### Sta A regular feature in a newspaper asks readers to respond via e-mail to a survey that requires a yes or no response. In the following day's newspaper, the percentage of yes and no responses are reported. discuss why we should 4. ### English 1. My father reads the newspaper every day. (What does it mean? 2. My father reads the same newspaper everyday. 3. My father reads a newspaper every day. Does #1 mean #2 or #3? Is 'the newspaper' the generic term? ) 1. ### History "The Daily News- March 14, 1904 Supreme Court Rules Northern Securities in Violation of Sherman Antitrust Act" Which statement best explains the significance of the newspaper headline? a.)The ruling provided a legal basis for 2. ### finance 2 questions 9. When Patricia sells her General Motors common stock at the same time that Brian purchases the same amount of General Motor's stock, General Motors receives: A. The spread between the bid and ask of the transaction B. The dollar 3. ### English e.g. My father reads the newspaper every day. (What is the meaning of the sentence above? #1 or #2?) 1. My father reads the same/specific newspaper every day /repeatedly. 2. My father reads newspapers every day. He reads newly 4. ### nutrition and wellness HI JISKHA, just got my first course for next year. this is my first writing assignment: Think about your personal food choices: How do you decide what you will eat on a daily basis? What sorts of things influence your decision?	1,076	4,177	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.9375	4	CC-MAIN-2021-49	latest	en	0.927123	# math. Almost everyone at the Bright Corporation reads a daily newspaper on a regular basis. In fact, altogether te company employs 2,940 people,of whom70% read The Daily News,45% read The Courier and 60% read The inquirer.. One fourth of the employees reads The Daily News and The Courier, and 30% read The Inquirer and The Courier, and 35% read The Daily News and The Inquirer. One tenth of the people read all three newspapers on a regular basis.. a. how many people read only The Daily news?. b. How many people read only The Courier?. c. How many people read on The Inquirer?. d. How many people don't read any newspaper?. 1. 👍. 2. 👎. 3. 👁. 1. Draw your 3-circle Venn diagram. Label the circles D,C,I. DCI = .1. so. DC only = .15 (.25-.10). CI only = .20 (.30-.10). DI only = .25 (.35-.10). D only = .20 (.70 - .15 - .10 - .25). C only = .00 (.45 - .15 - .10 - .20). I only = .05 (.60 - .25 - .10 - .20). Add up all the percentages and it is clear that only 90% of the employees read any papers.. That makes .92940 = 2646 readers.. So,. a. D only = .22940 = 588. b. C only = 0. c. I only = .052940 = 147. d. None = .102940 = 294. 1. 👍. 2. 👎. 2. 95% of the employees read any newspaper, therefore 5% don't read and newspaper. D should be 147. 1. 👍. 2. 👎. ## Similar Questions. 1. ### social studies (check my answers). What is the definition of civic-mindedness? paying attention to the needs of one's community treating others with respect following the newspaper on a daily basis following and showing respect for the law. 2. ### English. 1.	My father reads the newspaper every day. [Does this sentence mean that he reads the same newspaper every day{gain and again}? Or was 'the newspaper' in generic use?] 2. My father reads a newspaper every day. [What about this. 3. ### Math. in a survey of 290 newspaper readers, 181 of them read the Daily Times, 142 Read the Guardian 117 read punch and each reads at least one of the three papers if 75 read the Daily Times, and the Guardian, 60 read the Daily Times and. 4. ### English. What distinguishes a fact from an opinion? -A fact can be proven true. -A fact can be changed over time. -A fact can be supported with evidence. -A fact can be supported with examples.. 1. ### Art. Art can serve many purposes. here, an artist created bowls and plates that are used every day. what other artistic creations are used on a daily basis? clothing**** sculptures paintings drawings The **** is what i think my. 2. ### Mathematics. In a surveyor of 200 newspaper readers,181 of them read daily times,142 read the guardian,117 read the punch and each reads at least one of the three papers. If 75 read the daily times and the guardian,60 read the daily times and. 3. ### Sta. A regular feature in a newspaper asks readers to respond via e-mail to a survey that requires a yes or no response. In the following day's newspaper, the percentage of yes and no responses are reported. discuss why we should. 4. ### English. 1. My father reads the newspaper every day. (What does it mean? 2. My father reads the same newspaper everyday. 3. My father reads a newspaper every day. Does #1 mean #2 or #3? Is 'the newspaper' the generic term? ). 1. ### History. "The Daily News- March 14, 1904 Supreme Court Rules Northern Securities in Violation of Sherman Antitrust Act" Which statement best explains the significance of the newspaper headline? a.)The ruling provided a legal basis for. 2. ### finance 2 questions. 9. When Patricia sells her General Motors common stock at the same time that Brian purchases the same amount of General Motor's stock, General Motors receives: A. The spread between the bid and ask of the transaction B. The dollar. 3. ### English. e.g. My father reads the newspaper every day. (What is the meaning of the sentence above? #1 or #2?) 1. My father reads the same/specific newspaper every day /repeatedly. 2. My father reads newspapers every day. He reads newly. 4. ### nutrition and wellness. HI JISKHA, just got my first course for next year. this is my first writing assignment: Think about your personal food choices: How do you decide what you will eat on a daily basis? What sorts of things influence your decision?.
http://classic-puzzles.blogspot.com/2006/12/microsoft-interview-question-polar-bear.html	1,487,825,895,000,000,000	text/html	crawl-data/CC-MAIN-2017-09/segments/1487501171078.90/warc/CC-MAIN-20170219104611-00376-ip-10-171-10-108.ec2.internal.warc.gz	50,426,955	15,631	Classic Puzzles Wednesday, December 27, 2006 Microsoft Interview Question : Polar Bear One of the most asked and well known microsoft interview question is that of the walking bear.The question is still asked because a lot of people have either not heard of it or most of them don't know the correct solution yet. If a bear walks one mile south, turns left and walks one mile to the east and then turns left again and walks one mile north and arrives at its original position, what is the color of the bear. Well,from the very framing of the question it is evident that we r talking about the poles,and all polar bears are white.You can also very well argue that any bear or man walking the same path will reach the same starting position if he is at north pole.The question can also be extended and that's what we are interested in. The question is how many such points exists on the surface of the globe. Jim Miles said... Infinity. There is indeed the one point exactly on the north pole where walking 1 mile south, 1 mile east then 1 mile north returns you to the starting position but there is also a circle of infinity points in the southern hemisphere. Let us call this circle 'A'. It is formed by the points 1 mile north of another circle 'B' of circumference 1, parallel to the equator and between the equator and the south pole. Suppose you are on 'A'. Then going 1 mile south puts you on 'B'. Travelling 1 mile east lands you up exactly where you just were on the circle (because the circumference of 'B' is 1). Then travelling north puts you back where you started. However, 'A' is not the only circle of such points. In the same way that we constructed 'A' by taking the points 1 mile north of the circle 'B' of circumference 1, parallel to the equator, between the equator and the south pole we may take any circle 'A_n' of points 1 mile north of the circle 'B_n' of circumference 1/n (for any n in the natural numbers), parallel to the equator, between the equator and the south pole. Suman said... Thats a good question and i agree with jim miles solution. ---------------- My blog: http://justriddles.blogspot.com http://justfungames.blogspot.com ---------------- hugo bowne-anderson said... great, jim. so: as your n gets really, really big, the circle B_n gets closer and closer to the south pole. so in the limit as n goes to infinity, we find another solution!, which is: start at any point on the circle A' which is all points 1mile north of the south pole. then when you get to the south pole, walking east is doing nothing! Narendran Kumaragurunathan said... The bear is white because, there are no bears on south pole. Eighth Wonder said... There r no polar bears in Antarctica. Answer is north pole,which is exactly a point on earth. Thiep said... First of all. Thanks very much for your useful post. I just came across your blog and wanted to drop you a note telling you how impressed I was with the information you have posted here. Please let me introduce you some info related to this post and I hope that it is useful for community. Source: Microsoft interview questions Thanks again Ngo aashish said... To add: Somebody might think that such a circle (of unit circumference) can exist in northern hemisphere also all the points lying on circle one mile above(the circle of unit radius) will also be the answer but it is not possible because the distance of north pole from this circle would be 1/2*pi which is less than 1 so only one point in northern hemisphere and i.e north pole. aashish said... ohhhh sry instead of radius it should be circumference at one place Lily said... My husband became like a polar bear he is like an iceberg with me so I had to suggest him Viagra Online because I just need to be with my old sweetheart husband. power balance silly bandz Raymond Weil Watches concord papillon rolex datejust 36mm wedding dresses develop quality for discerning customers and Experience the comfort, free shipping. Buy evening dresses with a price guarantee and top rated customer service. guddu said... infinity...as we earth is as a hemisphere so we cannot say there is one point. lavesh said... Hi I got inspired by your blog and created my own puzzle blog Quiz Lauraine said... That bear should be white. Bcoz it started one mile south from north. So it lives in north. That's it should been white. Formspring Clone Script Formspring Backgrounds Chankey Pathak said... Color: White (North pole's bear are white and also there are no bears in south pole) sophia-yang said... AT THE FIRST SIGHT IF THIS ARTICLE, I THINK OF A SONG POLAR BEAR OUR FOREIGN TEACHER TOUGHT US EVER. IT IS A SO WARM MEMORY FOR ME NOW! Dell laptop keyboard Gateway laptop keyboard Rajesh said... excellent solution given by Jim! Hats off.. Ashutosh Mukherjee said... but y do we consider that bear statred off from north pole...not obviously soth but it can be any place and then how do we come to the conclusion Ashutosh Mukherjee said... and how do we arrive at this... we may take any circle 'A_n' of points 1 mile north of the circle 'B_n' of circumference 1/n (for any n in the natural numbers), parallel to the equator, between the equator and the south pole. Ian Webb said... It would be very difficult to find any bears at the Magnetic North Pole, considering it is in the middle of the sea! Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ? Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ? Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ? Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ? Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ? Realmatri.com said... You are on north pole, it can’t be south pole because being at south, you can’t go south. Eldhose Baby said... HemanT AgrawaL said... Nice puzzle.. my puzzle collection is here... Top Logical Puzzles Unknown said... There is one more point , barring that vicinity circle concept, where in with these three movements you will come to original point??Wait for ans Unknown said... There is one more point , barring that vicinity circle concept, where in with these three movements you will come to original point??Wait for ans	1,546	6,792	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.859375	4	CC-MAIN-2017-09	longest	en	0.952276	Classic Puzzles. Wednesday, December 27, 2006. Microsoft Interview Question : Polar Bear. One of the most asked and well known microsoft interview question is that of the walking bear.The question is still asked because a lot of people have either not heard of it or most of them don't know the correct solution yet.. If a bear walks one mile south, turns left and walks one mile to the east and then turns left again and walks one mile north and arrives at its original position, what is the color of the bear.. Well,from the very framing of the question it is evident that we r talking about the poles,and all polar bears are white.You can also very well argue that any bear or man walking the same path will reach the same starting position if he is at north. pole.The question can also be extended and that's what we are interested in.. The question is how many such points exists on the surface of the globe.. Jim Miles said.... Infinity.. There is indeed the one point exactly on the north pole where walking 1 mile south, 1 mile east then 1 mile north returns you to the starting position but there is also a circle of infinity points in the southern hemisphere.. Let us call this circle 'A'. It is formed by the points 1 mile north of another circle 'B' of circumference 1, parallel to the equator and between the equator and the south pole.. Suppose you are on 'A'. Then going 1 mile south puts you on 'B'. Travelling 1 mile east lands you up exactly where you just were on the circle (because the circumference of 'B' is 1). Then travelling north puts you back where you started.. However, 'A' is not the only circle of such points. In the same way that we constructed 'A' by taking the points 1 mile north of the circle 'B' of circumference 1, parallel to the equator, between the equator and the south pole we may take any circle 'A_n' of points 1 mile north of the circle 'B_n' of circumference 1/n (for any n in the natural numbers), parallel to the equator, between the equator and the south pole.. Suman said.... Thats a good question and i agree with jim miles solution.. ----------------. My blog:. http://justriddles.blogspot.com. http://justfungames.blogspot.com. ----------------. hugo bowne-anderson said.... great, jim.. so: as your n gets really, really big, the circle B_n gets closer and closer to the south pole.. so in the limit as n goes to infinity, we find another solution!, which is:. start at any point on the circle A' which is all points 1mile north of the south pole.. then when you get to the south pole, walking east is doing nothing!. Narendran Kumaragurunathan said.... The bear is white because, there are no bears on south pole.. Eighth Wonder said.... There r no polar bears in Antarctica.. Answer is north pole,which is exactly a point on earth.. Thiep said.... First of all. Thanks very much for your useful post.. I just came across your blog and wanted to drop you a note telling you how impressed I was with the information you have posted here.. Please let me introduce you some info related to this post and I hope that it is useful for community.. Source: Microsoft interview questions. Thanks again. Ngo. aashish said.... To add: Somebody might think that such a circle (of unit circumference) can exist in northern hemisphere also all the points lying on circle one mile above(the circle of unit radius) will also be the answer but it is not possible because the distance of north pole from this circle would be 1/2*pi which is less than 1 so only one point in northern hemisphere and i.e north pole.. aashish said.... ohhhh sry instead of radius it should be circumference at one place. Lily said.... My husband became like a polar bear he is like an iceberg with me so I had to suggest him Viagra Online because I just need to be with my old sweetheart husband.. power balance. silly bandz. Raymond Weil Watches.	concord papillon. rolex datejust 36mm. wedding dresses develop quality for discerning customers and Experience the comfort, free shipping.. Buy evening dresses with a price guarantee and top rated customer service.. guddu said.... infinity...as we earth is as a hemisphere so we cannot say there is one point.. lavesh said.... Hi I got inspired by your blog and created my own puzzle blog. Quiz. Lauraine said.... That bear should be white. Bcoz it started one mile south from north. So it lives in north. That's it should been white.. Formspring Clone Script. Formspring Backgrounds. Chankey Pathak said.... Color: White (North pole's bear are white and also there are no bears in south pole). sophia-yang said.... AT THE FIRST SIGHT IF THIS ARTICLE, I THINK OF A SONG POLAR BEAR OUR FOREIGN TEACHER TOUGHT US EVER. IT IS A SO WARM MEMORY FOR ME NOW!. Dell laptop keyboard. Gateway laptop keyboard. Rajesh said.... excellent solution given by Jim!. Hats off... Ashutosh Mukherjee said.... but y do we consider that bear statred off from north pole...not obviously soth but it can be any place and then how do we come to the conclusion. Ashutosh Mukherjee said.... and how do we arrive at this.... we may take any circle 'A_n' of points 1 mile north of the circle 'B_n' of circumference 1/n (for any n in the natural numbers), parallel to the equator, between the equator and the south pole.. Ian Webb said.... It would be very difficult to find any bears at the Magnetic North Pole, considering it is in the middle of the sea!. Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ?. Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ?. Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ?. Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ?. Amazing solution, Jim ! Just out of curiosity, how much distance south would one need to travel to reach the South Pole from the Circle B ( with a 1 mile circumference) ?. Realmatri.com said.... You are on north pole, it can’t be south pole because being at south, you can’t go south.. Eldhose Baby said.... HemanT AgrawaL said.... Nice puzzle.. my puzzle collection is here.... Top Logical Puzzles. Unknown said.... There is one more point , barring that vicinity circle concept, where in with these three movements you will come to original point??Wait for ans. Unknown said.... There is one more point , barring that vicinity circle concept, where in with these three movements you will come to original point??Wait for ans.
http://mathhelpforum.com/differential-geometry/170304-lebesgue-integrability-showing-limit-x-approaches-f-x-0-a-print.html	1,513,055,216,000,000,000	text/html	crawl-data/CC-MAIN-2017-51/segments/1512948515165.6/warc/CC-MAIN-20171212041010-20171212061010-00648.warc.gz	195,525,003	5,182	# lebesgue integrability and showing limit as x approaches +/-∞ of F(x) = 0. • Feb 5th 2011, 08:04 PM oblixps lebesgue integrability and showing limit as x approaches +/-∞ of F(x) = 0. a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help. • Feb 5th 2011, 09:51 PM TheEmptySet Quote: Originally Posted by oblixps a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help. for a) you will want to use sequential continuity. Let $x_n \to x$ and consider the function $\displaystyle F(x_n)=\lim_{n \to \infty}\int f \chi_{[0,x_n]}d\lambda$ Now since $f(x)\chi_{[0,x_n]} \le f(x)$ and $f(x) \in L^1$ Just use DCT and $F(x_n) \to F(x)$ For b) what would happen it the function didn't go to zero? • Feb 5th 2011, 11:40 PM chisigma Quote: Originally Posted by oblixps a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help. Examples of function L-integrable on R that doesn't tend to 0 if x tends to infinity are given in... http://www.mathhelpforum.com/math-he...tml#post597107 Kind regards $\chi$ $\sigma$ • Feb 6th 2011, 12:22 AM Tinyboss Quote: Originally Posted by chisigma Examples of function L-integrable on R that doesn't tend to 0 if x tends to infinity are given in... http://www.mathhelpforum.com/math-he...tml#post597107 Kind regards $\chi$ $\sigma$ Exactly...the function could do anything at all on a set of measure zero and not affect the Lebesgue integral. • Feb 7th 2011, 01:07 PM oblixps for part b) would the reason be along these lines: since F is L-integrable it can be written as the infinite sum of the integrals of L-integrable functions and in order for the infinite sum to converge the individual integrals in the sum must approach 0 as n approaches infinite. I'm a little confused though since it is x in F(x) that is approaching infinite. i have looked at the thread you have provided me and near the bottom i saw that F(x) must approach 0 since if it didn't the integral of \|F\| would not be finite. although i can intuitively see that, i am trying to reconcile that with the definitions provided in my book and that is what is causing me some confusion. my book does not use measure sets to motivate the lebesgue integral but defines it as: let f_k be a sequence of R integrable functions such that the infinite sum of the integral (-infinite, infinite) \|f_k\|dx < infinite, then the lebesgue integral of f = infinite sum of f_k is: integral of f(x) dx = infinite sum of integral of f_k dx. can you help me make sense of this problem given my book's definition? thanks in advance. • Feb 7th 2011, 04:21 PM Jose27 So, apparently condition b) indeed holds, but functions that satisfy the hypothesis are quite limited: Assume for the moment $f\geq 0$ then if there exists a measurable set $M \subset \mathbb{R}$ such that $\int_{M} f >0$ then $\int_{[0,\infty )} f >0$ and so $F$ can't be integrable there (take a sequence in $M$ tending to infinity and apply dominated convergence to obtain a contradiction, if $M$ is bounded it's easier still) and from this we deduce $F\notin L^1(\mathbb{R})$ contradicting the assumption. We conclude $f=0$ a.e. In the same way we deal with the case $f\leq 0$. Now, for any function $g$, we define $g^+=\max \{ g,0\}$ and $g^-=\min \{ g,0 \}$. It's a standard result that $g\in L^1(\mathbb{R})$ iff $g^+,g^-\in L^1(\mathbb{R})$, so we get $F(x)= \int_0^x f^+(t)dt + \int_0^x f^-(t)dt = F^+(x) + F^-(x)$ (this is easily seen to be the case because the integral is monotone), but $F^+ \in L^1(\mathbb{R})$ iff $f= 0$ a.e., and analogous for $F^-$. We therefore must have $f= 0$ a.e. if $F\in L^1(\mathbb{R})$, in which case the conclusion trivially holds. On the other hand if you ask that $F+c \in L^1(\mathbb{R})$ for some constant c, then the problem is more difficult (read I don't have a proof for this case), but certainly interesting. As an example take $f(t)=2\|t\|e^{-t^2}$ and $F(x)=1-e^{-x^2}$, then $F$ is not integrable but $F(x)-1$ is. Edit: There is a mistake in the argument, it only works if the hypothesis are satisfied by $G(x)= \int_0^x \|f(t)\|dt$. I'm not sure if the argument can be adapted. • Feb 8th 2011, 12:42 AM oblixps why does $\int_{M} f >0$ imply $\int_{[0,\infty )} f >0$? also i am not sure how to obtain the contradiction. so i take a sequence of functions f_k but each of them have to be bounded by an integrable function and i'm not sure how to pick that. i also have trouble seeing how because of this F(x) is not L-integrable. • Feb 8th 2011, 07:11 AM Tinyboss Are you sure (b) isn't something like, if f is Lebesgue integrable, then $\lim_{x\to\infty}\int_x^\infty f(t)dt=0$? • Feb 8th 2011, 10:19 AM oblixps yes i double checked. b) is: let f, F be as in part a. Show that if F(x) = $\int_0^x f(t)dt$ is L-integrable, then $lim_{x\to\infty} F(x) = 0$. i wasn't sure how to type it but, the limit is actually as x approaches +/- infinite.	2,011	6,851	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 41, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.609375	4	CC-MAIN-2017-51	longest	en	0.941461	# lebesgue integrability and showing limit as x approaches +/-∞ of F(x) = 0.. • Feb 5th 2011, 08:04 PM. oblixps. lebesgue integrability and showing limit as x approaches +/-∞ of F(x) = 0.. a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help.. • Feb 5th 2011, 09:51 PM. TheEmptySet. Quote:. Originally Posted by oblixps. a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help.. for a) you will want to use sequential continuity.. Let $x_n \to x$ and consider the function. $\displaystyle F(x_n)=\lim_{n \to \infty}\int f \chi_{[0,x_n]}d\lambda$. Now since $f(x)\chi_{[0,x_n]} \le f(x)$ and $f(x) \in L^1$. Just use DCT and $F(x_n) \to F(x)$. For b) what would happen it the function didn't go to zero?. • Feb 5th 2011, 11:40 PM. chisigma. Quote:. Originally Posted by oblixps. a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.. b)show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.. i am having trouble proving these statements. i'm not sure but i think part a) involves the property of differentiating under the integral sign which is justified by the dominated convergence theorem for lebesgue integrals. but the hypothesis of the differentiating under the integral sign property requires that the derivative of f (the integrand) exists for almost all x. i don't know if it satisfies this since the only information given is that f is L integrable. as for part b) i am stuck as well and don't know how to go about it. please help.. Examples of function L-integrable on R that doesn't tend to 0 if x tends to infinity are given in.... http://www.mathhelpforum.com/math-he...tml#post597107.	Kind regards. $\chi$ $\sigma$. • Feb 6th 2011, 12:22 AM. Tinyboss. Quote:. Originally Posted by chisigma. Examples of function L-integrable on R that doesn't tend to 0 if x tends to infinity are given in.... http://www.mathhelpforum.com/math-he...tml#post597107. Kind regards. $\chi$ $\sigma$. Exactly...the function could do anything at all on a set of measure zero and not affect the Lebesgue integral.. • Feb 7th 2011, 01:07 PM. oblixps. for part b) would the reason be along these lines: since F is L-integrable it can be written as the infinite sum of the integrals of L-integrable functions and in order for the infinite sum to converge the individual integrals in the sum must approach 0 as n approaches infinite. I'm a little confused though since it is x in F(x) that is approaching infinite. i have looked at the thread you have provided me and near the bottom i saw that F(x) must approach 0 since if it didn't the integral of \|F\| would not be finite. although i can intuitively see that, i am trying to reconcile that with the definitions provided in my book and that is what is causing me some confusion.. my book does not use measure sets to motivate the lebesgue integral but defines it as: let f_k be a sequence of R integrable functions such that the infinite sum of the integral (-infinite, infinite) \|f_k\|dx < infinite, then the lebesgue integral of f = infinite sum of f_k is:. integral of f(x) dx = infinite sum of integral of f_k dx.. can you help me make sense of this problem given my book's definition? thanks in advance.. • Feb 7th 2011, 04:21 PM. Jose27. So, apparently condition b) indeed holds, but functions that satisfy the hypothesis are quite limited:. Assume for the moment $f\geq 0$ then if there exists a measurable set $M \subset \mathbb{R}$ such that $\int_{M} f >0$ then $\int_{[0,\infty )} f >0$ and so $F$ can't be integrable there (take a sequence in $M$ tending to infinity and apply dominated convergence to obtain a contradiction, if $M$ is bounded it's easier still) and from this we deduce $F\notin L^1(\mathbb{R})$ contradicting the assumption. We conclude $f=0$ a.e. In the same way we deal with the case $f\leq 0$.. Now, for any function $g$, we define $g^+=\max \{ g,0\}$ and $g^-=\min \{ g,0 \}$. It's a standard result that $g\in L^1(\mathbb{R})$ iff $g^+,g^-\in L^1(\mathbb{R})$, so we get $F(x)= \int_0^x f^+(t)dt + \int_0^x f^-(t)dt = F^+(x) + F^-(x)$ (this is easily seen to be the case because the integral is monotone), but $F^+ \in L^1(\mathbb{R})$ iff $f= 0$ a.e., and analogous for $F^-$. We therefore must have $f= 0$ a.e. if $F\in L^1(\mathbb{R})$, in which case the conclusion trivially holds.. On the other hand if you ask that $F+c \in L^1(\mathbb{R})$ for some constant c, then the problem is more difficult (read I don't have a proof for this case), but certainly interesting. As an example take $f(t)=2\|t\|e^{-t^2}$ and $F(x)=1-e^{-x^2}$, then $F$ is not integrable but $F(x)-1$ is.. Edit: There is a mistake in the argument, it only works if the hypothesis are satisfied by $G(x)= \int_0^x \|f(t)\|dt$. I'm not sure if the argument can be adapted.. • Feb 8th 2011, 12:42 AM. oblixps. why does $\int_{M} f >0$ imply $\int_{[0,\infty )} f >0$? also i am not sure how to obtain the contradiction. so i take a sequence of functions f_k but each of them have to be bounded by an integrable function and i'm not sure how to pick that. i also have trouble seeing how because of this F(x) is not L-integrable.. • Feb 8th 2011, 07:11 AM. Tinyboss. Are you sure (b) isn't something like, if f is Lebesgue integrable, then $\lim_{x\to\infty}\int_x^\infty f(t)dt=0$?. • Feb 8th 2011, 10:19 AM. oblixps. yes i double checked. b) is:. let f, F be as in part a. Show that if F(x) = $\int_0^x f(t)dt$ is L-integrable, then $lim_{x\to\infty} F(x) = 0$.. i wasn't sure how to type it but, the limit is actually as x approaches +/- infinite.
https://www.enotes.com/homework-help/find-equation-line-which-passes-through-4-1-an-767643	1,695,554,743,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233506632.31/warc/CC-MAIN-20230924091344-20230924121344-00800.warc.gz	846,925,883	17,720	# Find the equation of the line which passes through (-4,1) and is at an angle of 135° with the positive direction of the x axis. The angle essentially gives the slope. If you consider, slope is a measure of the steepness of a line, a lot like a hill, which will go up at a certain angle. To use it for the slope, we need to take tangent function (from trigonometry) of the angle: `tan135 = -1` So, in the equation for the line, slope = m = -1 So, we have m = -1 with a point on the line (-4,1). Various ways to get the equation from there. We could put this information into the equation`y=mx+b`: `1 = -1*-4 + b` So, we can solve that for b: `1 = 4 + b` `b = -3` So, then, we have m and b. So, we can write the equation for the line: `y = -x - 3`	232	762	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.3125	4	CC-MAIN-2023-40	latest	en	0.935487	# Find the equation of the line which passes through (-4,1) and is at an angle of 135° with the positive direction of the x axis.. The angle essentially gives the slope. If you consider, slope is a measure of the steepness of a line, a lot like a hill, which will go up at a certain angle. To use it for the slope, we need to take tangent function (from trigonometry) of the angle:. `tan135 = -1`. So, in the equation for the line, slope = m = -1. So, we have m = -1 with a point on the line (-4,1).. Various ways to get the equation from there. We could put this information into the equation`y=mx+b`:.	`1 = -1*-4 + b`. So, we can solve that for b:. `1 = 4 + b`. `b = -3`. So, then, we have m and b. So, we can write the equation for the line:. `y = -x - 3`.
https://www.coursehero.com/file/6856769/lec-1/	1,519,256,268,000,000,000	text/html	crawl-data/CC-MAIN-2018-09/segments/1518891813818.15/warc/CC-MAIN-20180221222354-20180222002354-00492.warc.gz	890,100,903	28,387	{[ promptMessage ]} Bookmark it {[ promptMessage ]} # lec 1 - Tuesday January 3 Lecture 1 Integration by... This preview shows pages 1–2. Sign up to view the full content. Tuesday, January 3 Lecture 1: Integration by substitution (Refers to 6.1 in your text) After having practiced using the concepts of this lecture the student should be able to: define the differential of a function, integrate indefinite integrals by making appropriate change of variables (substitution), integrate definite integrals by appropriate change of variables. Summary of what we have learned about the notion of integration . - Our study of integration began with attempts at finding the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ]. To do this we introduced the notion of a Riemann sum. But computing areas in this way is inefficient. - The Fundamental theorem of calculus presented an alternate way to compute such numbers. This important theorem is presented into two parts. - The second part of the Fundamental theorem of calculus says that to find the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ] it suffices to find an anti-derivative of the function f ( x ), evaluating it at the limits of integration and subtracting the results. That is, if f ( x ) is continuous on [ a , b ] and F ( x ) is an anti- derivative of f ( x ) then - The first part of the Fundamental theorem of calculus says that, if f ( x ) is continuous on [ a , b ] and as x ranges over [ a , b ], then This can also be expressed by - The first part of the FTC appears to be unrelated to our initial inquiry. In these notes we used it to provide an easier proof of the second part of the FTC. But it is worth remembering it since we occasionally invoke it in particular situations. It is a more This preview has intentionally blurred sections. Sign up to view the full version. View Full Document This is the end of the preview. Sign up to access the rest of the document. {[ snackBarMessage ]} ### Page1 / 9 lec 1 - Tuesday January 3 Lecture 1 Integration by... This preview shows document pages 1 - 2. Sign up to view the full document. View Full Document Ask a homework question - tutors are online	513	2,267	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.875	4	CC-MAIN-2018-09	latest	en	0.926176	{[ promptMessage ]}. Bookmark it. {[ promptMessage ]}. # lec 1 - Tuesday January 3 Lecture 1 Integration by.... This preview shows pages 1–2. Sign up to view the full content.. Tuesday, January 3 Lecture 1: Integration by substitution (Refers to 6.1 in your text) After having practiced using the concepts of this lecture the student should be able to: define the differential of a function, integrate indefinite integrals by making appropriate change of variables (substitution), integrate definite integrals by appropriate change of variables. Summary of what we have learned about the notion of integration . - Our study of integration began with attempts at finding the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ]. To do this we introduced the notion of a Riemann sum. But computing areas in this way is inefficient. - The Fundamental theorem of calculus presented an alternate way to compute such numbers. This important theorem is presented into two parts. - The second part of the Fundamental theorem of calculus says that to find the area of the region bounded by the curve of f ( x ) and the x -axis over an interval [ a , b ] it suffices to find an anti-derivative of the function f ( x ), evaluating it at the limits of integration and subtracting the results. That is, if f ( x ) is continuous on [ a , b ] and F ( x ) is an anti- derivative of f ( x ) then - The first part of the Fundamental theorem of calculus says that, if f ( x ) is continuous on [ a , b ] and as x ranges over [ a , b ], then This can also be expressed by - The first part of the FTC appears to be unrelated to our initial inquiry. In these notes we used it to provide an easier proof of the second part of the FTC. But it is worth remembering it since we occasionally invoke it in particular situations.	It is a more. This preview has intentionally blurred sections. Sign up to view the full version.. View Full Document. This is the end of the preview. Sign up to access the rest of the document.. {[ snackBarMessage ]}. ### Page1 / 9. lec 1 - Tuesday January 3 Lecture 1 Integration by.... This preview shows document pages 1 - 2. Sign up to view the full document.. View Full Document. Ask a homework question - tutors are online.
https://www.brainkart.com/article/Kutzbach-criterion,-Grashoff-s-law-Kutzbach-criterion_6272/	1,685,231,913,000,000,000	text/html	crawl-data/CC-MAIN-2023-23/segments/1685224643388.45/warc/CC-MAIN-20230527223515-20230528013515-00088.warc.gz	753,347,541	7,344	Home \| \| Kinematics of Machinery \| Kutzbach criterion, Grashoff's law Kutzbach criterion # Kutzbach criterion, Grashoff's law Kutzbach criterion Fundamental Equation for 2-D Mechanisms: M = 3(L – 1) – 2J1 – J2 Kutzbach criterion, Grashoff's law Kutzbach criterion: · Fundamental Equation for 2-D Mechanisms: M = 3(L – 1) – 2J1 – J2 Can we intuitively derive Kutzbach’s modification of Grubler’s equation? Consider a rigid link constrained to move in a plane. How many degrees of freedom does the link have? (3: translation in x and y directions, rotation about z-axis) · If you pin one end of the link to the plane, how many degrees of freedom does it now have? · Add a second link to the picture so that you have one link pinned to the plane and one free to move in the plane. How many degrees of freedom exist between the two links? (4 is the correct answer) · Pin the second link to the free end of the first link. How many degrees of freedom do you now have? · How many degrees of freedom do you have each time you introduce a moving link? How many degrees of freedom do you take away when you add a simple joint? How many degrees of freedom would you take away by adding a half joint? Do the different terms in equation make sense in light of this knowledge? Grashoff's law: · Grashoff 4-bar linkage: A linkage that contains one or more links capable of undergoing a full rotation. A linkage is Grashoff if: S + L < P + Q (where: S = shortest link length, L = longest, P, Q = intermediate length links). Both joints of the shortest link are capable of 360 degrees of rotation in a Grashoff linkages. This gives us 4 possible linkages: crank-rocker (input rotates 360), rocker-crank-rocker (coupler rotates 360), rocker-crank (follower); double crank (all links rotate 360). Note that these mechanisms are simply the possible inversions (section 2.11, Figure 2-16) of a Grashoff mechanism. · Non Grashoff 4 bar: No link can rotate 360 if: S + L > P + Q Let’s examine why the Grashoff condition works: · Consider a linkage with the shortest and longest sides joined together. Examine the linkage when the shortest side is parallel to the longest side (2 positions possible, folded over on the long side and extended away from the long side). How long do P and Q have to be to allow the linkage to achieve these positions? · Consider a linkage where the long and short sides are not joined. Can you figure out the required lengths for P and Q in this type of mechanism Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail Mechanical : Kinematics of Machinery : Basics of Mechanisms : Kutzbach criterion, Grashoff's law Kutzbach criterion \|	668	2,766	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.78125	4	CC-MAIN-2023-23	longest	en	0.830054	Home \| \| Kinematics of Machinery \| Kutzbach criterion, Grashoff's law Kutzbach criterion. # Kutzbach criterion, Grashoff's law Kutzbach criterion. Fundamental Equation for 2-D Mechanisms: M = 3(L – 1) – 2J1 – J2. Kutzbach criterion, Grashoff's law Kutzbach criterion:. · Fundamental Equation for 2-D Mechanisms: M = 3(L – 1) – 2J1 – J2. Can we intuitively derive Kutzbach’s modification of Grubler’s equation? Consider a rigid link constrained to move in a plane. How many degrees of freedom does the link have? (3: translation in x and y directions, rotation about z-axis). · If you pin one end of the link to the plane, how many degrees of freedom does it now have?. · Add a second link to the picture so that you have one link pinned to the plane and one free to move in the plane. How many degrees of freedom exist between the two links? (4 is the correct answer). · Pin the second link to the free end of the first link. How many degrees of freedom do you now have?. · How many degrees of freedom do you have each time you introduce a moving link? How many degrees of freedom do you take away when you add a simple joint? How many degrees of freedom would you take away by adding a half joint? Do the different terms in equation make sense in light of this knowledge?.	Grashoff's law:. · Grashoff 4-bar linkage: A linkage that contains one or more links capable of undergoing a full rotation. A linkage is Grashoff if: S + L < P + Q (where: S = shortest link length, L = longest, P, Q = intermediate length links). Both joints of the shortest link are capable of 360 degrees of rotation in a Grashoff linkages. This gives us 4 possible linkages: crank-rocker (input rotates 360), rocker-crank-rocker (coupler rotates 360), rocker-crank (follower); double crank (all links rotate 360). Note that these mechanisms are simply the possible inversions (section 2.11, Figure 2-16) of a Grashoff mechanism.. · Non Grashoff 4 bar: No link can rotate 360 if: S + L > P + Q. Let’s examine why the Grashoff condition works:. · Consider a linkage with the shortest and longest sides joined together. Examine the linkage when the shortest side is parallel to the longest side (2 positions possible, folded over on the long side and extended away from the long side). How long do P and Q have to be to allow the linkage to achieve these positions?. · Consider a linkage where the long and short sides are not joined. Can you figure out the required lengths for P and Q in this type of mechanism. Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail. Mechanical : Kinematics of Machinery : Basics of Mechanisms : Kutzbach criterion, Grashoff's law Kutzbach criterion \|.
https://wiki.haskell.org/index.php?title=Euler_problems/1_to_10&diff=51650&oldid=51649	1,500,672,101,000,000,000	text/html	crawl-data/CC-MAIN-2017-30/segments/1500549423809.62/warc/CC-MAIN-20170721202430-20170721222430-00342.warc.gz	724,218,855	8,937	# Euler problems/1 to 10 (Difference between revisions) ## 1 Problem 1 Add all the natural numbers below 1000 that are multiples of 3 or 5. Two solutions using sum: ```problem_1 = sum [x \| x <- [1..999], x `mod` 3 == 0 \|\| x `mod` 5 == 0] import Data.List (union) problem_1' = sum (union [3,6..999] [5,10..999])``` Another solution which uses algebraic relationships: ```problem_1 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999 where sumStep s n = s * sumOnetoN (n `div` s) sumOnetoN n = n * (n+1) `div` 2``` ## 2 Problem 2 Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million. Solution: ```problem_2 = sum [ x \| x <- takeWhile (<= 1000000) fibs, even x] where fibs = 1 : 1 : zipWith (+) fibs (tail fibs)``` The following two solutions use the fact that the even-valued terms in the Fibonacci sequence themselves form a Fibonacci-like sequence that satisfies evenFib 0 = 0, evenFib 1 = 2, evenFib (n+2) = evenFib n + 4 * evenFib (n+1) . ```problem_2_v2 = sumEvenFibs \$ numEvenFibsLessThan 1000000 sumEvenFibs n = (evenFib n + evenFib (n+1) - 2) `div` 4 evenFib n = round \$ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5 numEvenFibsLessThan n = floor \$ (log (fromIntegral n - 0.5) + 0.5log 5) / log (2 + sqrt 5)``` The first two solutions work because 10^6 is small. The following solution also works for much larger numbers (up to at least 10^1000000 on my computer): ```problem_2 = sumEvenFibsLessThan 1000000 sumEvenFibsLessThan n = (a + b - 1) `div` 2 where n2 = n `div` 2 (a, b) = foldr f (0,1) . takeWhile ((<= n2) . fst) . iterate times2E \$ (1, 4) f x y \| fst z <= n2 = z \| otherwise = y where z = x `addE` y addE (a, b) (c, d) = (ad + bc - 4ac, ac + bd) where ac=ac times2E (a, b) = addE (a, b) (a, b)``` ## 3 Problem 3 Find the largest prime factor of 317584931803. Solution: ```primes = 2 : filter ((==1) . length . primeFactors) [3,5..] primeFactors n = factor n primes where factor n (p:ps) \| pp > n = [n] \| n `mod` p == 0 = p : factor (n `div` p) (p:ps) \| otherwise = factor n ps problem_3 = last (primeFactors 317584931803)``` Another solution, not using recursion, is: ```problem_3 = (m !! 0) `div` (m !! 1) where m = reverse \$ takeWhile (<=n) (scanl1 () [ x \| x <- 2:[3,5..], (n `mod` x) == 0 ]) n = 600851475143``` ## 4 Problem 4 Find the largest palindrome made from the product of two 3-digit numbers. Solution: ```problem_4 = maximum [ x \| y <- [100..999], z <- [y..999], let x = y * z, let s = show x, s == reverse s ]``` ## 5 Problem 5 What is the smallest number divisible by each of the numbers 1 to 20? Solution: ```--http://www.research.att.com/~njas/sequences/A003418 problem_5 = foldr1 lcm [1..20]``` ## 6 Problem 6 What is the difference between the sum of the squares and the square of the sums? Solution: ```fun n = a - b where a = (sum [1..n])^2 b = sum (map (^2) [1..n]) problem_6 = fun 100``` ## 7 Problem 7 Find the 10001st prime. Solution: ```--primes in problem_3 problem_7 = primes !! 10000``` ## 8 Problem 8 Discover the largest product of five consecutive digits in the 1000-digit number. Solution: ```import Data.Char groupsOf _ [] = [] groupsOf n xs = take n xs : groupsOf n ( tail xs ) problem_8 x = maximum . map product . groupsOf 5 \$ x main = do t <- readFile "p8.log" let digits = map digitToInt \$concat \$ lines t print \$ problem_8 digits``` ## 9 Problem 9 There is only one Pythagorean triplet, {a, b, c}, for which a + b + c = 1000. Find the product abc. Solution: ```triplets l = [[a,b,c] \| m <- [2..limit], n <- [1..(m-1)], let a = m^2 - n^2, let b = 2mn, let c = m^2 + n^2, a+b+c==l] where limit = floor . sqrt . fromIntegral \$ l problem_9 = product . head . triplets \$ 1000``` ## 10 Problem 10 Calculate the sum of all the primes below one million. Solution: ```--http://www.research.att.com/~njas/sequences/A046731 problem_10 = sum (takeWhile (< 1000000) primes)```	1,389	3,954	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.28125	4	CC-MAIN-2017-30	latest	en	0.715954	# Euler problems/1 to 10. (Difference between revisions). ## 1 Problem 1. Add all the natural numbers below 1000 that are multiples of 3 or 5.. Two solutions using sum:. ```problem_1 = sum [x \| x <- [1..999], x `mod` 3 == 0 \|\| x `mod` 5 == 0]. import Data.List (union). problem_1' = sum (union [3,6..999] [5,10..999])```. Another solution which uses algebraic relationships:. ```problem_1 = sumStep 3 999 + sumStep 5 999 - sumStep 15 999. where. sumStep s n = s * sumOnetoN (n `div` s). sumOnetoN n = n * (n+1) `div` 2```. ## 2 Problem 2. Find the sum of all the even-valued terms in the Fibonacci sequence which do not exceed one million.. Solution:. ```problem_2 =. sum [ x \| x <- takeWhile (<= 1000000) fibs,. even x]. where. fibs = 1 : 1 : zipWith (+) fibs (tail fibs)```. The following two solutions use the fact that the even-valued terms in the Fibonacci sequence themselves form a Fibonacci-like sequence that satisfies. evenFib 0 = 0, evenFib 1 = 2, evenFib (n+2) = evenFib n + 4 * evenFib (n+1). .. ```problem_2_v2 = sumEvenFibs \$ numEvenFibsLessThan 1000000. sumEvenFibs n = (evenFib n + evenFib (n+1) - 2) `div` 4. evenFib n = round \$ (2 + sqrt 5) ** (fromIntegral n) / sqrt 5. numEvenFibsLessThan n =. floor \$ (log (fromIntegral n - 0.5) + 0.5log 5) / log (2 + sqrt 5)```. The first two solutions work because 10^6 is small. The following solution also works for much larger numbers (up to at least 10^1000000 on my computer):. ```problem_2 = sumEvenFibsLessThan 1000000. sumEvenFibsLessThan n = (a + b - 1) `div` 2. where. n2 = n `div` 2. (a, b) = foldr f (0,1). . takeWhile ((<= n2) . fst). . iterate times2E \$ (1, 4). f x y \| fst z <= n2 = z. \| otherwise = y. where z = x `addE` y. addE (a, b) (c, d) = (ad + bc - 4ac, ac + bd). where ac=ac. times2E (a, b) = addE (a, b) (a, b)```. ## 3 Problem 3. Find the largest prime factor of 317584931803.. Solution:. ```primes = 2 : filter ((==1) . length . primeFactors) [3,5..]. primeFactors n = factor n primes. where. factor n (p:ps). \| p*p > n = [n]. \| n `mod` p == 0 = p : factor (n `div` p) (p:ps). \| otherwise = factor n ps. problem_3 = last (primeFactors 317584931803)```. Another solution, not using recursion, is:. ```problem_3 = (m !! 0) `div` (m !! 1). where.	m = reverse \$. takeWhile (<=n) (scanl1 () [ x \| x <- 2:[3,5..], (n `mod` x) == 0 ]). n = 600851475143```. ## 4 Problem 4. Find the largest palindrome made from the product of two 3-digit numbers.. Solution:. ```problem_4 = maximum [ x \| y <- [100..999],. z <- [y..999],. let x = y z,. let s = show x,. s == reverse s ]```. ## 5 Problem 5. What is the smallest number divisible by each of the numbers 1 to 20?. Solution:. ```--http://www.research.att.com/~njas/sequences/A003418. problem_5 = foldr1 lcm [1..20]```. ## 6 Problem 6. What is the difference between the sum of the squares and the square of the sums?. Solution:. ```fun n = a - b. where. a = (sum [1..n])^2. b = sum (map (^2) [1..n]). problem_6 = fun 100```. ## 7 Problem 7. Find the 10001st prime.. Solution:. ```--primes in problem_3. problem_7 = primes !! 10000```. ## 8 Problem 8. Discover the largest product of five consecutive digits in the 1000-digit number.. Solution:. ```import Data.Char. groupsOf _ [] = []. groupsOf n xs =. take n xs : groupsOf n ( tail xs ). problem_8 x = maximum . map product . groupsOf 5 \$ x. main = do t <- readFile "p8.log". let digits = map digitToInt \$concat \$ lines t. print \$ problem_8 digits```. ## 9 Problem 9. There is only one Pythagorean triplet, {a, b, c}, for which a + b + c = 1000. Find the product abc.. Solution:. ```triplets l = [[a,b,c] \| m <- [2..limit],. n <- [1..(m-1)],. let a = m^2 - n^2,. let b = 2mn,. let c = m^2 + n^2,. a+b+c==l]. where limit = floor . sqrt . fromIntegral \$ l. problem_9 = product . head . triplets \$ 1000```. ## 10 Problem 10. Calculate the sum of all the primes below one million.. Solution:. ```--http://www.research.att.com/~njas/sequences/A046731. problem_10 = sum (takeWhile (< 1000000) primes)```.
https://vectormap.net/map_design_tutorials/a-free-hand-drawing-with-simultaneous-projection-of-predicted-guideline/	1,618,465,068,000,000,000	text/html	crawl-data/CC-MAIN-2021-17/segments/1618038083007.51/warc/CC-MAIN-20210415035637-20210415065637-00176.warc.gz	695,452,873	23,773	# A free-hand drawing with simultaneous projection of predicted guideline Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing. In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point and a line on a plane. The locus of points in that plane that are equidistant from both the line and point is a parabola. In mathematics, an ellipse (from Greek ¿¿¿¿¿¿¿¿ elleipsis, a ‘falling short’) is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone’s axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are equidistant from a given point, the centre. The distance between any of the points and the centre is called the radius. Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. Thus, until seventeenth century, lines were defined like this: ‘The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [… … In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional or three-dimensional (space curves) Euclidean space are of interest. Drawing is a form of visual art that makes use of any number of drawing instruments to mark a two-dimensional medium. Common instruments include graphite pencils, pen and ink, inked brushes, wax color pencils, crayons, charcoal, chalk, pastels, various kinds of erasers, markers, styluses, and various metals. An artist who practices or works in drawing may be called a draftsman or draughtsman. A small amount of material is released onto the two dimensional medium, leaving a visible mark. Canvas is an extremely heavy-duty plain-woven fabric used for making sails, tents, marquees, backpacks, and other items for which sturdiness is required. It is also popularly used by artists as a painting surface, typically stretched across a wooden frame. It is also used in such fashion objects as handbags and shoes. Source.	621	2,956	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2021-17	latest	en	0.946529	# A free-hand drawing with simultaneous projection of predicted guideline. Graphical projection is a protocol by which an image of a three-dimensional object is projected onto a planar surface without the aid of mathematical calculation, used in technical drawing. In mathematics, a parabola is a conic section, created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point and a line on a plane. The locus of points in that plane that are equidistant from both the line and point is a parabola. In mathematics, an ellipse (from Greek ¿¿¿¿¿¿¿¿ elleipsis, a ‘falling short’) is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone’s axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are equidistant from a given point, the centre. The distance between any of the points and the centre is called the radius. Circles are simple closed curves which divide the plane into two regions: an interior and an exterior. The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects.	Thus, until seventeenth century, lines were defined like this: ‘The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [… … In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but which is not required to be straight. This entails that a line is a special case of curve, namely a curve with null curvature. Often curves in two-dimensional or three-dimensional (space curves) Euclidean space are of interest. Drawing is a form of visual art that makes use of any number of drawing instruments to mark a two-dimensional medium. Common instruments include graphite pencils, pen and ink, inked brushes, wax color pencils, crayons, charcoal, chalk, pastels, various kinds of erasers, markers, styluses, and various metals. An artist who practices or works in drawing may be called a draftsman or draughtsman. A small amount of material is released onto the two dimensional medium, leaving a visible mark. Canvas is an extremely heavy-duty plain-woven fabric used for making sails, tents, marquees, backpacks, and other items for which sturdiness is required. It is also popularly used by artists as a painting surface, typically stretched across a wooden frame. It is also used in such fashion objects as handbags and shoes. Source.
http://www.dailykos.com/story/2013/11/17/1256193/-Snarky-Opinion-on-Statistics?detail=hide	1,432,776,663,000,000,000	text/html	crawl-data/CC-MAIN-2015-22/segments/1432207929176.79/warc/CC-MAIN-20150521113209-00097-ip-10-180-206-219.ec2.internal.warc.gz	387,346,480	18,026	A personal opinion on statistics inspired by the excellent Kos diary "Vaccinate Yourself Against Statistics" which I recommend follows. Yeah, following is just my opinion, based on that dwem LaPlace's writing on probability and a stubborn streak about real facts. A Snarky Treat(ise) on Statistics Winning Powerball versus Committing Suicide Powerball Odds *Compared to U. S. Committing Suicide Odds ** 175,223,510 24,530 times more likely to commit suicide than win Jackpot 5,153,632 721 times more likely to commit suicide than win \$1 million 648,975 91 times more likely to commit suicide than win \$10,000 19,087 2.67 times more likely to commit suicide than win \$100 12,244 1.71 times more likely to commit suicide than win \$100 red ball 706 (only) 10 times more likely to win \$7 with red ball than commit suicide 360 (only) 20 times more likely to win \$7 without red than commit suicide 110 (only) 65 times more likely to win \$4 without red than commit suicide 55 (only) 130 times more likely to win \$4 with red ball than commit suicide "LIkely" based on 14 suicides per 100,000, the tenth leading cause of death in US. Suicide rate from http://www.bbc.co.uk/... Which reported U.S. Center for Disease Control Statistics. May 2, 2013 Powerball odds from http://www.powerball.com/... Calculations as follows: 14 per 100,000 equals 1 per 7,143 persons commit suicide. Comparison of odds = powerball odds (per 1) divided by suicide odds of 7,143 (per 1). Huh? In addition to the obvious, shows that comparison of statistical subjects has close to zero meaning for an individual, a way in which statistics mislead. For example, for a person who will definitely not commit suicide the comparison of statistics is invalid, his number will not change them. The powerball odds stay the same, the suicide odds stay the same. If he does win powerball then his odds of winning were 100%, there was just no way to know that in advance. If the individual does commit suicide, his odds were 100% in retrospect. At some point that person knew in advance. What are the odds that the sun will rise tomorrow? Rise versus not-rise, like flip of a coin, 50/50. I hear in the Anthropology Museum in Ciudad Mexico that the Aztec priests realized this 50/50 statistic and prayed without ceasing, that the sun would rise tomorrow, which worked, as the sun kept rising. Heart rending. Same as with suicide as above, there are other factors than the 50/50. Statistics are to inform opinions, not twist them. As seen here, comparing statistics does not work. The struck by lightning trope among thousands of other stats "more likely" is B.S.. When do stats work? In my opinion, only on an agreed common base, like cards, then only in a special way of guessing. Same for surveys, sports. Now if they rise the same way every day...for awhile...same base... One person believed that if she has a certain gene her "risk of getting alzheimer's 'increases'. Huh? Wait a minute. No, no, some reasoning missing here. You can figure it out, and other "stats" too. So more people have been killed by falling sandcastles than sharks? Did I get that right? Hm, if true what am I to conclude? Something fishy here. Some can check that out, saw it on BBC net. I won't bother. May your chi-square be one or two dots. May your sun rise tomorrow. #### Tags EMAIL TO A FRIEND X You must add at least one tag to this diary before publishing it. Add keywords that describe this diary. Separate multiple keywords with commas. Tagging tips - Search For Tags - Browse For Tags ? More Tagging tips: A tag is a way to search for this diary. If someone is searching for "Barack Obama," is this a diary they'd be trying to find? Use a person's full name, without any title. Senator Obama may become President Obama, and Michelle Obama might run for office. If your diary covers an election or elected official, use election tags, which are generally the state abbreviation followed by the office. CA-01 is the first district House seat. CA-Sen covers both senate races. NY-GOV covers the New York governor's race. 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Are you sure you want to save these changes to the published diary?	1,351	5,828	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.6875	4	CC-MAIN-2015-22	latest	en	0.853319	A personal opinion on statistics inspired by the excellent Kos diary "Vaccinate Yourself Against Statistics" which I recommend follows. Yeah, following is just my opinion, based on that dwem LaPlace's writing on probability and a stubborn streak about real facts.. A Snarky Treat(ise) on Statistics. Winning Powerball versus Committing Suicide. Powerball Odds *Compared to U. S. Committing Suicide Odds **. 175,223,510 24,530 times more likely to commit suicide than win Jackpot. 5,153,632 721 times more likely to commit suicide than win \$1 million. 648,975 91 times more likely to commit suicide than win \$10,000. 19,087 2.67 times more likely to commit suicide than win \$100. 12,244 1.71 times more likely to commit suicide than win \$100 red ball. 706 (only) 10 times more likely to win \$7 with red ball than commit suicide. 360 (only) 20 times more likely to win \$7 without red than commit suicide. 110 (only) 65 times more likely to win \$4 without red than commit suicide. 55 (only) 130 times more likely to win \$4 with red ball than commit suicide. "LIkely" based on 14 suicides per 100,000, the tenth leading cause of death in US.. Suicide rate from http://www.bbc.co.uk/.... Which reported U.S. Center for Disease Control Statistics. May 2, 2013. Powerball odds from http://www.powerball.com/.... Calculations as follows: 14 per 100,000 equals 1 per 7,143 persons commit suicide.. Comparison of odds = powerball odds (per 1) divided by suicide odds of 7,143 (per 1).. Huh? In addition to the obvious, shows that comparison of statistical subjects has close to zero meaning for an individual, a way in which statistics mislead. For example, for a person who will definitely not commit suicide the comparison of statistics is invalid, his number will not change them. The powerball odds stay the same, the suicide odds stay the same. If he does win powerball then his odds of winning were 100%, there was just no way to know that in advance. If the individual does commit suicide, his odds were 100% in retrospect. At some point that person knew in advance.. What are the odds that the sun will rise tomorrow? Rise versus not-rise, like flip of a coin, 50/50. I hear in the Anthropology Museum in Ciudad Mexico that the Aztec priests realized this 50/50 statistic and prayed without ceasing, that the sun would rise tomorrow, which worked, as the sun kept rising. Heart rending. Same as with suicide as above, there are other factors than the 50/50. Statistics are to inform opinions, not twist them.. As seen here, comparing statistics does not work. The struck by lightning trope among thousands of other stats "more likely" is B.S.. When do stats work? In my opinion, only on an agreed common base, like cards, then only in a special way of guessing. Same for surveys, sports. Now if they rise the same way every day...for awhile...same base.... One person believed that if she has a certain gene her "risk of getting alzheimer's 'increases'. Huh? Wait a minute. No, no, some reasoning missing here.	You can figure it out, and other "stats" too.. So more people have been killed by falling sandcastles than sharks? Did I get that right? Hm, if true what am I to conclude? Something fishy here. Some can check that out, saw it on BBC net. I won't bother.. May your chi-square be one or two dots. May your sun rise tomorrow.. #### Tags. EMAIL TO A FRIEND X. You must add at least one tag to this diary before publishing it.. Add keywords that describe this diary. Separate multiple keywords with commas.. Tagging tips - Search For Tags - Browse For Tags. ?. More Tagging tips:. A tag is a way to search for this diary. If someone is searching for "Barack Obama," is this a diary they'd be trying to find?. Use a person's full name, without any title. Senator Obama may become President Obama, and Michelle Obama might run for office.. If your diary covers an election or elected official, use election tags, which are generally the state abbreviation followed by the office. CA-01 is the first district House seat. CA-Sen covers both senate races. NY-GOV covers the New York governor's race.. Tags do not compound: that is, "education reform" is a completely different tag from "education". A tag like "reform" alone is probably not meaningful.. Consider if one or more of these tags fits your diary: Civil Rights, Community, Congress, Culture, Economy, Education, Elections, Energy, Environment, Health Care, International, Labor, Law, Media, Meta, National Security, Science, Transportation, or White House. If your diary is specific to a state, consider adding the state (California, Texas, etc). Keep in mind, though, that there are many wonderful and important diaries that don't fit in any of these tags. Don't worry if yours doesn't.. You can add a private note to this diary when hotlisting it:. Are you sure you want to remove this diary from your hotlist?. Are you sure you want to remove your recommendation? You can only recommend a diary once, so you will not be able to re-recommend it afterwards.. Rescue this diary, and add a note:. Are you sure you want to remove this diary from Rescue?. Choose where to republish this diary. The diary will be added to the queue for that group. Publish it from the queue to make it appear.. You must be a member of a group to use this feature.. Add a quick update to your diary without changing the diary itself:. Are you sure you want to remove this diary?. Unpublish Diary (The diary will be removed from the site and returned to your drafts for further editing.) Delete Diary (The diary will be removed.). Are you sure you want to save these changes to the published diary?.
http://www.jiskha.com/display.cgi?id=1394425835	1,498,412,085,000,000,000	text/html	crawl-data/CC-MAIN-2017-26/segments/1498128320545.67/warc/CC-MAIN-20170625170634-20170625190634-00680.warc.gz	563,008,688	3,621	# physics posted by . A piece of metal has a mass of 85cg and a volume of 450L. What is its density in g/cm3? What would be the mass of 300 cm3 of this metal? What would be the volume of 00 ounces of this metal? • physics - m = 85 kg? V = 450 L. V = 450L * 1000cm^3/L = 450,000cm^3 D = 85,000g/450,000cm^3 = 0.189 g/cm^3. Mass = 300cm^3 * 0.189g/cm^3 = 56.67 g = 0.0567 kg.	152	380	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.625	4	CC-MAIN-2017-26	latest	en	0.896695	# physics. posted by .. A piece of metal has a mass of 85cg and a volume of 450L. What is its density in g/cm3? What would be the mass of 300 cm3 of this metal? What would be the volume of 00 ounces of this metal?. • physics -. m = 85 kg?.	V = 450 L.. V = 450L * 1000cm^3/L = 450,000cm^3. D = 85,000g/450,000cm^3 = 0.189 g/cm^3.. Mass = 300cm^3 * 0.189g/cm^3 = 56.67 g =. 0.0567 kg.
https://unemployedtutors.org/you-are-fishing-off-a-bridge-and-feel-a-tug-on-the-vertical-line-this-time-your-lucky-catch-is-an-old-boot-2/	1,660,635,987,000,000,000	text/html	crawl-data/CC-MAIN-2022-33/segments/1659882572221.38/warc/CC-MAIN-20220816060335-20220816090335-00391.warc.gz	526,770,722	10,087	# You are fishing off a bridge and feel a tug on the vertical line. This time, your lucky catch is an old boot. \| You are fishing off a bridge and feel a tug on the vertical line. This time, your lucky catch is an old boot. (a) Assume that the boot is not punctured, so that as you lift it out of the water at constant speed, you haul up one bootful, or 7500 cm^3, of water along with the boot. If the neoprene rubber making up the boot has volume 435 cm^3 and density 1240 kg/m^3, then what is the tension on your fishing line after you pull the boot out of the water? Answer in Newton. Do NOT give me the answer 1.02N on yahoo, that is for part b..button {background-color: #4CAF50;border: none;color: white;padding: 10px 20px;text-align: center;text-decoration: none;display: inline-block;font-size: 16px;margin: 4px 2px;cursor: pointer;border-radius: 10px;} “Are you looking for this answer? We can Help click Order Now”	250	926	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.546875	4	CC-MAIN-2022-33	latest	en	0.845599	# You are fishing off a bridge and feel a tug on the vertical line. This time, your lucky catch is an old boot. \|. You are fishing off a bridge and feel a tug on the vertical line. This time, your lucky catch is an old boot.	(a) Assume that the boot is not punctured, so that as you lift it out of the water at constant speed, you haul up one bootful, or 7500 cm^3, of water along with the boot. If the neoprene rubber making up the boot has volume 435 cm^3 and density 1240 kg/m^3, then what is the tension on your fishing line after you pull the boot out of the water? Answer in Newton. Do NOT give me the answer 1.02N on yahoo, that is for part b..button {background-color: #4CAF50;border: none;color: white;padding: 10px 20px;text-align: center;text-decoration: none;display: inline-block;font-size: 16px;margin: 4px 2px;cursor: pointer;border-radius: 10px;} “Are you looking for this answer? We can Help click Order Now”.
https://www.racehotwheels.com/2016/01/hot-wheels-stem-lesson-plan-scientific.html	1,718,998,959,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198862157.88/warc/CC-MAIN-20240621191840-20240621221840-00805.warc.gz	823,944,256	21,691	## Thursday, January 28, 2016 ### Introduction and background After I have taught/reviewed the scientific method with my students, we do a lab using Hot Wheels cars to give them practice using the scientific method. The question they are trying to answer is: On which surface will my Hot Wheels car roll furthest after going down a ramp: 60 grit sandpaper, carpet, or cedar fencing? You could do this lab with any number or type of surfaces that you choose. Also, depending on the time you have and resources available you could let the students completely design the experiment, coming up with the surfaces and the set-up. If this is the case, you could reword the question to: On which surface will my Hot Wheels car roll furthest after going down a ramp. For our lab, due to time constraints, I have designed the initial set-up of the lab and the surfaces for the cars. ### The Lab The purpose of the lab is: To show understanding of the scientific method by designing a lab sheet (and possibly the lab itself), completing the experiment, then analyzing the data and reporting on the lab. After showing the students the initial set-up, I have the students use a word processor to design a lab sheet that includes the following headings: • Problem/Question • Hypothesis • Independent and Dependent Variables • Constants • Procedure • Data • Analysis • Conclusion The students need to have information under each heading through the Procedure as well as a data table that they can fill out during the experiment done and printed out before I will let them proceed with the experiment. An example may look like this. #### Hot Wheels Scientific Method Lab Problem/Question On which surface will my Hot Wheels car roll furthest after going down a ramp: 60 grit sandpaper, carpet, or cedar fencing? Hypothesis My hypothesis is that the Hot Wheels car will roll the furthest on the 60 grit sandpaper. Independent and Dependent Variables The independent variable in this experiment is the surfaces the car is traveling on. The dependent variable is the distance the car goes. Constants Some things that need to remain constant are: The car I use. Where/how high the car starts on the ramp. How I release the car. How I measure the distance including to the front or back of the car and the units I use. Procedure Line the back of the car up with back edge of the track. Without pushing the car, release it so that it rolls down the orange track. Let it roll on the surface until it comes to a stop. Measure from the end of the orange track to the back of the car and record the distance in centimeters. Repeat 5 times for each surface. Use the same car for each trial and each surface. Data Analysis Conclusion Once they have this much done and printed out, I let them start the experiment. They should record the data on in their table. When they have finished the experiment, it's back to the computer for them to do their analysis and write up their conclusion. When the lab paper is finished I have them print it out and turn it in. ### Notes • Remember, this lab is not really about what surface the cars go further on. The students are not graded on getting the "right" surface. It's about the scientific method. How well did they organize their lab sheet? How well did they pay attention to the constants. How well did they perform the procedure. How well did they analyze the experiment and discuss the problems. These are the things that I pay attention to and grade them on. • The set-up I used is not perfect by design. I like that there are problems with the set-up and hope that the students notice these and address them in their analysis and conclusion. One of the potential problems is that on the wood and the sandpaper, sometimes the cars veer off the surface. The most common ways the students try to deal with these are to just do more trials until they get their specified number on the surface; or they will put "railings" on the edge to keep the car on the surface. Hopefully they realize that rubbing on their "railing" will cause the car not to go as far and perhaps skew their results. I usually don't talk to them about this, but do look to see if they discussed how their solution may have affected their data. Using yard sticks to guide the cars Another issue with the experiment is that I've taped several pieces of sandpaper together. Where they come together the seams aren't always smooth and sometime there are little ridges. That could also affect their data. Again, I don't talk to them about it. I just watch to see if they notice. There are a few other small possible problems with this set-up, but the key is, do the students notice, what if anything do they do to solve them, and most importantly, do they talk about them in their analysis and conclusion. This is how they learn about experiments and what make a good one. • Make sure that all the constants listed are addressed in the procedure. • I don't mention how many trials they should do. I let them decided. This let's me know how well they understand the scientific method and how important multiple trials are. I always have a few who only do one trial. Most do 2 or 3. Once in a while I get students who will do 5 or more.	1,155	5,275	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.03125	4	CC-MAIN-2024-26	latest	en	0.923837	## Thursday, January 28, 2016. ### Introduction and background. After I have taught/reviewed the scientific method with my students, we do a lab using Hot Wheels cars to give them practice using the scientific method.. The question they are trying to answer is: On which surface will my Hot Wheels car roll furthest after going down a ramp: 60 grit sandpaper, carpet, or cedar fencing?. You could do this lab with any number or type of surfaces that you choose. Also, depending on the time you have and resources available you could let the students completely design the experiment, coming up with the surfaces and the set-up. If this is the case, you could reword the question to: On which surface will my Hot Wheels car roll furthest after going down a ramp.. For our lab, due to time constraints, I have designed the initial set-up of the lab and the surfaces for the cars.. ### The Lab. The purpose of the lab is: To show understanding of the scientific method by designing a lab sheet (and possibly the lab itself), completing the experiment, then analyzing the data and reporting on the lab.. After showing the students the initial set-up, I have the students use a word processor to design a lab sheet that includes the following headings:. • Problem/Question. • Hypothesis. • Independent and Dependent Variables. • Constants. • Procedure. • Data. • Analysis. • Conclusion. The students need to have information under each heading through the Procedure as well as a data table that they can fill out during the experiment done and printed out before I will let them proceed with the experiment. An example may look like this.. #### Hot Wheels Scientific Method Lab. Problem/Question. On which surface will my Hot Wheels car roll furthest after going down a ramp: 60 grit sandpaper, carpet, or cedar fencing?. Hypothesis. My hypothesis is that the Hot Wheels car will roll the furthest on the 60 grit sandpaper.. Independent and Dependent Variables.	The independent variable in this experiment is the surfaces the car is traveling on. The dependent variable is the distance the car goes.. Constants. Some things that need to remain constant are:. The car I use.. Where/how high the car starts on the ramp.. How I release the car.. How I measure the distance including to the front or back of the car and the units I use.. Procedure. Line the back of the car up with back edge of the track. Without pushing the car, release it so that it rolls down the orange track. Let it roll on the surface until it comes to a stop. Measure from the end of the orange track to the back of the car and record the distance in centimeters. Repeat 5 times for each surface. Use the same car for each trial and each surface.. Data. Analysis. Conclusion. Once they have this much done and printed out, I let them start the experiment. They should record the data on in their table. When they have finished the experiment, it's back to the computer for them to do their analysis and write up their conclusion. When the lab paper is finished I have them print it out and turn it in.. ### Notes. • Remember, this lab is not really about what surface the cars go further on. The students are not graded on getting the "right" surface. It's about the scientific method. How well did they organize their lab sheet? How well did they pay attention to the constants. How well did they perform the procedure. How well did they analyze the experiment and discuss the problems. These are the things that I pay attention to and grade them on.. • The set-up I used is not perfect by design. I like that there are problems with the set-up and hope that the students notice these and address them in their analysis and conclusion. One of the potential problems is that on the wood and the sandpaper, sometimes the cars veer off the surface. The most common ways the students try to deal with these are to just do more trials until they get their specified number on the surface; or they will put "railings" on the edge to keep the car on the surface. Hopefully they realize that rubbing on their "railing" will cause the car not to go as far and perhaps skew their results. I usually don't talk to them about this, but do look to see if they discussed how their solution may have affected their data. Using yard sticks to guide the cars. Another issue with the experiment is that I've taped several pieces of sandpaper together. Where they come together the seams aren't always smooth and sometime there are little ridges. That could also affect their data. Again, I don't talk to them about it. I just watch to see if they notice. There are a few other small possible problems with this set-up, but the key is, do the students notice, what if anything do they do to solve them, and most importantly, do they talk about them in their analysis and conclusion. This is how they learn about experiments and what make a good one.. • Make sure that all the constants listed are addressed in the procedure.. • I don't mention how many trials they should do. I let them decided. This let's me know how well they understand the scientific method and how important multiple trials are. I always have a few who only do one trial. Most do 2 or 3. Once in a while I get students who will do 5 or more.
https://madwirebuild.com/solved-how-much-time-does-it-take-an-electromagnetic-gentle/	1,695,519,064,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233506539.13/warc/CC-MAIN-20230923231031-20230924021031-00655.warc.gz	427,240,786	17,971	Solved How Much Time Does It Take An Electromagnetic Gentle Okay, So this query requested, how long wouldn’t it soak up minutes for radio wave to journey from Venice to the earth? Now we all know that according query, it’s 28 million miles between Venus and Earth. So we need to make certain that those air in the identical items. All right, so we know that there’s for every one mile, there are 1.sixty one kilometers. Okay, so now if we have been to cease right here, we now have been kilometers, however we wanted in meters. So we know that for each kilometer based mostly on the the tens scale of the metric system, there will be one thousand meters, so that can give us after we plug that in. In the earlier couple of years scientists have found that the water molecules in the environment additionally generate waves of light busiest travel days around christmas 2018. Compute the angular momentum of the earth arising from the next motions. Earth’s orbital movement around the solar. Any radio wave is transferring at the velocity of sunshine, whatever the frequency, so the frequency must be superfluous info. AM radio waves have larger wavelengths (some as massive as a kilometer!). It is normally easier to get reception for AM waves in mountain areas compared to FM waves which have shorter wavelengths. Explain why this is so utilizing your data of diffraction. Describe how the motions of the particles that make up an object change when the object’s temperature will increase. Astronomers observe the chromosphere of the Sun with a filter that passes the pink hydrogen spectral line of wavelength 656.three nm, known as the $\mathrm _$ line. The filter consists of a clear dielectric of thickness d held between two partially aluminized glass plates. The filter is saved at a constant temperature. The dielectric may also pass what near-visible wavelength? The pace of sunshine is 299,792,458 metres per second – which is 1,079,252,849km/hour. The Earth is the source of every wave that has ever been created, however that’s not all that makes the Earth particular. It is the supply of the next largest wave, but the Earth is the source of the second largest wave, which is the wave that’s traveling to the moon. Scientist typically agree that earth is getting warmer as a end result of what call the green house effectt. Why is the Curie temperature such a significant property of magnetic materials? • 69	509	2,426	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2023-40	latest	en	0.935842	Solved How Much Time Does It Take An Electromagnetic Gentle. Okay, So this query requested, how long wouldn’t it soak up minutes for radio wave to journey from Venice to the earth? Now we all know that according query, it’s 28 million miles between Venus and Earth. So we need to make certain that those air in the identical items. All right, so we know that there’s for every one mile, there are 1.sixty one kilometers. Okay, so now if we have been to cease right here, we now have been kilometers, however we wanted in meters. So we know that for each kilometer based mostly on the the tens scale of the metric system, there will be one thousand meters, so that can give us after we plug that in.. In the earlier couple of years scientists have found that the water molecules in the environment additionally generate waves of light busiest travel days around christmas 2018. Compute the angular momentum of the earth arising from the next motions. Earth’s orbital movement around the solar.. Any radio wave is transferring at the velocity of sunshine, whatever the frequency, so the frequency must be superfluous info. AM radio waves have larger wavelengths (some as massive as a kilometer!). It is normally easier to get reception for AM waves in mountain areas compared to FM waves which have shorter wavelengths. Explain why this is so utilizing your data of diffraction.	Describe how the motions of the particles that make up an object change when the object’s temperature will increase. Astronomers observe the chromosphere of the Sun with a filter that passes the pink hydrogen spectral line of wavelength 656.three nm, known as the $\mathrm _$ line. The filter consists of a clear dielectric of thickness d held between two partially aluminized glass plates. The filter is saved at a constant temperature. The dielectric may also pass what near-visible wavelength?. The pace of sunshine is 299,792,458 metres per second – which is 1,079,252,849km/hour. The Earth is the source of every wave that has ever been created, however that’s not all that makes the Earth particular. It is the supply of the next largest wave, but the Earth is the source of the second largest wave, which is the wave that’s traveling to the moon. Scientist typically agree that earth is getting warmer as a end result of what call the green house effectt. Why is the Curie temperature such a significant property of magnetic materials?. • 69.
https://mathspace.co/textbooks/syllabuses/Syllabus-1000/topics/Topic-19901/subtopics/Subtopic-263735/?textbookIntroActiveTab=overview	1,638,505,041,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964362589.37/warc/CC-MAIN-20211203030522-20211203060522-00263.warc.gz	453,096,952	54,266	# 3.04 12 and 24 hour time Lesson ### Representing time There are two main ways we represent time over a day. $24$24 hour time represents time as one block of $24$24 hours.$12$12 hour time represents a day as two blocks of $12$12 hours each. The hours before midday add an "am" to show it's the morning and the hours after midday add on a "pm" so we know it's the afternoon. There is also variation in how $12$12 and $24$24 hour time is displayed.$12$12 hour time may use "am", "a.m.", "AM" or similar variations, while $24$24 hour time could represent $03:15$03:15 as $3:15$3:15or $0315$0315 which is more common in the military Below is a diagram showing the conversion between $12$12 and $24$24 time. In all cases the number of minutes will be the same. ### Calculating time The conversions between the different units of time are: Time calculations are similar to our normal addition and subtraction so we can use similar strategies. However, we will also use these time unit conversions, such as $60$60 minutes in an hour and $24$24 hours in a day. #### Worked Example What will the time be $2$2 hours and $40$40 minutes after $10:45$10:45 am? Think: We can approach this problem in a few ways. We can add the number of minutes on first and then the number of hours. Do: If we add $40$40 minutes onto $10:45$10:45 we can only add $15$15 minutes on before we get to $11:00$11:00 am. We then can add the remaining $\left(40-15\right)=25$(4015)=25 minutes to get to $11:25$11:25 am. We can then add $2$2 hours on, which is $13:25$13:25 which would be correct in $24$24 hour time, but we know this is $12$12 hour time because the time has "am". So, the time will be $1:25$1:25 pm. Reflect: There are many different strategies we can take to get to this answer, just like regular addition and subtraction. Use the clock applet above to play around with adding or subtracting different amounts of time. We can also change the starting time using input box. #### Practice questions ##### Question 1 Write $9$9 days in hours. ##### Question 2 Which of the following represents $30$30 minutes after $11$11$:$:$40$40 pm? 1. $12$12$:$:$10$10am A $11$11$:$:$10$10 pm B $11$11$:$:$70$70 pm C $12$12$:$:$10$10 pm D $12$12$:$:$10$10am A $11$11$:$:$10$10 pm B $11$11$:$:$70$70 pm C $12$12$:$:$10$10 pm D ##### Question 3 Kate leaves her home at $11$11$:$:$15$15 am to visit her grandmother in a different state. She arrives at her grandmother's house at $5$5$:$:$15$15 pm. 1. How long was Kate's trip in hours? ### Outcomes #### MA4-7NA operates with ratios and rates, and explores their graphical representation #### MA4-15MG performs calculations of time that involve mixed units, and interprets time zones	812	2,740	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.71875	5	CC-MAIN-2021-49	latest	en	0.924089	# 3.04 12 and 24 hour time. Lesson. ### Representing time. There are two main ways we represent time over a day. $24$24 hour time represents time as one block of $24$24 hours.$12$12 hour time represents a day as two blocks of $12$12 hours each. The hours before midday add an "am" to show it's the morning and the hours after midday add on a "pm" so we know it's the afternoon.. There is also variation in how $12$12 and $24$24 hour time is displayed.$12$12 hour time may use "am", "a.m.", "AM" or similar variations, while $24$24 hour time could represent $03:15$03:15 as $3:15$3:15or $0315$0315 which is more common in the military. Below is a diagram showing the conversion between $12$12 and $24$24 time. In all cases the number of minutes will be the same.. ### Calculating time. The conversions between the different units of time are:. Time calculations are similar to our normal addition and subtraction so we can use similar strategies. However, we will also use these time unit conversions, such as $60$60 minutes in an hour and $24$24 hours in a day.. #### Worked Example. What will the time be $2$2 hours and $40$40 minutes after $10:45$10:45 am?. Think: We can approach this problem in a few ways. We can add the number of minutes on first and then the number of hours.. Do: If we add $40$40 minutes onto $10:45$10:45 we can only add $15$15 minutes on before we get to $11:00$11:00 am. We then can add the remaining $\left(40-15\right)=25$(4015)=25 minutes to get to $11:25$11:25 am.. We can then add $2$2 hours on, which is $13:25$13:25 which would be correct in $24$24 hour time, but we know this is $12$12 hour time because the time has "am". So, the time will be $1:25$1:25 pm.. Reflect: There are many different strategies we can take to get to this answer, just like regular addition and subtraction.. Use the clock applet above to play around with adding or subtracting different amounts of time. We can also change the starting time using input box.. #### Practice questions. ##### Question 1. Write $9$9 days in hours.. ##### Question 2. Which of the following represents $30$30 minutes after $11$11$:$:$40$40 pm?. 1.	$12$12$:$:$10$10am. A. $11$11$:$:$10$10 pm. B. $11$11$:$:$70$70 pm. C. $12$12$:$:$10$10 pm. D. $12$12$:$:$10$10am. A. $11$11$:$:$10$10 pm. B. $11$11$:$:$70$70 pm. C. $12$12$:$:$10$10 pm. D. ##### Question 3. Kate leaves her home at $11$11$:$:$15$15 am to visit her grandmother in a different state. She arrives at her grandmother's house at $5$5$:$:$15$15 pm.. 1. How long was Kate's trip in hours?. ### Outcomes. #### MA4-7NA. operates with ratios and rates, and explores their graphical representation. #### MA4-15MG. performs calculations of time that involve mixed units, and interprets time zones.
https://greprepclub.com/forum/b-2a-14727.html	1,566,746,662,000,000,000	text/html	crawl-data/CC-MAIN-2019-35/segments/1566027330750.45/warc/CC-MAIN-20190825151521-20190825173521-00342.warc.gz	488,677,539	28,828	It is currently 25 Aug 2019, 07:24 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # b = 2a + 1 Author Message TAGS: Founder Joined: 18 Apr 2015 Posts: 7810 Followers: 145 Kudos [?]: 1608 [0], given: 7073 b = 2a + 1 [#permalink] 16 Aug 2019, 01:53 Expert's post 00:00 Question Stats: 100% (00:07) correct 0% (00:00) wrong based on 1 sessions $$b = 2a + 1$$ Quantity A Quantity B $$2b$$ $$4a+1$$ A)The quantity in Column A is greater. B)The quantity in Column B is greater. C)The two quantities are equal. D)The relationship cannot be determined from the information given. Kudos for the right answer and explanation [Reveal] Spoiler: OA _________________ Intern Joined: 15 Aug 2019 Posts: 31 Followers: 0 Kudos [?]: 20 [0], given: 3 Re: b = 2a + 1 [#permalink] 16 Aug 2019, 08:46 b= 2a +1 => 2(b) = 2(2a+1)= 41+2 So ans is C Re: b = 2a + 1 [#permalink] 16 Aug 2019, 08:46 Display posts from previous: Sort by	445	1,362	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2019-35	latest	en	0.813143	It is currently 25 Aug 2019, 07:24. ### GMAT Club Daily Prep. #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.. Customized. for You. we will pick new questions that match your level based on your Timer History. Track. every week, we’ll send you an estimated GMAT score based on your performance. Practice. Pays. we will pick new questions that match your level based on your Timer History. # b = 2a + 1. Author Message. TAGS:. Founder. Joined: 18 Apr 2015. Posts: 7810. Followers: 145. Kudos [?]: 1608 [0], given: 7073. b = 2a + 1 [#permalink] 16 Aug 2019, 01:53. Expert's post. 00:00.	Question Stats:. 100% (00:07) correct 0% (00:00) wrong based on 1 sessions. $$b = 2a + 1$$. Quantity A Quantity B $$2b$$ $$4a+1$$. A)The quantity in Column A is greater.. B)The quantity in Column B is greater.. C)The two quantities are equal.. D)The relationship cannot be determined from the information given.. Kudos for the right answer and explanation. [Reveal] Spoiler: OA. _________________. Intern. Joined: 15 Aug 2019. Posts: 31. Followers: 0. Kudos [?]: 20 [0], given: 3. Re: b = 2a + 1 [#permalink] 16 Aug 2019, 08:46. b= 2a +1. => 2(b) = 2(2a+1)= 41+2. So ans is C. Re: b = 2a + 1 [#permalink] 16 Aug 2019, 08:46. Display posts from previous: Sort by.
https://web2.0calc.com/questions/probability_28977	1,603,716,951,000,000,000	text/html	crawl-data/CC-MAIN-2020-45/segments/1603107891228.40/warc/CC-MAIN-20201026115814-20201026145814-00288.warc.gz	583,689,175	6,413	+0 # probability 0 84 1 +107 If two standard six-sided dice are tossed, what is the probability that a 5 is rolled on at least one of the two dice? Express your answer as a common fraction. Jul 24, 2020 #1 +27848 +2 There are 36 possible rolls 5 x = 5 rolls x 5 = 5 rolls 5 5 = 1 roll 11/36 Jul 24, 2020 edited by ElectricPavlov Jul 24, 2020 edited by ElectricPavlov Jul 24, 2020	151	413	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.640625	4	CC-MAIN-2020-45	latest	en	0.886752	+0. # probability. 0. 84. 1. +107. If two standard six-sided dice are tossed, what is the probability that a 5 is rolled on at least one of the two dice? Express your answer as a common fraction.. Jul 24, 2020. #1. +27848.	+2. There are 36 possible rolls. 5 x = 5 rolls. x 5 = 5 rolls. 5 5 = 1 roll. 11/36. Jul 24, 2020. edited by ElectricPavlov Jul 24, 2020. edited by ElectricPavlov Jul 24, 2020.
http://mathhelpforum.com/trigonometry/158404-ferrish-wheel-question.html	1,481,456,710,000,000,000	text/html	crawl-data/CC-MAIN-2016-50/segments/1480698544672.33/warc/CC-MAIN-20161202170904-00277-ip-10-31-129-80.ec2.internal.warc.gz	178,818,563	11,006	1. ## Ferrish Wheel Question The height above the ground of one of the seats of a Ferris wheel, in metres, can be modelled by the function $h(t) = 8 + 7(sin15(t))$ where $t$ is measured in seconds. a) What is the maximum and minimum height reached by any seat? Sub in t = 90 and t = 270 to obtain the max and the min, 15m and 1m. b) How long does it take for the seat on this ride to rotate back to its starting point? How exactly am I supposed to figure this out? I know I'm supposed to isolate for $t$, but I can only do that if I have a value for $h(t)$. Any help would be appreciated. 2. Originally Posted by RogueDemon The height above the ground of one of the seats of a Ferris wheel, in metres, can be modelled by the function $h(t) = 8 + 7(sin15(t))$ where $t$ is measured in seconds. a) What is the maximum and minimum height reached by any seat? Sub in t = 90 and t = 270 to obtain the max and the min, 15m and 1m. Mr F says: Correct. But easier to just say that the min and max value of sin is +1 and -1 ..... b) How long does it take for the seat on this ride to rotate back to its starting point? How exactly am I supposed to figure this out? I know I'm supposed to isolate for $t$, but I can only do that if I have a value for $h(t)$. Any help would be appreciated. b) The required time is the period .... 3. Would the period be 180 degrees? 4. Originally Posted by RogueDemon Would the period be 180 degrees? No, the period is a time and will be in seconds and is equal to the time it takes to rotate once 5. Originally Posted by RogueDemon Would the period be 180 degrees? No. For starters, a modelling problem like this one always assumes that the argumne tof the trig function is measured in radians not degrees. Also, your classnotes or textbook should have a formula for the period of things like sin(kt) ..... (The formula is period = 2pi/k).	494	1,879	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.84375	4	CC-MAIN-2016-50	longest	en	0.947788	1. ## Ferrish Wheel Question. The height above the ground of one of the seats of a Ferris wheel, in metres, can be modelled by the function $h(t) = 8 + 7(sin15(t))$ where $t$ is measured in seconds.. a) What is the maximum and minimum height reached by any seat?. Sub in t = 90 and t = 270 to obtain the max and the min, 15m and 1m.. b) How long does it take for the seat on this ride to rotate back to its starting point?. How exactly am I supposed to figure this out? I know I'm supposed to isolate for $t$, but I can only do that if I have a value for $h(t)$.. Any help would be appreciated.. 2. Originally Posted by RogueDemon. The height above the ground of one of the seats of a Ferris wheel, in metres, can be modelled by the function $h(t) = 8 + 7(sin15(t))$ where $t$ is measured in seconds.. a) What is the maximum and minimum height reached by any seat?. Sub in t = 90 and t = 270 to obtain the max and the min, 15m and 1m.. Mr F says: Correct. But easier to just say that the min and max value of sin is +1 and -1 ...... b) How long does it take for the seat on this ride to rotate back to its starting point?.	How exactly am I supposed to figure this out? I know I'm supposed to isolate for $t$, but I can only do that if I have a value for $h(t)$.. Any help would be appreciated.. b) The required time is the period ..... 3. Would the period be 180 degrees?. 4. Originally Posted by RogueDemon. Would the period be 180 degrees?. No, the period is a time and will be in seconds and is equal to the time it takes to rotate once. 5. Originally Posted by RogueDemon. Would the period be 180 degrees?. No. For starters, a modelling problem like this one always assumes that the argumne tof the trig function is measured in radians not degrees. Also, your classnotes or textbook should have a formula for the period of things like sin(kt) ..... (The formula is period = 2pi/k).
https://mulloverthings.com/what-kind-of-mathematics-is-used-in-artificial-intelligence/	1,642,565,778,000,000,000	text/html	crawl-data/CC-MAIN-2022-05/segments/1642320301263.50/warc/CC-MAIN-20220119033421-20220119063421-00406.warc.gz	444,035,285	6,273	MullOverThings Useful tips for everyday # What kind of mathematics is used in artificial intelligence? ## What kind of mathematics is used in artificial intelligence? The three main branches of mathematics that constitute a thriving career in AI are Linear algebra, calculus, and Probability. Linear Algebra is the field of applied mathematics which is something AI experts can’t live without. You will never become a good AI specialist without mastering this field. ## What is algorithm in artificial intelligence? In machine learning, an algorithm is a set of rules given to an AI program to help it learn on its own. In machine learning, an algorithm is a set of rules or instructions given to an AI program, neural network, or other machine to help it learn on its own. ## Do I need math for AI? To become skilled at Machine Learning and Artificial Intelligence, you need to know: Linear algebra (essential to understanding most ML/AI approaches) Basic differential calculus (with a bit of multi-variable calculus) Basic Statistics (ML/AI use a lot of concepts from statistics) ## How are mathematics and statistics used in AI? AI algorithms based on Mathematics and Statistics, in this article explain importance of Mathematics in AI. Maths behind AI Algorithms is tough to understand and need a steep learning curve. AI algorithms uses Mathematical subjects even though concepts taken from other disciplines (Example: Biological Neuron for Artificial Neural Networks). ## What makes an AI algorithm different from a human algorithm? This is called model-based learning, and it allows AI to make better decisions than humans because it can take many more factors into account and analyze them in milliseconds. An algorithm is like following a recipe. ## What kind of math do you need to learn artificial intelligence? Artificial Intelligence is a very broad field and it covers many and very deep areas of computer science, mathematics, hardware design, and even biology and psychology. What math do you need? ## How does machine learning work in artificial intelligence? Machine learning is a subfield of AI – machines use inputs and by doing mathematics logic, generate output. However, Artificial Intelligence Algorithms use both output and input to generate new data output after getting new inputs.	432	2,324	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.609375	4	CC-MAIN-2022-05	latest	en	0.939147	MullOverThings. Useful tips for everyday. # What kind of mathematics is used in artificial intelligence?. ## What kind of mathematics is used in artificial intelligence?. The three main branches of mathematics that constitute a thriving career in AI are Linear algebra, calculus, and Probability. Linear Algebra is the field of applied mathematics which is something AI experts can’t live without. You will never become a good AI specialist without mastering this field.. ## What is algorithm in artificial intelligence?. In machine learning, an algorithm is a set of rules given to an AI program to help it learn on its own. In machine learning, an algorithm is a set of rules or instructions given to an AI program, neural network, or other machine to help it learn on its own.. ## Do I need math for AI?. To become skilled at Machine Learning and Artificial Intelligence, you need to know: Linear algebra (essential to understanding most ML/AI approaches) Basic differential calculus (with a bit of multi-variable calculus) Basic Statistics (ML/AI use a lot of concepts from statistics). ## How are mathematics and statistics used in AI?.	AI algorithms based on Mathematics and Statistics, in this article explain importance of Mathematics in AI. Maths behind AI Algorithms is tough to understand and need a steep learning curve. AI algorithms uses Mathematical subjects even though concepts taken from other disciplines (Example: Biological Neuron for Artificial Neural Networks).. ## What makes an AI algorithm different from a human algorithm?. This is called model-based learning, and it allows AI to make better decisions than humans because it can take many more factors into account and analyze them in milliseconds. An algorithm is like following a recipe.. ## What kind of math do you need to learn artificial intelligence?. Artificial Intelligence is a very broad field and it covers many and very deep areas of computer science, mathematics, hardware design, and even biology and psychology. What math do you need?. ## How does machine learning work in artificial intelligence?. Machine learning is a subfield of AI – machines use inputs and by doing mathematics logic, generate output. However, Artificial Intelligence Algorithms use both output and input to generate new data output after getting new inputs.
https://se.mathworks.com/help/symbolic/atan2.html	1,713,781,805,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296818105.48/warc/CC-MAIN-20240422082202-20240422112202-00028.warc.gz	450,649,900	20,085	# atan2 ## Syntax ``P = atan2(Y,X)`` ## Description example ````P = atan2(Y,X)` computes the four-quadrant inverse tangent (arctangent) of `Y` and `X`.Symbolic arguments `X` and `Y` are assumed to be real, and `atan2(Y,X)` returns values in the interval `[-pi,pi]`.``` ## Examples collapse all Compute the arctangents of these parameters. Because these numbers are not symbolic objects, you get floating-point results. `P = [atan2(1,1), atan2(pi,4), atan2(Inf,Inf)]` ```P = 1×3 0.7854 0.6658 0.7854 ``` Compute the arctangents of these parameters which are converted to symbolic objects. `P = [atan2(sym(1),1), atan2(sym(pi),sym(4)), atan2(Inf,sym(Inf))]` ```P = $\left(\begin{array}{ccc}\frac{\pi }{4}& \mathrm{atan}\left(\frac{\pi }{4}\right)& \frac{\pi }{4}\end{array}\right)$``` Compute the limits of this symbolic expression. ```syms x P_minusinf = limit(atan2(x^2/(1 + x),x),x,-Inf)``` ```P_minusinf = $-\frac{3 \pi }{4}$``` `P_plusinf = limit(atan2(x^2/(1 + x),x),x,Inf)` ```P_plusinf = $\frac{\pi }{4}$``` Compute the arctangents of the elements of matrices `Y` and `X`. ```Y = sym([3 sqrt(3); 1 1]); X = sym([sqrt(3) 3; 1 0]); P = atan2(Y,X)``` ```P = $\left(\begin{array}{cc}\frac{\pi }{3}& \frac{\pi }{6}\\ \frac{\pi }{4}& \frac{\pi }{2}\end{array}\right)$``` ## Input Arguments collapse all y-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of `Y` must be real. Inputs `Y` and `X` must either be the same size or have sizes that are compatible (for example, `Y` is an `M`-by-`N` matrix and `X` is a scalar or `1`-by-`N` row vector). For more information, see Compatible Array Sizes for Basic Operations. x-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of `X` must be real. Inputs `Y` and `X` must either be the same size or have sizes that are compatible (for example, `Y` is an `M`-by-`N` matrix and `X` is a scalar or `1`-by-`N` row vector). For more information, see Compatible Array Sizes for Basic Operations. collapse all If X ≠ 0 and Y ≠ 0, then `$\text{atan2}\left(Y,X\right)=\text{atan}\left(\frac{Y}{X}\right)+\frac{\pi }{2}\text{sign}\left(Y\right)\left(1-\text{sign}\left(X\right)\right)$` Results returned by `atan2(Y,X)` belong to the closed interval `[-pi,pi]`. Meanwhile, results returned by `atan(Y/X)` belong to the closed interval `[-pi/2,pi/2]`. ## Tips • Calling `atan2` for numbers (or vectors or matrices of numbers) that are not symbolic objects invokes the MATLAB® `atan2` function. • If one of the arguments `X` and `Y` is a vector or a matrix, and another one is a scalar, then `atan2` expands the scalar into a vector or a matrix of the same length with all elements equal to that scalar. • If `X = 0` and `Y > 0`, then `atan2(Y,X)` returns `pi/2`. If `X = 0` and `Y < 0`, then `atan2(Y,X)` returns `-pi/2`. If `X = Y = 0`, then `atan2(Y,X)` returns `0`. ## Alternatives For complex `Z = X + Y*i`, the call `atan2(Y,X)` is equivalent to `angle(Z)`. ## Version History Introduced in R2013a	1,043	3,263	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 5, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.640625	4	CC-MAIN-2024-18	latest	en	0.634868	# atan2. ## Syntax. ``P = atan2(Y,X)``. ## Description. example. ````P = atan2(Y,X)` computes the four-quadrant inverse tangent (arctangent) of `Y` and `X`.Symbolic arguments `X` and `Y` are assumed to be real, and `atan2(Y,X)` returns values in the interval `[-pi,pi]`.```. ## Examples. collapse all. Compute the arctangents of these parameters. Because these numbers are not symbolic objects, you get floating-point results.. `P = [atan2(1,1), atan2(pi,4), atan2(Inf,Inf)]`. ```P = 1×3 0.7854 0.6658 0.7854 ```. Compute the arctangents of these parameters which are converted to symbolic objects.. `P = [atan2(sym(1),1), atan2(sym(pi),sym(4)), atan2(Inf,sym(Inf))]`. ```P = $\left(\begin{array}{ccc}\frac{\pi }{4}& \mathrm{atan}\left(\frac{\pi }{4}\right)& \frac{\pi }{4}\end{array}\right)$```. Compute the limits of this symbolic expression.. ```syms x P_minusinf = limit(atan2(x^2/(1 + x),x),x,-Inf)```. ```P_minusinf = $-\frac{3 \pi }{4}$```. `P_plusinf = limit(atan2(x^2/(1 + x),x),x,Inf)`. ```P_plusinf = $\frac{\pi }{4}$```. Compute the arctangents of the elements of matrices `Y` and `X`.. ```Y = sym([3 sqrt(3); 1 1]); X = sym([sqrt(3) 3; 1 0]); P = atan2(Y,X)```. ```P = $\left(\begin{array}{cc}\frac{\pi }{3}& \frac{\pi }{6}\\ \frac{\pi }{4}& \frac{\pi }{2}\end{array}\right)$```.	## Input Arguments. collapse all. y-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of `Y` must be real.. Inputs `Y` and `X` must either be the same size or have sizes that are compatible (for example, `Y` is an `M`-by-`N` matrix and `X` is a scalar or `1`-by-`N` row vector). For more information, see Compatible Array Sizes for Basic Operations.. x-coordinates, specified as a symbolic number, variable, expression, or function, or as a vector, matrix, or array of symbolic numbers, variables, expressions, or functions. All numerical elements of `X` must be real.. Inputs `Y` and `X` must either be the same size or have sizes that are compatible (for example, `Y` is an `M`-by-`N` matrix and `X` is a scalar or `1`-by-`N` row vector). For more information, see Compatible Array Sizes for Basic Operations.. collapse all. If X ≠ 0 and Y ≠ 0, then. `$\text{atan2}\left(Y,X\right)=\text{atan}\left(\frac{Y}{X}\right)+\frac{\pi }{2}\text{sign}\left(Y\right)\left(1-\text{sign}\left(X\right)\right)$`. Results returned by `atan2(Y,X)` belong to the closed interval `[-pi,pi]`. Meanwhile, results returned by `atan(Y/X)` belong to the closed interval `[-pi/2,pi/2]`.. ## Tips. • Calling `atan2` for numbers (or vectors or matrices of numbers) that are not symbolic objects invokes the MATLAB® `atan2` function.. • If one of the arguments `X` and `Y` is a vector or a matrix, and another one is a scalar, then `atan2` expands the scalar into a vector or a matrix of the same length with all elements equal to that scalar.. • If `X = 0` and `Y > 0`, then `atan2(Y,X)` returns `pi/2`.. If `X = 0` and `Y < 0`, then `atan2(Y,X)` returns `-pi/2`.. If `X = Y = 0`, then `atan2(Y,X)` returns `0`.. ## Alternatives. For complex `Z = X + Y*i`, the call `atan2(Y,X)` is equivalent to `angle(Z)`.. ## Version History. Introduced in R2013a.
http://mathhelpforum.com/trigonometry/66283-equation-unit-circle-non-center-point-print.html	1,527,079,065,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794865651.2/warc/CC-MAIN-20180523121803-20180523141803-00599.warc.gz	181,932,809	4,313	# Equation of unit circle, from a non-center point? • Dec 29th 2008, 10:49 AM mdunnbass Equation of unit circle, from a non-center point? Hi everyone, This is my first post here, so please feel free to ignore/delete/relocate this post if it is in the wrong place or has been covered before. I was idly wondering about this the other day, and thought I would seek out the insight of those who might know. If we take a Unit circle, with it's center located at the origin and radius r=1, the equation of the circle is written as cos^2(theta) + sin^2(theta) = 1 where theta is defined as angle POR, where P is a point on the circle, O is the origin, and R is, for the sake of simplicity, (x = 1, y = 0) Is there a simple way to write the equation of the same circle, with respect to a different angle, phi, defined as PQR, where Q is any random point along y=0, such as (x=-0.5, y=0)? I have been trying to figure it out in terms of generating triangle POQ, and invoking the laws of sines and cosines, but I inevitably going in circles (pun not intended) and wind up with something along the lines of x = x. Any insights would be greatly appreciated. Thanks, Matt • Dec 29th 2008, 11:15 AM Jhevon Quote: Originally Posted by mdunnbass Hi everyone, This is my first post here, so please feel free to ignore/delete/relocate this post if it is in the wrong place or has been covered before. I was idly wondering about this the other day, and thought I would seek out the insight of those who might know. If we take a Unit circle, with it's center located at the origin and radius r=1, the equation of the circle is written as cos^2(theta) + sin^2(theta) = 1 where theta is defined as angle POR, where P is a point on the circle, O is the origin, and R is, for the sake of simplicity, (x = 1, y = 0) Is there a simple way to write the equation of the same circle, with respect to a different angle, phi, defined as PQR, where Q is any random point along y=0, such as (x=-0.5, y=0)? I have been trying to figure it out in terms of generating triangle POQ, and invoking the laws of sines and cosines, but I inevitably going in circles (pun not intended) and wind up with something along the lines of x = x. Any insights would be greatly appreciated. Thanks, Matt when a circle is defined parametrically in this way, to change the center, you simply add the values you want to each component. that is, a circle with radius $\displaystyle r$ centered at $\displaystyle (h,k)$ can be parametrized by: $\displaystyle x = r \cos \theta + h$ and $\displaystyle y = r \sin \theta + k$ .........see if you can figure out why this works so the circle you want is given by: $\displaystyle x = \cos \theta - 0.5$, and $\displaystyle y = \sin \theta$ you can check that this yields the circle $\displaystyle (x + 0.5)^2 + y^2 = 1$ • Dec 29th 2008, 11:23 AM mdunnbass Quote: Originally Posted by Jhevon when a circle is defined parametrically in this way, to change the center, you simply add the values you want to each component. that is, a circle with radius $\displaystyle r$ centered at $\displaystyle (h,k)$ can be parametrized by: $\displaystyle x = r \cos \theta + h$ and $\displaystyle y = r \sin \theta + k$ .........see if you can figure out why this works so the circle you want is given by: $\displaystyle x = \cos \theta - 0.5$, and $\displaystyle y = \sin \theta$ you can check that this yields the circle $\displaystyle (x + 0.5)^2 + y^2 = 1$ Wait, I thought that was for a circle where the center of the circle was not at the origin, but rather at x=-0.5 I'm asking about a circle with its center at the origin, but the angle $\displaystyle \phi$ (not $\displaystyle \theta$) is NOT residing at the origin. Hence, the radius $\displaystyle r$ in this case is not constant, but rather, when $\displaystyle \phi = 0,$ r=1.5. and when $\displaystyle \phi = \pi,$ r=0.5 • Dec 29th 2008, 11:28 AM Jhevon Quote: Originally Posted by mdunnbass Wait, I thought that was for a circle where the center of the circle was not at the origin, but rather at x=-0.5 indeed, it is. the center here is (-0.5, 0). i thought that is what you were asking for. my apologies for misunderstanding. Quote: I'm asking about a circle with its center at the origin, but the angle $\displaystyle \phi$ (not $\displaystyle \theta$) is NOT residing at the origin. Hence, the radius $\displaystyle r$ in this case is not constant, but rather, when $\displaystyle \phi = 0,$ r=1.5. and when $\displaystyle \phi = \pi,$ r=0.5 ok, so it seems you want to define a circle in terms of an angle based at another point, say from (-0.5,0) that still describes the circle centered at the origin with radius 1? ...why? anyway, i will think about it and get back to you, if no one else has by the time i get back. i have to leave for a while. • Dec 29th 2008, 11:31 AM mdunnbass Quote: Originally Posted by Jhevon ok, so it seems you want to define a circle in terms of an angle based at another point, say from (-0.5,0) that still describes the circle centered at the origin with radius 1? Exactly. Quote: ...why? idle curiosity. (Evilgrin) • Mar 21st 2009, 04:45 PM Off-center circle equation I was looking for the solution to this problem to help me solve a homework problem, so I came here in hopes of finding it. After a few hours, I finally worked out a solution, but it didn't end up helping me with my homework problem. (Doh) I started by drawing a circle of radius $\displaystyle r$ and defining a new "origin" at a distance $\displaystyle m$ from the center on the x-axis. Then, I selected an arbitrary point P on the circle and drew a radius from the center and a quasi-radius, $\displaystyle rho$, from the new origin, as well as dropping a perpendicular to the x-axis. The radius made an angle with the x-axis I called $\displaystyle alpha$, and $\displaystyle rho$ made an angle I called $\displaystyle beta$. Then I found the following relations: $\displaystyle rsin(alpha) = rhosin(beta)$ $\displaystyle rcos(alpha) = m + rhocos(beta)$ Solving for rho in both cases and setting the results equal to each other, I found alpha in terms of beta, which I plugged back into the first equation. The solution is: $\displaystyle rho = r(sqrt(1-(m/rsin(beta))^2)-m/r*cos(beta))$	1,698	6,285	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.78125	4	CC-MAIN-2018-22	latest	en	0.974936	# Equation of unit circle, from a non-center point?. • Dec 29th 2008, 10:49 AM. mdunnbass. Equation of unit circle, from a non-center point?. Hi everyone,. This is my first post here, so please feel free to ignore/delete/relocate this post if it is in the wrong place or has been covered before.. I was idly wondering about this the other day, and thought I would seek out the insight of those who might know.. If we take a Unit circle, with it's center located at the origin and radius r=1, the equation of the circle is written as. cos^2(theta) + sin^2(theta) = 1. where theta is defined as angle POR, where P is a point on the circle, O is the origin, and R is, for the sake of simplicity, (x = 1, y = 0). Is there a simple way to write the equation of the same circle, with respect to a different angle, phi, defined as PQR, where Q is any random point along y=0, such as (x=-0.5, y=0)?. I have been trying to figure it out in terms of generating triangle POQ, and invoking the laws of sines and cosines, but I inevitably going in circles (pun not intended) and wind up with something along the lines of x = x.. Any insights would be greatly appreciated.. Thanks,. Matt. • Dec 29th 2008, 11:15 AM. Jhevon. Quote:. Originally Posted by mdunnbass. Hi everyone,. This is my first post here, so please feel free to ignore/delete/relocate this post if it is in the wrong place or has been covered before.. I was idly wondering about this the other day, and thought I would seek out the insight of those who might know.. If we take a Unit circle, with it's center located at the origin and radius r=1, the equation of the circle is written as. cos^2(theta) + sin^2(theta) = 1. where theta is defined as angle POR, where P is a point on the circle, O is the origin, and R is, for the sake of simplicity, (x = 1, y = 0). Is there a simple way to write the equation of the same circle, with respect to a different angle, phi, defined as PQR, where Q is any random point along y=0, such as (x=-0.5, y=0)?. I have been trying to figure it out in terms of generating triangle POQ, and invoking the laws of sines and cosines, but I inevitably going in circles (pun not intended) and wind up with something along the lines of x = x.. Any insights would be greatly appreciated.. Thanks,. Matt. when a circle is defined parametrically in this way, to change the center, you simply add the values you want to each component.. that is, a circle with radius $\displaystyle r$ centered at $\displaystyle (h,k)$ can be parametrized by:. $\displaystyle x = r \cos \theta + h$ and $\displaystyle y = r \sin \theta + k$ .........see if you can figure out why this works. so the circle you want is given by: $\displaystyle x = \cos \theta - 0.5$, and $\displaystyle y = \sin \theta$. you can check that this yields the circle $\displaystyle (x + 0.5)^2 + y^2 = 1$. • Dec 29th 2008, 11:23 AM. mdunnbass. Quote:. Originally Posted by Jhevon. when a circle is defined parametrically in this way, to change the center, you simply add the values you want to each component.. that is, a circle with radius $\displaystyle r$ centered at $\displaystyle (h,k)$ can be parametrized by:. $\displaystyle x = r \cos \theta + h$ and $\displaystyle y = r \sin \theta + k$ .........see if you can figure out why this works. so the circle you want is given by: $\displaystyle x = \cos \theta - 0.5$, and $\displaystyle y = \sin \theta$. you can check that this yields the circle $\displaystyle (x + 0.5)^2 + y^2 = 1$.	Wait, I thought that was for a circle where the center of the circle was not at the origin, but rather at x=-0.5. I'm asking about a circle with its center at the origin, but the angle $\displaystyle \phi$ (not $\displaystyle \theta$) is NOT residing at the origin. Hence, the radius $\displaystyle r$ in this case is not constant, but rather, when $\displaystyle \phi = 0,$ r=1.5. and when $\displaystyle \phi = \pi,$ r=0.5. • Dec 29th 2008, 11:28 AM. Jhevon. Quote:. Originally Posted by mdunnbass. Wait, I thought that was for a circle where the center of the circle was not at the origin, but rather at x=-0.5. indeed, it is. the center here is (-0.5, 0). i thought that is what you were asking for. my apologies for misunderstanding.. Quote:. I'm asking about a circle with its center at the origin, but the angle $\displaystyle \phi$ (not $\displaystyle \theta$) is NOT residing at the origin. Hence, the radius $\displaystyle r$ in this case is not constant, but rather, when $\displaystyle \phi = 0,$ r=1.5. and when $\displaystyle \phi = \pi,$ r=0.5. ok, so it seems you want to define a circle in terms of an angle based at another point, say from (-0.5,0) that still describes the circle centered at the origin with radius 1? ...why? anyway, i will think about it and get back to you, if no one else has by the time i get back. i have to leave for a while.. • Dec 29th 2008, 11:31 AM. mdunnbass. Quote:. Originally Posted by Jhevon. ok, so it seems you want to define a circle in terms of an angle based at another point, say from (-0.5,0) that still describes the circle centered at the origin with radius 1?. Exactly.. Quote:. ...why?. idle curiosity.. (Evilgrin). • Mar 21st 2009, 04:45 PM. Off-center circle equation. I was looking for the solution to this problem to help me solve a homework problem, so I came here in hopes of finding it. After a few hours, I finally worked out a solution, but it didn't end up helping me with my homework problem. (Doh). I started by drawing a circle of radius $\displaystyle r$ and defining a new "origin" at a distance $\displaystyle m$ from the center on the x-axis. Then, I selected an arbitrary point P on the circle and drew a radius from the center and a quasi-radius, $\displaystyle rho$, from the new origin, as well as dropping a perpendicular to the x-axis. The radius made an angle with the x-axis I called $\displaystyle alpha$, and $\displaystyle rho$ made an angle I called $\displaystyle beta$.. Then I found the following relations:. $\displaystyle rsin(alpha) = rhosin(beta)$. $\displaystyle rcos(alpha) = m + rhocos(beta)$. Solving for rho in both cases and setting the results equal to each other, I found alpha in terms of beta, which I plugged back into the first equation. The solution is:. $\displaystyle rho = r(sqrt(1-(m/rsin(beta))^2)-m/r*cos(beta))$.
http://solution-dailybrainteaser.blogspot.com/2013/08/logical-playing-cards-puzzle.html	1,503,571,658,000,000,000	text/html	crawl-data/CC-MAIN-2017-34/segments/1502886133449.19/warc/CC-MAIN-20170824101532-20170824121532-00300.warc.gz	394,518,440	13,473	## Search Puzzles ### Logical Playing Cards Puzzle Logical Playing Cards Puzzle Solution - 28 August From a pack of 52 cards , i placed 4 cards on the table. I will give you 4 clues about the cards: Clue 1: Card on left cannot be greater than card on the right. Clue 2: Difference between 1st card and 3rd card is 8. Clue 3: There is no card of ace. Clue 4: There is no face cards (queen,king,jacks). Clue 5: Difference between 2nd card and 4th card is 7. Identify four cards ? Solution 2 3 10 10 The 1st card has to be 2 and last card has be 10 as there is no other way difference can be 8. => 2 ? 10 ? because of clue 4, we know 4th card is 10 => 2 ? 10 10 because of clue 5, we know 1st card is 3 => 2 3 10 10	238	722	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2017-34	longest	en	0.960263	## Search Puzzles. ### Logical Playing Cards Puzzle. Logical Playing Cards Puzzle Solution - 28 August. From a pack of 52 cards , i placed 4 cards on the table.. I will give you 4 clues about the cards:. Clue 1: Card on left cannot be greater than card on the right.. Clue 2: Difference between 1st card and 3rd card is 8.. Clue 3: There is no card of ace.. Clue 4: There is no face cards (queen,king,jacks).. Clue 5: Difference between 2nd card and 4th card is 7.	Identify four cards ?. Solution. 2 3 10 10. The 1st card has to be 2 and last card has be 10 as there is no other way difference can be 8.. => 2 ? 10 ?. because of clue 4, we know 4th card is 10. => 2 ? 10 10. because of clue 5, we know 1st card is 3. => 2 3 10 10.
http://www.maths.lth.se/matstat/wafo/documentation/wafodoc/wafo/cycles/cc2amp.html	1,558,295,587,000,000,000	text/html	crawl-data/CC-MAIN-2019-22/segments/1558232255092.55/warc/CC-MAIN-20190519181530-20190519203530-00501.warc.gz	302,537,366	2,749	Home > wafo > cycles > cc2amp.m # cc2amp ## PURPOSE Calculates the amplitudes from a cycle count. [amp]=cc2amp(cc) ## DESCRIPTION ``` CC2AMP Calculates the amplitudes from a cycle count. CALL: amp = cc2amp(cc); amp = a vector with amplitudes. cc = a two column matrix with cycles. The amplitude of a cycle is defined as (Max-min)/2 Example: [mM,Mm] = tp2mm(TP); amp = cc2amp(mM); whisto(amp); ## CROSS-REFERENCE INFORMATION This function calls: error Display message and abort function. This function is called by: Chapter4 % CHAPTER4 contains the commands used in Chapter 4 of the tutorial itmkurs_lab1 Script to computer exercises 1 test_cycles Quick test of the routines in module 'cycles' ## SOURCE CODE ```001 function [amp]=cc2amp(cc) 002 %CC2AMP Calculates the amplitudes from a cycle count. 003 % 004 % CALL: amp = cc2amp(cc); 005 % 006 % amp = a vector with amplitudes. 007 % cc = a two column matrix with cycles. 008 % 009 % The amplitude of a cycle is defined as (Max-min)/2 010 % 011 % Example: 012 % x=load('sea.dat'); TP = dat2tp(x); 013 % [mM,Mm] = tp2mm(TP); 014 % amp = cc2amp(mM); 015 % whisto(amp); 016 % 018 019 % Tested on Matlab 6.0 020 % 021 % History: 022 % Revised by jr 01-Apr-2001 023 % - histo-> whisto in example 024 % - minor changes help text 025 % Corrected by PJ 09-Dec-1999 026 % Now calculates amplitudes (before ranges) 027 % Created by PJ (Pär Johannesson) 01-Nov-1999 028 029 % Check input arguments 030 031 ni = nargin; 032 no = nargout; 033 error(nargchk(1,1,ni)); 034 035 % Calculate amplitudes 036 037 amp = (cc(:,2)-cc(:,1))/2; 038 %amp = abs(cc(:,2)-cc(:,1)); 039``` Mathematical Statistics Centre for Mathematical Sciences Lund University with Lund Institute of Technology Comments or corrections to the WAFO group Generated on Thu 06-Oct-2005 02:21:16 for WAFO by m2html © 2003	559	1,866	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.515625	4	CC-MAIN-2019-22	latest	en	0.578992	Home > wafo > cycles > cc2amp.m. # cc2amp. ## PURPOSE. Calculates the amplitudes from a cycle count.. [amp]=cc2amp(cc). ## DESCRIPTION. ``` CC2AMP Calculates the amplitudes from a cycle count.. CALL: amp = cc2amp(cc);. amp = a vector with amplitudes.. cc = a two column matrix with cycles.. The amplitude of a cycle is defined as (Max-min)/2. Example:. [mM,Mm] = tp2mm(TP);. amp = cc2amp(mM);. whisto(amp);. ## CROSS-REFERENCE INFORMATION. This function calls:. error Display message and abort function.. This function is called by:. Chapter4 % CHAPTER4 contains the commands used in Chapter 4 of the tutorial itmkurs_lab1 Script to computer exercises 1 test_cycles Quick test of the routines in module 'cycles'. ## SOURCE CODE. ```001 function [amp]=cc2amp(cc). 002 %CC2AMP Calculates the amplitudes from a cycle count.. 003 %. 004 % CALL: amp = cc2amp(cc);. 005 %. 006 % amp = a vector with amplitudes.. 007 % cc = a two column matrix with cycles.. 008 %. 009 % The amplitude of a cycle is defined as (Max-min)/2. 010 %. 011 % Example:. 012 % x=load('sea.dat'); TP = dat2tp(x);.	013 % [mM,Mm] = tp2mm(TP);. 014 % amp = cc2amp(mM);. 015 % whisto(amp);. 016 %. 018. 019 % Tested on Matlab 6.0. 020 %. 021 % History:. 022 % Revised by jr 01-Apr-2001. 023 % - histo-> whisto in example. 024 % - minor changes help text. 025 % Corrected by PJ 09-Dec-1999. 026 % Now calculates amplitudes (before ranges). 027 % Created by PJ (Pär Johannesson) 01-Nov-1999. 028. 029 % Check input arguments. 030. 031 ni = nargin;. 032 no = nargout;. 033 error(nargchk(1,1,ni));. 034. 035 % Calculate amplitudes. 036. 037 amp = (cc(:,2)-cc(:,1))/2;. 038 %amp = abs(cc(:,2)-cc(:,1));. 039```. Mathematical Statistics. Centre for Mathematical Sciences. Lund University with Lund Institute of Technology. Comments or corrections to the WAFO group. Generated on Thu 06-Oct-2005 02:21:16 for WAFO by m2html © 2003.
http://mathhelpforum.com/calculus/275792-find-curl-vector-field.html	1,511,241,600,000,000,000	text/html	crawl-data/CC-MAIN-2017-47/segments/1510934806316.80/warc/CC-MAIN-20171121040104-20171121060104-00351.warc.gz	200,519,018	10,596	# Thread: Find Curl of Vector Field 1. ## Find Curl of Vector Field Find the curl of vector field. F(x, y, z) = xy i + yz j + xy k ∇xF = x^2 • z i + y^2•x j + z^2 k ∇xF = -y i - z j - x k Who is right and why? 2. ## Re: Find Curl of Vector Field Originally Posted by USNAVY Find the curl of vector field. F(x, y, z) = xy i + yz j + xy k ∇xF = x^2 • z i + y^2•x j + z^2 k ∇xF = -y i - z j - x k Who is right and why? Show some effort on your part. 3. ## Re: Find Curl of Vector Field I actually have a suspicion that there is a typo in the question. I think the last term of the vector field should be xz k instead of xy k. When it is xz k, then my answer agrees with the textbook. 4. ## Re: Find Curl of Vector Field Originally Posted by USNAVY I actually have a suspicion that there is a typo in the question. I think the last term of the vector field should be xz k instead of xy k. When it is xz k, then my answer agrees with the textbook. Well find $\nabla F$ and show it. Then explain how it differs from the text. 5. ## Re: Find Curl of Vector Field The book's answer is correct. I can't imagine how you got your answer! Your answer has polynomials of higher power than the original vector while differentiating always reduces the power of a polynomial. $\nabla\times F= \left\|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} & \frac{\partial F}{\partial z} \\ xy & yz & xz \end{array}\right\|= (0- x)\vec{i}-(z- 0)\vec{j}+ (0- x)\vec{k}= -x\vec{i}- \vec{j}- \vec{k}$.	497	1,556	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.875	4	CC-MAIN-2017-47	longest	en	0.825003	# Thread: Find Curl of Vector Field. 1. ## Find Curl of Vector Field. Find the curl of vector field.. F(x, y, z) = xy i + yz j + xy k. ∇xF = x^2 • z i + y^2•x j + z^2 k. ∇xF = -y i - z j - x k. Who is right and why?. 2. ## Re: Find Curl of Vector Field. Originally Posted by USNAVY. Find the curl of vector field.. F(x, y, z) = xy i + yz j + xy k. ∇xF = x^2 • z i + y^2•x j + z^2 k. ∇xF = -y i - z j - x k. Who is right and why?. Show some effort on your part.. 3.	## Re: Find Curl of Vector Field. I actually have a suspicion that there is a typo in the question. I think the last term of the vector field should be xz k instead of xy k. When it is xz k, then my answer agrees with the textbook.. 4. ## Re: Find Curl of Vector Field. Originally Posted by USNAVY. I actually have a suspicion that there is a typo in the question. I think the last term of the vector field should be xz k instead of xy k. When it is xz k, then my answer agrees with the textbook.. Well find $\nabla F$ and show it. Then explain how it differs from the text.. 5. ## Re: Find Curl of Vector Field. The book's answer is correct. I can't imagine how you got your answer! Your answer has polynomials of higher power than the original vector while differentiating always reduces the power of a polynomial.. $\nabla\times F= \left\|\begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} & \frac{\partial F}{\partial z} \\ xy & yz & xz \end{array}\right\|= (0- x)\vec{i}-(z- 0)\vec{j}+ (0- x)\vec{k}= -x\vec{i}- \vec{j}- \vec{k}$.
http://www.qtpschool.com/2011/03/woking-with-arrays.html	1,534,615,343,000,000,000	text/html	crawl-data/CC-MAIN-2018-34/segments/1534221213693.23/warc/CC-MAIN-20180818173743-20180818193743-00568.warc.gz	592,403,382	20,189	## Wednesday, March 2 ### Find the highest number in the array 'This code demonstrate how to find the highest number in a numaric array. Dim num, i, Length1 num=array(34,12,98,43,89,49,56) Length1 = UBound(num) 'Find the length of array For i= 1 to Length1 If (num(i) < num(0)) Then 'to find lowest number, just change it to > num(0)=num(i) End If Next MsgBox num(0) 'Highest Number 1. thanks Abhikansh.. keep it up!! :) 2. outstanding work....keep it up.... 3. Super!! 4. i tried to pass numbers(5,10,15) in array to find the biggest number and getting as 5. but it should be 15. pls help me in this. num=array("5","10","15") Length1 = UBound(num) For i= 1 to Length1 If (num(i)>num(0)) Then num(0)=num(i) End If Next MsgBox num(0) 5. @raghav, plz dont use double qoutes while paasing values in array.. 6. superb! whatever search here i can get solution here thanks dude 7. plz help me... how to capture screen shot during runtime... what is syntax and explain plz 8. @satish k plz search CaptureBitmap in qtp help 9. Good Job :) 10. Cool! 11. find the max value from array Dim x,y,max x=array(50,20,30,40,90) z=UBound(x) max=0 For i = 0 To z If x(i)>max Then max=x(i) End If Next MsgBox max 12. Yo can try this one also: Dim a, temp, i, j, firsthighest, secondhighest, thirdhighest a = Array(35,44,99,66,98,76) 'Sorting the array in Ascending order for i=0 to UBound(a) for j=0 to UBound(a) if (strcomp(a(i), a(j), 1)<0) then temp = a(i) a(i) = a(j) a(j) = temp end if Next Next firsthighest = (UBound(a)) msgbox "First highest mark is: "&a(Firsthighest) secondhighest = (UBound(a)-1) msgbox "Second highest mark is: "&a(secondhighest) 13. I dint get these lines..... num(i) is actually an number or index? why for loop started from 1 rather 0 For i= 1 to Length1 If (num(i)>num(0)) Then num(0)=num(i) End If Next 14. The above code should be start little bit we need to correct For i = 0 to Length 15. How can i sort larger number to lowest? but user input the numbers in arry	659	2,011	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.59375	4	CC-MAIN-2018-34	longest	en	0.760021	## Wednesday, March 2. ### Find the highest number in the array. 'This code demonstrate how to find the highest number in a numaric array.. Dim num, i, Length1. num=array(34,12,98,43,89,49,56). Length1 = UBound(num) 'Find the length of array. For i= 1 to Length1. If (num(i) < num(0)) Then 'to find lowest number, just change it to >. num(0)=num(i). End If. Next. MsgBox num(0) 'Highest Number. 1. thanks Abhikansh... keep it up!! :). 2. outstanding work....keep it up..... 3. Super!!. 4. i tried to pass numbers(5,10,15) in array to find the biggest number and getting as 5. but it should be 15. pls help me in this.. num=array("5","10","15"). Length1 = UBound(num). For i= 1 to Length1. If (num(i)>num(0)) Then. num(0)=num(i). End If. Next. MsgBox num(0). 5. @raghav,. plz dont use double qoutes while paasing values in array... 6. superb! whatever search here i can get solution here thanks dude. 7. plz help me... how to capture screen shot during runtime... what is syntax and explain plz. 8. @satish k. plz search CaptureBitmap in qtp help. 9. Good Job :). 10. Cool!.	11. find the max value from array. Dim x,y,max. x=array(50,20,30,40,90). z=UBound(x). max=0. For i = 0 To z. If x(i)>max Then. max=x(i). End If. Next. MsgBox max. 12. Yo can try this one also:. Dim a, temp, i, j, firsthighest, secondhighest, thirdhighest. a = Array(35,44,99,66,98,76). 'Sorting the array in Ascending order. for i=0 to UBound(a). for j=0 to UBound(a). if (strcomp(a(i), a(j), 1)<0) then. temp = a(i). a(i) = a(j). a(j) = temp. end if. Next. Next. firsthighest = (UBound(a)). msgbox "First highest mark is: "&a(Firsthighest). secondhighest = (UBound(a)-1). msgbox "Second highest mark is: "&a(secondhighest). 13. I dint get these lines...... num(i) is actually an number or index? why for loop started from 1 rather 0. For i= 1 to Length1. If (num(i)>num(0)) Then. num(0)=num(i). End If. Next. 14. The above code should be start little bit we need to correct. For i = 0 to Length. 15. How can i sort larger number to lowest? but user input the numbers in arry.
huygens-fokker.org	1,722,758,202,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640393185.10/warc/CC-MAIN-20240804071743-20240804101743-00437.warc.gz	244,323,934	10,833	# Logarithmic Interval Measures The first persons to use logarithms for calculation of interval sizes were Bonaventura Cavalieri (1639), Juan Caramel de Lobkowitz (1647), and Lemme Rossi (1666). Also Christiaan Huygens was among the first to do this. Earlier calculations of equal intervals sizes, like those of Simon Stevin (1585) and Marin Mersenne (1636) were done by square and cubic roots. The logarithm, invented by John Napier in 1614, is a mathematical operation that turns a multiplication into an addition, and by the same definition, raising to a higher power into multiplication. Because stacking two intervals involves multiplication of frequency ratios, this is equivalent to addition of the logarithmic measures of these ratios. • cent: 1/1200 part of an octave Defined by Alexander John Ellis (1814-1890) (see photo) in 1884 and presented in the article "The Musical Scales of Various Nations" as well as in the appendix of his English translation of Hermann von Helmholtz' book Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. One cent is one hundredth part of the semitone in 12-tone Equal Temperament, a centisemitone. The frequency ratio represented by one cent is the 1200th root of 2. So the 12-tET whole tone is 200 cents, the minor third 300 cents, the major third 400 cents, etc. Rounding to the nearest cent is sufficiently accurate for practical purposes. Cents are the most universally used interval measure. They have the advantage that familiar intervals have an easily rememberable value. Ellis, whose real name was Sharpe, is also regarded as the founder of the field of ethnomusicology. He invented cents originally for the purpose of expressing non-Western scales. In a different form, this measure was invented earlier by Heinrich Bellermann (1832-1903), namely the logarithm with base twelfth root of 2, which has a size a hundred times larger than a cent, the same size as an equal tempered semitone. Contrary to cents, it was meant to be used for Western classical music. This measure was also endorsed by Paul von Jankó. Bellermann received little acknowledgement for it, because people adopted Ellis's cents much more enthusiastically. It goes back to Isaac Newton, who also expressed intervals in terms of 1/12 of an octave in 1665. • centitone: see Iring • comma Not a fixed measure as such but often used as an interval unit. The syntonic (Didymic) and Pythagorean (ditonic) commas have almost the same size. In the past they were often confused and their difference was often neglected. The word coma is Latin for hair. There are 55.79763 syntonic commas and 51.15087 Pythagorean commas in the octave. Therefore one step in 53-tET is often named a comma because it's in the middle of them and 53-tET is a very accurate approximation to 5-limit just intonation scales. The major whole tone 9/8 is 9 commas and the minor whole tone 10/9 is 8 commas. The chromatic and diatonic semitones are 4 and 5 commas. The 1/53 octave comma was discovered by Jean Galle as reported by Marin Mersenne in 1637. Nicolaus Mercator gave a more mathematically precise description of it in 1660. This comma has also been wrongly attributed to William Holder. In regular meantone temperaments where the tempering of the fifth is expressed in a fraction of a comma, the comma is the syntonic comma. In well-temperaments, like Werckmeister's, where the cycle of fifths closes in a circle, the temperings are usually expressed in fractions of a Pythagorean comma. The measure 1/55 of an octave is called a Sauveur comma and 1/50 of an octave could be called a Henfling comma. • decaméride: 1/3010 part of an octave Defined by Joseph Sauveur in 1696 as one tenth of an eptaméride. • Delfi unit: 1/665 part of an octave Used in Byzantine music theory? Approximately 1/12 part of the syntonic comma and 1/13 part of the Pythagorean comma. • demi-heptaméride: 1/602 part of an octave Defined by Joseph Sauveur in 1696 as one half of an eptaméride. • diesis (plural: dieses or dieseis) Like comma, also an interval. The name was used by ancient Greek theorists like Aristoxenos for several different intervals. Marchettus de Padua (Marchetto Padoano), in his Lucidarium written in 1317/1318, was the first to use it as a standard measure. He divided the whole tone in 5 parts, called a diesis. The chromatic semitone (semitonium enharmonicum) was 2 dieses, the diatonic semitone (semitonium diatonicum) 3 dieses and another chromatic semitone of 4 dieses was meant for some augmentations like C-C giving a very high leading tone. See also this essay. Later the minor or enharmonic diesis became the difference between an octave and three pure major thirds. There are 29.22634 of it in an octave. Adriaan Fokker used this name for the step of 31-tone equal temperament, where it is also equal to 1/5 part of a whole tone, because of its similar size. This 1/31 part of an octave called normal diesis by Fokker is a convenient measure in which to express 7-limit just intervals. • Dröbisch Angle: 1/360 part of an octave The Angle was proposed by Moritz Dröbisch in the 19th century as a cycle of 360 degrees to the octave. Andrew Pikler has suggested this name in his article "Logarithmic Frequency Systems" (1966). • Ellis or El: absolute cent Not an interval measure, but a logarithmic absolute pitch measure. Proposed by Robert Stuckey and described in the paper Ellis at Belfast by Jeremy Montagu. It's the number of cents relative to the 64-foot C (16.3516 Hz), with the number of octaves written as a subscript in front of the number of cents. For example the A at 440 Hz is shown as 4900. The abbreviation is El, as Hertz are abbreviated to Hz. • eptaméride or heptaméride: 1/301 part of an octave Both spellings used by Sauveur. See méride and savart. Sauveur's rule to find the number of eptamérides of intervals smaller than 7/6 is as follows: multiply the difference of numerator and denominator with 875 and divide by the sum of numerator and denominator and round the result to the nearest integer. This is known as the bimodular method of approximating logarithms and can be used for other measures as well. • farab: 1/144 part of an octave Measure proposed by al-Farabi in the 10th century, from Mehdi Barkeshli, "Perfect scale of Farabi and his proposed scales", Iranian music, a collection of articles. A decifarab is 1/10th of a farab unit, with the advantage that 1440 is a number with many divisors. • flu: 1/46032 part of an octave Measure proposed by Gene Ward Smith for use with (nano-)temperaments. The Pythagorean comma is nearly 900 flus (899.9259), the syntonic comma 825 (824.9812) and the schisma 75 flus (74.94472), alas not divisible by 2. It's an alternative to the Temperament Unit which, although used with 5-limit temperaments, is not 5-limit consistent, whereas flus measure intervals consistently up to the 9-limit. • Grad (degree of tempering): 1/12 part of a Pythagorean comma, 1/613.810469983 part of an octave One Grad is the difference between a pure fifth of 1.95500086539 cents and the 12-tET fifth of 700.0 cents and makes it a useful measure to describe temperings in a well-temperament. The name comes from Andreas Werckmeister (1645-1706) who used it to denote different divisions of a comma. It has a negligible difference to 1/11 part of a syntonic comma. Jan van Biezen calls this measure a Werckmeister, symbol Wm. Organ builders sometimes use this symbol, and approximate its size by 2 cents. Another useful property is that the minor diesis is almost exactly 21 Grad (21.002). So the tempering of the major third in 12-tET is very close to 7 units. And likewise that of the major sixth in 12-tET very close to 8 units. Deviations from equal temperament of three consecutive major thirds when expressed in Grad always add up to 21. This measure was also used by Neidhardt and Sorge and later by Mark Lindley in their temperament diagrammes. A Grad is slightly larger than a schisma, 1.000655 times it. The 1/21 part of a minor diesis is 1.000094 Grads. • Harmos: 1/1728 part of an octave A twelve-based measure suggested by Paul Beaver: 1728 is 123. In the 1960s he asked John Chalmers to compute a table of Harmos, which he did later in decimal and duodecimal notation. • Hekt: 1/1300 part of a pure twelfth Defined by Heinz Bohlen as the hundredth part of a step of the equal tempered version of the Bohlen-Pierce scale: the 13th root of 3/1. Hekt are therefore the BP analogon to cents. See H. Bohlen, "13-Tonstufen in der Duodezime", Acustica vol. 39, 1978. There are 820.2086796 Hekt to the octave. • imp: 1/31920 part of an octave Able to accurately express intervals up to 41-limit. Also 1/1680th part of a 19-tET step. • iota: 1/1700 part of an octave Proposed by Margo Schulter on the Tuning List in 2002 and indicated by the Greek letter iota. The classic chromatic semitone 25/24 for example is 100.12 iotas. It's useful for comparing just intervals with 17-tone equal tempered ones. • Iring: 1/600 part of an octave The Iring unit was defined by Widogast Iring in his 1898 Die reine Stimmung in der Musik. He noted that the twelfth part of the Pythagorean comma and the schisma have almost the same size, both about 1/614 part of the octave. To get round numbers, he took for this size one 600th part of an octave. The size of the major second can then be rounded to 102 and the just major third to 193. The perfect fifth is 351 Iring units. The size of the Iring unit is twice the size of the cent. It is also about the smallest difference in pitch that untrained ears can hear. The same unit was later defined in 1932 by Joseph Yasser in his book A Theory of Evolving Tonality by dividing the equally tempered whole tone in 100 parts and calling it the centitone. • jinn: 1/16808 part of an octave This measure was proposed by Gene Ward Smith because it is consistent up until prime number 37 which means rounding to the nearest integer jinn will (usually) give the correct answers to this limit. • jot: 1/30103 part of an octave This name was given by Augustus De Morgan (1806-1871). The 10-base (Briggs) logarithm of 2 (10log 2) is 0.30102999566 so multiplied by 10000 this makes almost exactly 30103. Expressing intervals in this measure has the advantage of being able to calculate interval combinations without using logarithms, because rounding to the nearest integer jot will (usually) give the correct answers, at least for the prime numbers up to 11. And the jot values can be looked up in a 10-base logarithm table. A similar measure is the savart. • mem: 1/205 part of an octave This name was chosen by Andrew Aaron Hunt. One mem is equal to 12 minas and very close to 3 schismas and 4 Hekt. It has good approximations for the minor third (54), major third (66) and perfect fifth (120). • méride: 1/43 part of an octave This name was chosen by Joseph Sauveur (1653-1716) in 1696. The méride and eptaméride were the first logarithmic interval measures proposed. Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. He had set the comma to one step, then found a range of 2, 3 or 4 steps for the chromatic semitone, corresponding to 31, 43 and 55 tones per octave. He found 43 to be optimal because 4 steps is almost exactly a 16/15 minor second and 7 steps almost exactly the geometric mean of three 9/8 and two 10/9 whole tones. The chromatic scale contained in 43-tET is virtually identical to 1/5-comma meantone tuning. • MIDI Tuning Standard unit: 1/196608 part of an octave This divides the 12-tET semitone into 214 = 16384 parts which resolution makes sufficiently accurate tuning of electronic instruments possible. It's in the MIDI Tuning Specification 1.0. There are other MIDI tuning units which differ per manufacturer, for example Yamaha has models tuned in 1/768 or 1/1024 parts of an octave. There's also the MIDI Pitch Bend message, which can carry the values -8192 .. 8191, so when the range (which is variable) is the standard range of +/- 200 cents, then the unit is 1/49152 part of an octave or 0.024414 cents. • millioctave: 1/1000 part of an octave Named and used by Arthur Joachim von Oettingen (1836-1920) in his book Das duale Harmoniesystem (1913). Alfred Jonqičre indicated the millioctave with the Greek letter mu. It was first used however by John Herschel in the book which he wrote with George Bidell Airy On Sound and Atmospheric Vibrations with the Mathematical Elements of Music (1871). Sometimes millioctaves are propagated as a "value-free" substitute for cents, not having the 12-tET bias, because the round cent numbers may lead people to the false belief that the intervals are perfectly in tune. However using these millioctaves introduces a 10-tET bias, which is a much less familiar tuning. Often the cent values of just intervals are easy to remember by their deviation from the 12-tET multiple of 100, for example the pure fifth is 702 cents, with millioctaves this is harder: 585 millioctaves compared to 583.333. Another advantage of cents is the size of the schisma: almost 2 cents against 1.63 M.O. • mil or prima: 1/12276 part of an octave Suggested by Erv Wilson, Gene Ward Smith and Gavin Putland. It's 1/1023 part of 100 cents so close to 1/1000 semitone hence the name mil. It's very close to three times the size of the Temperament Unit and therefore has the same kind of advantages. The Pythagorean comma is 240 prima and the syntonic comma 220. • mina: 1/2460 part of an octave Named by Dave Keenan and George Secor as an abbreviation of "schismina". It's 0.487805 cents or 1/205 part of a 100 cent semitone, and selected because 2460-tone equal temperament is consistent up to the 28th harmonic and its step is therefore a useful measure to express high-limit just ratios in, and getting very little roundoff errors. They use it in the development of their Sagittal notation system. Although for that purpose the exact size of one mina is 1/233 part of a Pythagorean apotome, or 0.487918 cents or 1/2459.427234 octave. • morion (plural: moria): 1/72 part of an octave Defined as 1/30 part of a fourth by the theorist Cleonides around 100 AD for description of Greek tetrachords. Likewise Aristoxenos used a cipher of 12 parts to a whole tone. This measure is surrounded by controversy, because it's very unclear what Aristoxenos' measurements exactly are. Moria is Greek for molecules or small pieces. 72-tone equal temperament is a good approximation to many just intonation scales because the prime numbers 2, 3, 5, 7 and 11 are very well represented with deviations not exceeding 3 cents. • purdal: 1/9900 part of an octave This measure was suggested by Tútim Dennsuul. It is a highly composite number. Therefore many tempered intervals can be expressed with high precision in this measure, but also many just intervals. One Purdal is 4/33 cent. • harrop: 1/271 part of a pure twelfth One harrop is 7.0183 cents and there are close to 171 harrops in an octave. It was proposed by Todd Harrop as an equal tempered equivalent to the quark. The name quark was chosen by Heinz Bohlen and is derived from a set of supposedly four JI mini-intervals in the 7 cents range. Two of them have been identified by him: a = 6.990 cents (3, 4, -5) and b = 7.200 cents (-8, -3, 7). The other two still need to be determined, but can be replaced by the small semitone (3, -2, 0). Harrops are practical for showing interval relationships in the JI BP scale as whole numbers, e.g. the various JI semitone sizes measure between 19 and 24 harrops. Any interval and all its enharmonics can be represented by an integer without rounding inconsistencies. See also The 271-Tone BP Scale. • savart: 1/301 part of an octave This measure was defined by Joseph Sauveur (1653-1716) in 1696 as eptaméride, one seventh part of a méride. Later in the 20th century its name became savart, after the French physicist Félix Savart (1791-1841) who also advocated it. In French acoustical literature it's still used now and then. It is close to 100 times the base-10 logarithm of 2 and therefore almost as accurate as jots in calculations. So Sauveur proposed it because 301=7×43 and Savart because 301(.03) = 100×10log 2. Later the name savart was used in the book The Physics of Music by Alexander Wood to denote the slightly different value of 1/300 part of an octave. This would make it more practical for expressing 12-tET intervals. In some literature the savart is taken to be the 100/30103 part of an octave, making it exactly 100 jots. • schisma Like comma, also an interval: the difference between the Pythagorean and syntonic comma. Because it is so small it is also useful as a measure. The syntonic comma is 11.008 schismas, the Pythagorean comma 12.008, and the minor diesis 21.016 schismas, so practically 11, 12 and 21. There are also temperaments with the fifth tempered by a fraction of a schisma. There are 614.21264 schismas in an octave. A similar useful unit is 1/612 part of an octave, or one step of 612-tone equal temperament, because this temperament has extremely accurate approximations of fifth and thirds, and because 612 is divisible by 12. So one step is also very close to 1/12 of a Pythagorean comma and the schisma. • secor: 116.69 cents More an interval than a measure, it is almost 7/72 part of an octave. Proposed by George Secor in a 1975 Xenharmonikon article as a generator for scales which are nearly 11-limit just. See also this page. • Temperament Unit: 1/720 part of a Pythagorean comma This measure was developed by organ builder John Brombaugh to describe very small intervals as integer values. In this measure, the syntonic comma is almost exactly 660 Temperament Units and the schisma 60. Because 720 is divisible by all numbers from 2 to 6 and more, most temperaments can be described by only integer values. In a well-temperament, -720 TU must be distributed over the cycle of fifths. One Grad is 60 TU. There are 36828.62820 Temperament Units in an octave. See also this page. • tina: 1/8539 part of an octave Named by Dave Keenan and George Secor. It provides good approximations for 41-limit primes except 37. One schisma is approximately 14 tinas. • Türk sent (Turkish cent): 1/10600 part of an octave The tuning of Turkish classical music is Pythagorean which lends itself well to be approximated by 53-tone equal temperament, see comma. The comma plays an important role in this music and the smallest step of 53-tET is in between and approximately the same as the syntonic and Pythagorean comma. The Turkish theorist Ekrem Karadeniz has made a not very useful further subdivision into 200 parts, making 10600 to the octave. The notation systems of Arel, Ezgi and Yektâ Bey have special accidentals for the 1/53 octave comma (discovered by Jean Galle) and multiples of it.	4,864	18,898	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2024-33	latest	en	0.917019	# Logarithmic Interval Measures. The first persons to use logarithms for calculation of interval sizes were Bonaventura Cavalieri (1639), Juan Caramel de Lobkowitz (1647), and Lemme Rossi (1666). Also Christiaan Huygens was among the first to do this. Earlier calculations of equal intervals sizes, like those of Simon Stevin (1585) and Marin Mersenne (1636) were done by square and cubic roots.. The logarithm, invented by John Napier in 1614, is a mathematical operation that turns a multiplication into an addition, and by the same definition, raising to a higher power into multiplication. Because stacking two intervals involves multiplication of frequency ratios, this is equivalent to addition of the logarithmic measures of these ratios.. • cent: 1/1200 part of an octave. Defined by Alexander John Ellis (1814-1890) (see photo) in 1884 and presented in the article "The Musical Scales of Various Nations" as well as in the appendix of his English translation of Hermann von Helmholtz' book Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. One cent is one hundredth part of the semitone in 12-tone Equal Temperament, a centisemitone. The frequency ratio represented by one cent is the 1200th root of 2. So the 12-tET whole tone is 200 cents, the minor third 300 cents, the major third 400 cents, etc. Rounding to the nearest cent is sufficiently accurate for practical purposes. Cents are the most universally used interval measure. They have the advantage that familiar intervals have an easily rememberable value. Ellis, whose real name was Sharpe, is also regarded as the founder of the field of ethnomusicology. He invented cents originally for the purpose of expressing non-Western scales.. In a different form, this measure was invented earlier by Heinrich Bellermann (1832-1903), namely the logarithm with base twelfth root of 2, which has a size a hundred times larger than a cent, the same size as an equal tempered semitone. Contrary to cents, it was meant to be used for Western classical music. This measure was also endorsed by Paul von Jankó. Bellermann received little acknowledgement for it, because people adopted Ellis's cents much more enthusiastically. It goes back to Isaac Newton, who also expressed intervals in terms of 1/12 of an octave in 1665.. • centitone: see Iring. • comma. Not a fixed measure as such but often used as an interval unit. The syntonic (Didymic) and Pythagorean (ditonic) commas have almost the same size. In the past they were often confused and their difference was often neglected. The word coma is Latin for hair. There are 55.79763 syntonic commas and 51.15087 Pythagorean commas in the octave. Therefore one step in 53-tET is often named a comma because it's in the middle of them and 53-tET is a very accurate approximation to 5-limit just intonation scales. The major whole tone 9/8 is 9 commas and the minor whole tone 10/9 is 8 commas. The chromatic and diatonic semitones are 4 and 5 commas. The 1/53 octave comma was discovered by Jean Galle as reported by Marin Mersenne in 1637. Nicolaus Mercator gave a more mathematically precise description of it in 1660. This comma has also been wrongly attributed to William Holder.. In regular meantone temperaments where the tempering of the fifth is expressed in a fraction of a comma, the comma is the syntonic comma. In well-temperaments, like Werckmeister's, where the cycle of fifths closes in a circle, the temperings are usually expressed in fractions of a Pythagorean comma.. The measure 1/55 of an octave is called a Sauveur comma and 1/50 of an octave could be called a Henfling comma.. • decaméride: 1/3010 part of an octave. Defined by Joseph Sauveur in 1696 as one tenth of an eptaméride.. • Delfi unit: 1/665 part of an octave. Used in Byzantine music theory? Approximately 1/12 part of the syntonic comma and 1/13 part of the Pythagorean comma.. • demi-heptaméride: 1/602 part of an octave. Defined by Joseph Sauveur in 1696 as one half of an eptaméride.. • diesis (plural: dieses or dieseis). Like comma, also an interval. The name was used by ancient Greek theorists like Aristoxenos for several different intervals. Marchettus de Padua (Marchetto Padoano), in his Lucidarium written in 1317/1318, was the first to use it as a standard measure. He divided the whole tone in 5 parts, called a diesis. The chromatic semitone (semitonium enharmonicum) was 2 dieses, the diatonic semitone (semitonium diatonicum) 3 dieses and another chromatic semitone of 4 dieses was meant for some augmentations like C-C giving a very high leading tone. See also this essay.. Later the minor or enharmonic diesis became the difference between an octave and three pure major thirds. There are 29.22634 of it in an octave. Adriaan Fokker used this name for the step of 31-tone equal temperament, where it is also equal to 1/5 part of a whole tone, because of its similar size. This 1/31 part of an octave called normal diesis by Fokker is a convenient measure in which to express 7-limit just intervals.. • Dröbisch Angle: 1/360 part of an octave. The Angle was proposed by Moritz Dröbisch in the 19th century as a cycle of 360 degrees to the octave. Andrew Pikler has suggested this name in his article "Logarithmic Frequency Systems" (1966).. • Ellis or El: absolute cent. Not an interval measure, but a logarithmic absolute pitch measure. Proposed by Robert Stuckey and described in the paper Ellis at Belfast by Jeremy Montagu. It's the number of cents relative to the 64-foot C (16.3516 Hz), with the number of octaves written as a subscript in front of the number of cents. For example the A at 440 Hz is shown as 4900. The abbreviation is El, as Hertz are abbreviated to Hz.. • eptaméride or heptaméride: 1/301 part of an octave. Both spellings used by Sauveur. See méride and savart. Sauveur's rule to find the number of eptamérides of intervals smaller than 7/6 is as follows: multiply the difference of numerator and denominator with 875 and divide by the sum of numerator and denominator and round the result to the nearest integer. This is known as the bimodular method of approximating logarithms and can be used for other measures as well.. • farab: 1/144 part of an octave. Measure proposed by al-Farabi in the 10th century, from Mehdi Barkeshli, "Perfect scale of Farabi and his proposed scales", Iranian music, a collection of articles.. A decifarab is 1/10th of a farab unit, with the advantage that 1440 is a number with many divisors.. • flu: 1/46032 part of an octave. Measure proposed by Gene Ward Smith for use with (nano-)temperaments. The Pythagorean comma is nearly 900 flus (899.9259), the syntonic comma 825 (824.9812) and the schisma 75 flus (74.94472), alas not divisible by 2. It's an alternative to the Temperament Unit which, although used with 5-limit temperaments, is not 5-limit consistent, whereas flus measure intervals consistently up to the 9-limit.. • Grad (degree of tempering): 1/12 part of a Pythagorean comma, 1/613.810469983 part of an octave. One Grad is the difference between a pure fifth of 1.95500086539 cents and the 12-tET fifth of 700.0 cents and makes it a useful measure to describe temperings in a well-temperament. The name comes from Andreas Werckmeister (1645-1706) who used it to denote different divisions of a comma. It has a negligible difference to 1/11 part of a syntonic comma. Jan van Biezen calls this measure a Werckmeister, symbol Wm. Organ builders sometimes use this symbol, and approximate its size by 2 cents. Another useful property is that the minor diesis is almost exactly 21 Grad (21.002). So the tempering of the major third in 12-tET is very close to 7 units. And likewise that of the major sixth in 12-tET very close to 8 units. Deviations from equal temperament of three consecutive major thirds when expressed in Grad always add up to 21. This measure was also used by Neidhardt and Sorge and later by Mark Lindley in their temperament diagrammes. A Grad is slightly larger than a schisma, 1.000655 times it. The 1/21 part of a minor diesis is 1.000094 Grads.. • Harmos: 1/1728 part of an octave. A twelve-based measure suggested by Paul Beaver: 1728 is 123. In the 1960s he asked John Chalmers to compute a table of Harmos, which he did later in decimal and duodecimal notation.. • Hekt: 1/1300 part of a pure twelfth. Defined by Heinz Bohlen as the hundredth part of a step of the equal tempered version of the Bohlen-Pierce scale: the 13th root of 3/1. Hekt are therefore the BP analogon to cents. See H. Bohlen, "13-Tonstufen in der Duodezime", Acustica vol. 39, 1978. There are 820.2086796 Hekt to the octave.. • imp: 1/31920 part of an octave. Able to accurately express intervals up to 41-limit. Also 1/1680th part of a 19-tET step.. • iota: 1/1700 part of an octave. Proposed by Margo Schulter on the Tuning List in 2002 and indicated by the Greek letter iota. The classic chromatic semitone 25/24 for example is 100.12 iotas. It's useful for comparing just intervals with 17-tone equal tempered ones.. • Iring: 1/600 part of an octave.	The Iring unit was defined by Widogast Iring in his 1898 Die reine Stimmung in der Musik. He noted that the twelfth part of the Pythagorean comma and the schisma have almost the same size, both about 1/614 part of the octave. To get round numbers, he took for this size one 600th part of an octave. The size of the major second can then be rounded to 102 and the just major third to 193. The perfect fifth is 351 Iring units. The size of the Iring unit is twice the size of the cent. It is also about the smallest difference in pitch that untrained ears can hear.. The same unit was later defined in 1932 by Joseph Yasser in his book A Theory of Evolving Tonality by dividing the equally tempered whole tone in 100 parts and calling it the centitone.. • jinn: 1/16808 part of an octave. This measure was proposed by Gene Ward Smith because it is consistent up until prime number 37 which means rounding to the nearest integer jinn will (usually) give the correct answers to this limit.. • jot: 1/30103 part of an octave. This name was given by Augustus De Morgan (1806-1871). The 10-base (Briggs) logarithm of 2 (10log 2) is 0.30102999566 so multiplied by 10000 this makes almost exactly 30103. Expressing intervals in this measure has the advantage of being able to calculate interval combinations without using logarithms, because rounding to the nearest integer jot will (usually) give the correct answers, at least for the prime numbers up to 11. And the jot values can be looked up in a 10-base logarithm table. A similar measure is the savart.. • mem: 1/205 part of an octave. This name was chosen by Andrew Aaron Hunt. One mem is equal to 12 minas and very close to 3 schismas and 4 Hekt. It has good approximations for the minor third (54), major third (66) and perfect fifth (120).. • méride: 1/43 part of an octave. This name was chosen by Joseph Sauveur (1653-1716) in 1696. The méride and eptaméride were the first logarithmic interval measures proposed. Sauveur favoured 43-tone equal temperament because the small intervals are well represented in it. He had set the comma to one step, then found a range of 2, 3 or 4 steps for the chromatic semitone, corresponding to 31, 43 and 55 tones per octave. He found 43 to be optimal because 4 steps is almost exactly a 16/15 minor second and 7 steps almost exactly the geometric mean of three 9/8 and two 10/9 whole tones. The chromatic scale contained in 43-tET is virtually identical to 1/5-comma meantone tuning.. • MIDI Tuning Standard unit: 1/196608 part of an octave. This divides the 12-tET semitone into 214 = 16384 parts which resolution makes sufficiently accurate tuning of electronic instruments possible. It's in the MIDI Tuning Specification 1.0.. There are other MIDI tuning units which differ per manufacturer, for example Yamaha has models tuned in 1/768 or 1/1024 parts of an octave.. There's also the MIDI Pitch Bend message, which can carry the values -8192 .. 8191, so when the range (which is variable) is the standard range of +/- 200 cents, then the unit is 1/49152 part of an octave or 0.024414 cents.. • millioctave: 1/1000 part of an octave. Named and used by Arthur Joachim von Oettingen (1836-1920) in his book Das duale Harmoniesystem (1913). Alfred Jonqičre indicated the millioctave with the Greek letter mu. It was first used however by John Herschel in the book which he wrote with George Bidell Airy On Sound and Atmospheric Vibrations with the Mathematical Elements of Music (1871). Sometimes millioctaves are propagated as a "value-free" substitute for cents, not having the 12-tET bias, because the round cent numbers may lead people to the false belief that the intervals are perfectly in tune. However using these millioctaves introduces a 10-tET bias, which is a much less familiar tuning. Often the cent values of just intervals are easy to remember by their deviation from the 12-tET multiple of 100, for example the pure fifth is 702 cents, with millioctaves this is harder: 585 millioctaves compared to 583.333. Another advantage of cents is the size of the schisma: almost 2 cents against 1.63 M.O.. • mil or prima: 1/12276 part of an octave. Suggested by Erv Wilson, Gene Ward Smith and Gavin Putland. It's 1/1023 part of 100 cents so close to 1/1000 semitone hence the name mil. It's very close to three times the size of the Temperament Unit and therefore has the same kind of advantages. The Pythagorean comma is 240 prima and the syntonic comma 220.. • mina: 1/2460 part of an octave. Named by Dave Keenan and George Secor as an abbreviation of "schismina". It's 0.487805 cents or 1/205 part of a 100 cent semitone, and selected because 2460-tone equal temperament is consistent up to the 28th harmonic and its step is therefore a useful measure to express high-limit just ratios in, and getting very little roundoff errors. They use it in the development of their Sagittal notation system. Although for that purpose the exact size of one mina is 1/233 part of a Pythagorean apotome, or 0.487918 cents or 1/2459.427234 octave.. • morion (plural: moria): 1/72 part of an octave. Defined as 1/30 part of a fourth by the theorist Cleonides around 100 AD for description of Greek tetrachords. Likewise Aristoxenos used a cipher of 12 parts to a whole tone. This measure is surrounded by controversy, because it's very unclear what Aristoxenos' measurements exactly are. Moria is Greek for molecules or small pieces. 72-tone equal temperament is a good approximation to many just intonation scales because the prime numbers 2, 3, 5, 7 and 11 are very well represented with deviations not exceeding 3 cents.. • purdal: 1/9900 part of an octave. This measure was suggested by Tútim Dennsuul. It is a highly composite number. Therefore many tempered intervals can be expressed with high precision in this measure, but also many just intervals. One Purdal is 4/33 cent.. • harrop: 1/271 part of a pure twelfth. One harrop is 7.0183 cents and there are close to 171 harrops in an octave. It was proposed by Todd Harrop as an equal tempered equivalent to the quark. The name quark was chosen by Heinz Bohlen and is derived from a set of supposedly four JI mini-intervals in the 7 cents range. Two of them have been identified by him: a = 6.990 cents (3, 4, -5) and b = 7.200 cents (-8, -3, 7). The other two still need to be determined, but can be replaced by the small semitone (3, -2, 0). Harrops are practical for showing interval relationships in the JI BP scale as whole numbers, e.g. the various JI semitone sizes measure between 19 and 24 harrops. Any interval and all its enharmonics can be represented by an integer without rounding inconsistencies. See also The 271-Tone BP Scale.. • savart: 1/301 part of an octave. This measure was defined by Joseph Sauveur (1653-1716) in 1696 as eptaméride, one seventh part of a méride. Later in the 20th century its name became savart, after the French physicist Félix Savart (1791-1841) who also advocated it. In French acoustical literature it's still used now and then. It is close to 100 times the base-10 logarithm of 2 and therefore almost as accurate as jots in calculations. So Sauveur proposed it because 301=7×43 and Savart because 301(.03) = 100×10log 2. Later the name savart was used in the book The Physics of Music by Alexander Wood to denote the slightly different value of 1/300 part of an octave. This would make it more practical for expressing 12-tET intervals. In some literature the savart is taken to be the 100/30103 part of an octave, making it exactly 100 jots.. • schisma. Like comma, also an interval: the difference between the Pythagorean and syntonic comma. Because it is so small it is also useful as a measure. The syntonic comma is 11.008 schismas, the Pythagorean comma 12.008, and the minor diesis 21.016 schismas, so practically 11, 12 and 21. There are also temperaments with the fifth tempered by a fraction of a schisma. There are 614.21264 schismas in an octave. A similar useful unit is 1/612 part of an octave, or one step of 612-tone equal temperament, because this temperament has extremely accurate approximations of fifth and thirds, and because 612 is divisible by 12. So one step is also very close to 1/12 of a Pythagorean comma and the schisma.. • secor: 116.69 cents. More an interval than a measure, it is almost 7/72 part of an octave. Proposed by George Secor in a 1975 Xenharmonikon article as a generator for scales which are nearly 11-limit just. See also this page.. • Temperament Unit: 1/720 part of a Pythagorean comma. This measure was developed by organ builder John Brombaugh to describe very small intervals as integer values. In this measure, the syntonic comma is almost exactly 660 Temperament Units and the schisma 60. Because 720 is divisible by all numbers from 2 to 6 and more, most temperaments can be described by only integer values. In a well-temperament, -720 TU must be distributed over the cycle of fifths. One Grad is 60 TU. There are 36828.62820 Temperament Units in an octave. See also this page.. • tina: 1/8539 part of an octave. Named by Dave Keenan and George Secor. It provides good approximations for 41-limit primes except 37. One schisma is approximately 14 tinas.. • Türk sent (Turkish cent): 1/10600 part of an octave. The tuning of Turkish classical music is Pythagorean which lends itself well to be approximated by 53-tone equal temperament, see comma. The comma plays an important role in this music and the smallest step of 53-tET is in between and approximately the same as the syntonic and Pythagorean comma. The Turkish theorist Ekrem Karadeniz has made a not very useful further subdivision into 200 parts, making 10600 to the octave. The notation systems of Arel, Ezgi and Yektâ Bey have special accidentals for the 1/53 octave comma (discovered by Jean Galle) and multiples of it.
http://www.dotnetperls.com/number-python	1,590,450,678,000,000,000	text/html	crawl-data/CC-MAIN-2020-24/segments/1590347390437.8/warc/CC-MAIN-20200525223929-20200526013929-00412.warc.gz	170,568,779	4,847	Python Number: random, float and divmod Use numbers and numeric operators. See the add, multiply, subtract and divide constructs. Numbers. It is 2 PM. It is 50 degrees F. There is a beauty in numbers. Human beings use them to describe our world. Programs too are built of numbers. Operators. Tiny functions called operators act upon numbers. With operators and operands (the values operated upon), we make expressions and statements. Division. We have two division operators. With one slash, we divide two numbers. And with two slashes "//" we divide and round down the result. Operator 1: The "/" operator leaves the fractional part of the result intact. It does not matter if the two operands are fractional or not. Operator 2: The "//" operator divides in the same way. But it also rounds the result down to the nearest integer. Python program that divides numbers a = 100 b = 7 # Divide 100 by 7. print(a / b) # Discard fractional part of result. print(a // b) Output 14.285714285714286 14 Integral division. This operator does not round up if the value is closer to the higher value. This means 5 // 3 will give 1, even though 5 / 3 gives 1.6, which is closer to 2 than to 1. Python common line division >>> 5//3 1 >>> 5/3 1.6666666666666667 Float converts data to floating-point numbers. It acts on string values (like "10.0") or integers (like 10). On strings, it handles signs (positive or negative) and infinity values ("inf"). Conversion: Float is similar to other built-ins like int or str. Python simplifies common conversions. ConvertBuilt-ins Python program that uses float # Float converts a string into a float number. value = "3.14" number = float(value) print(number) print(number == 3.14) print(value == "3.14") print() # Float also converts an integer into a float number. integer = 100 floating = float(integer) print(floating) print(integer) Output 3.14 True True 100.0 100 Int. Like float, int() converts from strings and other number types. It returns an integer (a number with nothing past the decimal—no fractional part).Int Note: Int will cause an error if we try to convert a floating-point number within a string (like "123.4"). Python program that uses int # Convert a string to int. input = "123" result = int(input) print(result) # Use int to convert from floating to integral. input = 456.9 result = int(input) print(result) Output 123 456 Hex converts an integer into a hexadecimal number. This form of number can be useful in interoperating with other programs and systems. We see the hex representations of 10 and 100. Python program that uses hex # Convert this value to a hex. value = 10 result = hex(value) print(result) # Convert another value. value = 100 result = hex(value) print(result) Output 0xa 0x64 Octal numbers use not a base 10 like we are used to, but a base 8. So they only contain the digits 0 through 7. With oct() we convert a base 10 number into its octal representation. Python program that uses oct # Convert 74 into octal which is 112. number = 74 octal = oct(number) print(octal) Output 0o112 Bits. In computers, numbers are presented with bits, as binary. With the bin() built-in, we get a string representation of an integer. Zeros on the left of the representation are discarded. Negative: The sign bit is represented by the 0 before the lowercase "b." A -1 has a leading minus sign. Python program that uses bin number = 100 # Convert this number to a binary string. result = bin(number) print(result) Output 0b1100100 More bin examples bin(-1) -0b1 bin(0) 0b0 Complex. Complexity is not just in our computer programs. We also encounter complex numbers. These numbers have two components—real and imaginary. Imaginary unit: Complex numbers include an "imaginary unit" that is separate from the real parts. These numbers never become bored. Info: In Python we have the complex() built-in function. These numbers can be added, subtracted, and manipulated in other ways. Python program that uses complex numbers # Create two complex numbers. complexA = complex(3, 10) complexB = complex(5, 15) # Add the two together. # ... The result is also complex. complexC = complexA + complexB print(complexC) Output (8+25j) Benchmark, division. Division is a slow operation on processors. In Python we have both the "/" and "//" operators. Is there some optimization in the latter one? My benchmark tests this. Version 1: This version of the code uses the single-slash operator to perform a division. Version 2: Here we use the double-slash operator to perform division and get an integer-only result. Result: We find that the "//" operator is slower than the "/" operator. It adds steps beyond the regular operator. Python program that times division import time a = 1000 b = 223 c = 0 print (time.time()) # Version 1: normal division i = 0 while i < 10000000: c = a / b i += 1 print (time.time()) # Version 2: integer result division i = 0 while i < 10000000: c = a // b i += 1 print (time.time()) Output 1345843075.764 1345843077.922 (/ = 2.158 s) 1345843080.448 (// = 2.526 s) Divmod, modulo. The divmod function is built into the Python language. It computes two kinds of division at once: it does an integral division and a modulo division. It returns a tuple.Divmod Random numbers can be generated in Python with the randint method. But for a random selection in a list or other collection, random.choice is an ideal option.Random Pow. The pow built-in, or two asterisks, means exponentiation. We can raise a number to a certain power with clear syntax. The two syntax forms are equivalent.pow Bool converts an expression into True or False. It is similar to the if-statement, which also evaluates expressions. Bool is a value—often languages store False as 0 and True as 1.bool Is a number prime? We implement a prime-testing method with a def and a for-loop. Some arithmetic optimizations are applied to this method—we test the number's square.Prime Number Numeric operations are everywhere. All memory accesses in programs use numeric computations. An access to an element of a list, at an index, requires multiplications to locate memory. Compilers handle these. A simple program is an illusion. All programs involve complex numeric computation. All levels of programming, from the metal to object models, are numeric. Home Dot Net Perls © 2007-2020 Sam Allen. Every person is special and unique. Send bug reports to info@dotnetperls.com.	1,554	6,418	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2020-24	latest	en	0.783335	Python Number: random, float and divmod. Use numbers and numeric operators. See the add, multiply, subtract and divide constructs.. Numbers. It is 2 PM. It is 50 degrees F. There is a beauty in numbers. Human beings use them to describe our world. Programs too are built of numbers.. Operators. Tiny functions called operators act upon numbers. With operators and operands (the values operated upon), we make expressions and statements.. Division. We have two division operators. With one slash, we divide two numbers. And with two slashes "//" we divide and round down the result.. Operator 1: The "/" operator leaves the fractional part of the result intact. It does not matter if the two operands are fractional or not.. Operator 2: The "//" operator divides in the same way. But it also rounds the result down to the nearest integer.. Python program that divides numbers a = 100 b = 7 # Divide 100 by 7. print(a / b) # Discard fractional part of result. print(a // b) Output 14.285714285714286 14. Integral division. This operator does not round up if the value is closer to the higher value. This means 5 // 3 will give 1, even though 5 / 3 gives 1.6, which is closer to 2 than to 1.. Python common line division >>> 5//3 1 >>> 5/3 1.6666666666666667. Float converts data to floating-point numbers. It acts on string values (like "10.0") or integers (like 10). On strings, it handles signs (positive or negative) and infinity values ("inf").. Conversion: Float is similar to other built-ins like int or str. Python simplifies common conversions.. ConvertBuilt-ins. Python program that uses float # Float converts a string into a float number. value = "3.14" number = float(value) print(number) print(number == 3.14) print(value == "3.14") print() # Float also converts an integer into a float number. integer = 100 floating = float(integer) print(floating) print(integer) Output 3.14 True True 100.0 100. Int. Like float, int() converts from strings and other number types. It returns an integer (a number with nothing past the decimal—no fractional part).Int. Note: Int will cause an error if we try to convert a floating-point number within a string (like "123.4").. Python program that uses int # Convert a string to int. input = "123" result = int(input) print(result) # Use int to convert from floating to integral. input = 456.9 result = int(input) print(result) Output 123 456. Hex converts an integer into a hexadecimal number. This form of number can be useful in interoperating with other programs and systems. We see the hex representations of 10 and 100.. Python program that uses hex # Convert this value to a hex. value = 10 result = hex(value) print(result) # Convert another value. value = 100 result = hex(value) print(result) Output 0xa 0x64. Octal numbers use not a base 10 like we are used to, but a base 8. So they only contain the digits 0 through 7. With oct() we convert a base 10 number into its octal representation.. Python program that uses oct # Convert 74 into octal which is 112. number = 74 octal = oct(number) print(octal) Output 0o112. Bits. In computers, numbers are presented with bits, as binary.	With the bin() built-in, we get a string representation of an integer. Zeros on the left of the representation are discarded.. Negative: The sign bit is represented by the 0 before the lowercase "b." A -1 has a leading minus sign.. Python program that uses bin number = 100 # Convert this number to a binary string. result = bin(number) print(result) Output 0b1100100 More bin examples bin(-1) -0b1 bin(0) 0b0. Complex. Complexity is not just in our computer programs. We also encounter complex numbers. These numbers have two components—real and imaginary.. Imaginary unit: Complex numbers include an "imaginary unit" that is separate from the real parts. These numbers never become bored.. Info: In Python we have the complex() built-in function. These numbers can be added, subtracted, and manipulated in other ways.. Python program that uses complex numbers # Create two complex numbers. complexA = complex(3, 10) complexB = complex(5, 15) # Add the two together. # ... The result is also complex. complexC = complexA + complexB print(complexC) Output (8+25j). Benchmark, division. Division is a slow operation on processors. In Python we have both the "/" and "//" operators. Is there some optimization in the latter one? My benchmark tests this.. Version 1: This version of the code uses the single-slash operator to perform a division.. Version 2: Here we use the double-slash operator to perform division and get an integer-only result.. Result: We find that the "//" operator is slower than the "/" operator. It adds steps beyond the regular operator.. Python program that times division import time a = 1000 b = 223 c = 0 print (time.time()) # Version 1: normal division i = 0 while i < 10000000: c = a / b i += 1 print (time.time()) # Version 2: integer result division i = 0 while i < 10000000: c = a // b i += 1 print (time.time()) Output 1345843075.764 1345843077.922 (/ = 2.158 s) 1345843080.448 (// = 2.526 s). Divmod, modulo. The divmod function is built into the Python language. It computes two kinds of division at once: it does an integral division and a modulo division. It returns a tuple.Divmod. Random numbers can be generated in Python with the randint method. But for a random selection in a list or other collection, random.choice is an ideal option.Random. Pow. The pow built-in, or two asterisks, means exponentiation. We can raise a number to a certain power with clear syntax. The two syntax forms are equivalent.pow. Bool converts an expression into True or False. It is similar to the if-statement, which also evaluates expressions. Bool is a value—often languages store False as 0 and True as 1.bool. Is a number prime? We implement a prime-testing method with a def and a for-loop. Some arithmetic optimizations are applied to this method—we test the number's square.Prime Number. Numeric operations are everywhere. All memory accesses in programs use numeric computations. An access to an element of a list, at an index, requires multiplications to locate memory.. Compilers handle these. A simple program is an illusion. All programs involve complex numeric computation. All levels of programming, from the metal to object models, are numeric.. Home. Dot Net Perls. © 2007-2020 Sam Allen. Every person is special and unique. Send bug reports to info@dotnetperls.com.
https://m.jagranjosh.com/articles/ugc-net-november-2017-paper-i-set-d-previous-year-paper-1543475497-1?ref=next_art	1,638,657,482,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964363125.46/warc/CC-MAIN-20211204215252-20211205005252-00251.warc.gz	464,613,447	42,450	# UGC NET November 2017 Paper-I Set-D Previous Year Paper with Answers Created On: Nov 22, 2019 12:55 IST UGC NET November 2017 Paper-I Set-D Previous Year Paper with Answers For cracking NTA UGC NET December 2019 Exam, candidates must practice the previous year papers of the different subjects for which they are applying this year. It will help them in improving their speed of attempting maximum questions in minimum time with accuracy. So, in this article we have shared the UGC NET November 2017 Paper I (Set-D) Previous Year Paper alongwith their answers. Let's first look at the Exam Pattern of Paper-I NTA UGC NET 2019 Exam: NTA UGC NET Dec 2019 Paper-I Exam Pattern Part Sections Questions Marks I Teaching Aptitude 5 10 II Research Aptitude 5 10 III Reading Comprehension 5 10 IV Communication 5 10 V Reasoning (including Maths) 5 10 VI Logical Reasoning 5 10 VII Data Interpretation 5 10 VIII Information & Communication Technology (ICT) 5 10 IX People & Environment 5 10 X Higher Education System: Governance, Polity & Administration 5 10 Total 50 100 ## UGC NET November 2017 Paper I (Set-D) Previous Year Paper with Answers 1. Just as melting ice - cubes do not cause a glass of water to overflow, melting sea - ice does not increase oceanic volume. What type of argument is it? (1) Psychological (2) Statistical (3) Analogical (4) Hypothetical 2. A postman walked 20 m straight from his office, turned right and walked 10 m. After turning left he walked 10 m and after turning right walked 20 m. He again turned right and walked 70 m. How far he is from his office? (1) 60 m. (2) 20 m. (3) 50 m. (4) 40 m. 3. Given below are two premises (a and b). From those two premises four conclusions (i), (ii), (iii) and (iv) are drawn. Select the code that states the conclusion/conclusions drawn validly (taking the premises singularly or jointly). Premises: (a) All bats are mammals. (b) No birds are bats. Conclusions: (i) No birds are mammals. (ii) Some birds are not mammals. (iii) No bats are birds. (iv) All mammals are bats. Code: (1) (iii) only (2) (iii) and (iv) only (3) (i) only (4) (i) and (ii) only 4. A good communicator begins his/her presentation with a: (1) Repetitive phrase (2) Ice-breaker (3) Complex question (4) Non-sequitur 5. The classroom communication should essentially be: (1) Abstract (2) Non-descriptive (3) Contrived (4) Empathetic 6. In the series 1, 6, 15, 28, 45, ............ the next term will be: (1) 56 (2) 84 (3) 66 (4) 76 7. Ajay is a friend of Rakesh. Pointing to an old man Ajay asked Rakesh who is he? Rakesh said “His son is my son’s uncle”. The old man is related to Rakesh as: (1) Father (2) Uncle (3) Grandfather (4) Father-in-law 8. The spatial audio reproduction in a classroom can reduce the students’: (1) Motivation for excellence (2) Interest in technology - orientation (4) Respect for the teacher 9. It is Truism to say that no one was there when life first appeared on earth. Any assertion about life’s origin, thus, should be treated as a theory. The above two statements constitute: (1) An argument (2) A conjecture (3) A historical explanation (4) A narrative 10. In a classroom, the probability of message reception can be enhanced by: (2) Using high decibel audio tools (3) Establishing a viewpoint (4) Exposing the ignorance of students Click here to know the latest Exam Pattern and Syllabus of NTA UGC NET December 2019 Exam 11. Given below are four statements. Among them two are related in such a way that they can both be true but they cannot both be false. Select the code that indicates those two statements: Statements: (a) Honest people never suffer. (b) Almost all honest people do suffer. (c) Honest people hardly suffer. (d) Each and every honest person suffers. Code: (1) (a) and (d) (2) (b) and (c) (3) (a) and (b) (4) (a) and (c) 12. In certain code, “COVALENT” is coded as BWPDUOFM. The code of “ELEPHANT” will be: (1) QFMFUOBI (2) EPHNTEAS (3) MFUIQRTW 13. The interaction between a teacher and students creates a zone of proximal: (1) Development (2) Distortion (3) Difference (4) Confusion 14. A deductive argument is invalid if: (1) Its premises are all false but its conclusion is true. (2) Its premises are all true but its conclusion is false. (3) Its premises and conclusion are all true. (4) Its premises and conclusion are all false. 15. The next term in the series ABD, DGK, HMS, MTB, .......... is: (1) PSK (2) RUH (3) NSA (4) SBL Answer the questions 16 to 20 based on the data given in the table below. Table: Number of registered vehicles in India and India’s population. Year Total vehicles (Lakhs) Two wheelers (Lakhs) Cars, Jeeps, Taxis (Lakhs) Buses (Lakhs) Goods vehicles (Lakhs) Others (Lakhs) Population (India) (Millions) 1961 6.65 0.88 3.1 0.57 1.68 0.42 439.23 1971 18.65 5.76 6.82 0.94 3.43 1.70 548.15 1981 53.19 26.18 11.60 1.62 5.54 8.97 683.32 1991 213.74 142.00 29.54 3.31 13.56 25.33 846.42 2001 549.91 385.56 70.58 6.34 29.48 57.95 1028.73 2011 1417.58 1018.65 191.23 16.04 70.64 121.02 1210.19 16. In which year the decadal growth (%) in number of cars surpassed that of the two wheelers? (1) 1981 (2) 2011 (3) 1991 (4) 2001 17. What was the average decadal growth in the number of cars during 1961 - 2011? (1) ~ 217% (2) ~ 157% (3) ~ 131% (4) ~ 68% 18. In the year 2001, out of total number of vehicles, the number of passenger vehicles (4 wheelers) accounted for: (1) ~ 31% (2) ~ 43% (3) ~ 14% (4) ~ 24% 19. What was the per capita ownership of two wheelers in India in the year 2011? (1) ~ 0.84% (2) ~ 0.068% (3) ~ 0.084% (4) ~ 0.0084% 20. The maximum decadal growth in population of India is registered in the period: (1) 2001 - 2011 (2) 1981 - 1991 (3) 1961 - 1971 (4) 1991 - 2001 Click here to know the UGC NET Exam Previous Year Cut-Off Marks Read the passage carefully and answer question numbers from 21 to 25. 21. The traditional knowledge should be used through: (1) Synergy between government and local interventions (2) Modern technology (3) Its dissemination (4) Improvement in national circumstances 22. Given below are the factors of vulnerability of poor people to climate change. Select the code that contains the correct answer. (a) Their dependence on natural resources (b) Geographical attributes (c) Lack of financial resources Code: (1) (a), (b), (c) and (d) (2) (c) only (3) (a), (b) and (c) (4) (b), (c) and (d) 23. Adaptation as a process enables societies to cope with: (a) An uncertain future (c) Negative impact of climate change (d) Positive impact of climate change Select the most appropriate answer from the following code: (1) (b), (c) and (d) (2) (c) only (3) (a), (b), (c) and (d) (4) (a) and (c) 24. To address the challenge of climate change, developing countries urgently require: (3) Imposition of climate change tax (4) Implementation of national adaptation policy at their level 25. The main focus of the passage is on: (2) Social dimensions of climate change (3) Combining traditional knowledge with appropriate technology (4) Co-ordination between regional and national efforts 26. Which of the following come(s) within the ambit of the term ‘corruption’? (a) Misuse of official position (b) Deviation from rules, laws and norms (c) Non-action when action is required (d) Harm to public good Select the correct answer from the code given below: (1) (a), (b) and (d) (2) (a), (b), (c) and (d) (3) (a) only (4) (a) and (b) only 27. Which of the following universities has received the Visitor’s Award for the best Central University in India in Feb. 2017? (1) Tezpur University (3) Jawaharlal Nehru University (4) Banaras Hindu University 28. What is the name for a webpage address? (1) Protocol (2) URL (3) Domain (4) Directory 29. Occurrence of natural hazards is affected by: (a) Land use changes (b) Drainage and construction (c) Ozone depletion (d) Climate change Choose the correct answer from the code given below: (1) (a), (b) and (d) (2) (b), (c) and (d) (3) (a), (c) and (d) (4) (a), (b) and (c) 30. The data storage hierarchy consists of: (1) Bits, bytes, records, fields, files and databases (2) Bits, bytes, fields, files, records and databases (3) Bytes, bits, fields, records, files and databases (4) Bits, bytes, fields, records, files and databases 31. What is the full form of USB as used in computer related activities? (1) Universal Serial Bus (2) United Serial Bus (3) Ultra Security Block (4) Universal Security Block 32. Which of the following are the goals of higher education in India? (a) Access (b) Equity (c) Quality and Excellence (d) Relevance (e) Value based education (f) Compulsory and free education Select the correct answer from the code given below: (1) (a), (b), (c), (d) and (e) (2) (a), (b), (c), (d), (e) and (f) (3) (a), (b) and (e) only (4) (a), (b), (e) and (f) 33. Which of the following represents billion characters? (1) Kilobytes (2) Gigabytes (3) Terabytes (4) Megabytes 34. Who among the following can be removed by the President without Parliament’s resolution? (1) Chief Election Commissioner (2) Comptroller and Auditor - General (3) Judge of a High Court (4) Governor of a State 35. Which of the following pollutants is the major cause of respiratory diseases? (1) Carbon monoxide (2) Volatile organic compounds (3) Suspended fine particles (4) Nitrogen oxides 36. Which of the following domains is used for - profit businesses? (1) .edu (2) .com (3) .org (4) .net 37. Which of the following pollutant gases is not produced both naturally and as a result of industrial activity? (1) Methane (2) Carbon dioxide (3) Chlorofluoro carbons (4) Nitrous oxide 38. Which of the following has been ranked the best college in the country (2017) as per the National Institutional Ranking Framework (NIRF)? (1) Fergusson College, Pune (2) Maharaja’s College, Mysore (3) Miranda House, Delhi (4) St. Stephen’s College, Delhi 39. Assertion (A): In urban areas, smog episodes occur frequently in winters. Reason (R): In winters, a lot of biomass is burnt by people for heating purposes or to keep themselves warm. Choose the correct answer from the code given below: (1) (A) is true and (R) is false (2) Both (A) and (R) are false (3) Both (A) and (R) are true and (R) is the correct explanation of (A) (4) Both (A) and (R) are true but (R) is not the correct explanation of (A) 40. Among the following fuels of energy, which is the most environment friendly? (1) CNG (2) Hydrogen (3) Ethanol (4) Biogas Click here to know the Eligibility Criteria for NTA UGC NET December 2019 Exam 41. In which of the following arrangements a wider spectrum of ideas and issues may be made possible? (1) Conference (2) Symposium (3) Research Article (4) Workshop mode 42. A researcher attempts to evaluate the effect of method of feeding on anxiety - proneness of children. Which method of research would be appropriate for this? (1) Ex-post-facto method (2) Survey method (3) Case study method (4) Experimental method 43. Which of the following is susceptible to the issue of research ethics? (1) Choice of sampling techniques (2) Reporting of research findings (3) Inaccurate application of statistical techniques (4) Faulty research design 44. Which one of the following is a key behaviour in effective teaching? (1) Instructional variety (2) Questioning (3) Using student ideas and contribution (4) Structuring 45. From the list given below identify the learner characteristics which would facilitate teaching learning system to become effective. Choose the correct code to indicate your answer. (a) Prior experience of learner (b) Learner’s family lineage (c) Aptitude of the learner (d) Learner’s stage of development (e) Learner’s food habits and hobbies (f) Learner’s religious affiliation Code: (1) (a), (d) and (e) (2) (b), (c) and (f) (3) (a), (c) and (d) (4) (d), (e) and (f) 46. Which of the following set of statements best represents the nature and objective of teaching and learning? (a) Teaching is like selling and learning is like buying. (b) Teaching is a social act while learning is a personal act. (c) Teaching implies learning whereas learning does not imply teaching. (d) Teaching is a kind of delivery of knowledge while learning is like receiving it. (e) Teaching is an interaction and is triadic in nature whereas learning is an active engagement in a subject domain. Code: (1) (a), (b) and (c) (2) (a), (b) and (d) (3) (a), (d) and (e) (4) (b), (c) and (e) 47. Which of the following research types focuses on ameliorating the prevailing situations? (1) Action Research (2) Experimental Research (3) Fundamental Research (4) Applied Research 48. In finalizing a thesis writing format which of the following would form part of supplementary pages? (1) Conclusions of the study (2) Bibliography and Appendices (3) List of tables and figures 49. Assertion (A): All teaching implies learning. Reason (R): Learning to be useful must be derived from teaching. Choose the correct answer from the following: (1) (A) is true, but (R) is false. (2) (A) is false, but (R) is true. (3) Both (A) and (R) are true and (R) is the correct explanation of (A). (4) Both (A) and (R) are true but (R) is not the correct explanation of (A). 50. On the basis of summative tests, a teacher is interpreting his/her students, performance in terms of their wellness life style evident in behaviour. This will be called: (1) Norm - referenced testing (2) Criterion - referenced testing (3) Formative testing (4) Continuous and comprehensive evaluation Q. No Ans. No. Q. No Ans. No. 1 3 26 2 2 3 27 3 3 1 28 2 4 2 29 1 5 4 30 4 6 3 31 1 7 1 32 1 8 3 33 2 9 1 34 4 10 3 35 3 11 2 36 2 12 1 37 3 13 1 38 3 14 2 39 4 15 4 40 2 16 2 41 1 17 3 42 1 18 3 43 2 19 9 44 1 20 3 45 3 21 1 46 4 22 3 47 1 23 3 48 2 24 4 49 1 25 1 50 2 Practice makes the man perfect! The more you will practice, the more accuracy you will gain which will eventually lead you to a high score in the exam. Practice will help you in avoiding silly mistakes and making unnecessary guess works while attempting NTA UGC NET December 2019 Exam. Therefore, practicing previous year papers will help you in achieving accuracy and high score in NTA UGC NET December 2019 Exam. Comment (0) ### Post Comment 0 + 7 = Post Disclaimer: Comments will be moderated by Jagranjosh editorial team. Comments that are abusive, personal, incendiary or irrelevant will not be published. Please use a genuine email ID and provide your name, to avoid rejection.	4,309	14,714	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.890625	4	CC-MAIN-2021-49	latest	en	0.887718	# UGC NET November 2017 Paper-I Set-D Previous Year Paper with Answers. Created On: Nov 22, 2019 12:55 IST. UGC NET November 2017 Paper-I Set-D Previous Year Paper with Answers. For cracking NTA UGC NET December 2019 Exam, candidates must practice the previous year papers of the different subjects for which they are applying this year. It will help them in improving their speed of attempting maximum questions in minimum time with accuracy. So, in this article we have shared the UGC NET November 2017 Paper I (Set-D) Previous Year Paper alongwith their answers. Let's first look at the Exam Pattern of Paper-I NTA UGC NET 2019 Exam:. NTA UGC NET Dec 2019 Paper-I Exam Pattern Part Sections Questions Marks I Teaching Aptitude 5 10 II Research Aptitude 5 10 III Reading Comprehension 5 10 IV Communication 5 10 V Reasoning (including Maths) 5 10 VI Logical Reasoning 5 10 VII Data Interpretation 5 10 VIII Information & Communication Technology (ICT) 5 10 IX People & Environment 5 10 X Higher Education System: Governance, Polity & Administration 5 10 Total 50 100. ## UGC NET November 2017 Paper I (Set-D) Previous Year Paper with Answers. 1. Just as melting ice - cubes do not cause a glass of water to overflow, melting sea - ice does not increase oceanic volume.. What type of argument is it?. (1) Psychological. (2) Statistical. (3) Analogical. (4) Hypothetical. 2. A postman walked 20 m straight from his office, turned right and walked 10 m. After turning left he walked 10 m and after turning right walked 20 m. He again turned right and walked 70 m. How far he is from his office?. (1) 60 m.. (2) 20 m.. (3) 50 m.. (4) 40 m.. 3. Given below are two premises (a and b). From those two premises four conclusions (i), (ii), (iii) and (iv) are drawn. Select the code that states the conclusion/conclusions drawn validly (taking the premises singularly or jointly).. Premises:. (a) All bats are mammals.. (b) No birds are bats.. Conclusions:. (i) No birds are mammals.. (ii) Some birds are not mammals.. (iii) No bats are birds.. (iv) All mammals are bats.. Code:. (1) (iii) only. (2) (iii) and (iv) only. (3) (i) only. (4) (i) and (ii) only. 4. A good communicator begins his/her presentation with a:. (1) Repetitive phrase. (2) Ice-breaker. (3) Complex question. (4) Non-sequitur. 5. The classroom communication should essentially be:. (1) Abstract. (2) Non-descriptive. (3) Contrived. (4) Empathetic. 6. In the series 1, 6, 15, 28, 45, ............ the next term will be:. (1) 56. (2) 84. (3) 66. (4) 76. 7. Ajay is a friend of Rakesh. Pointing to an old man Ajay asked Rakesh who is he? Rakesh said “His son is my son’s uncle”. The old man is related to Rakesh as:. (1) Father. (2) Uncle. (3) Grandfather. (4) Father-in-law. 8. The spatial audio reproduction in a classroom can reduce the students’:. (1) Motivation for excellence. (2) Interest in technology - orientation. (4) Respect for the teacher. 9. It is Truism to say that no one was there when life first appeared on earth. Any assertion about life’s origin, thus, should be treated as a theory.. The above two statements constitute:. (1) An argument. (2) A conjecture. (3) A historical explanation. (4) A narrative. 10. In a classroom, the probability of message reception can be enhanced by:. (2) Using high decibel audio tools. (3) Establishing a viewpoint. (4) Exposing the ignorance of students. Click here to know the latest Exam Pattern and Syllabus of NTA UGC NET December 2019 Exam. 11. Given below are four statements. Among them two are related in such a way that they can both be true but they cannot both be false. Select the code that indicates those two statements:. Statements:. (a) Honest people never suffer.. (b) Almost all honest people do suffer.. (c) Honest people hardly suffer.. (d) Each and every honest person suffers.. Code:. (1) (a) and (d). (2) (b) and (c). (3) (a) and (b). (4) (a) and (c). 12. In certain code, “COVALENT” is coded as BWPDUOFM. The code of “ELEPHANT” will be:. (1) QFMFUOBI. (2) EPHNTEAS. (3) MFUIQRTW. 13. The interaction between a teacher and students creates a zone of proximal:. (1) Development. (2) Distortion. (3) Difference. (4) Confusion. 14. A deductive argument is invalid if:. (1) Its premises are all false but its conclusion is true.. (2) Its premises are all true but its conclusion is false.. (3) Its premises and conclusion are all true.. (4) Its premises and conclusion are all false.. 15. The next term in the series ABD, DGK, HMS, MTB, .......... is:. (1) PSK. (2) RUH. (3) NSA. (4) SBL. Answer the questions 16 to 20 based on the data given in the table below.. Table: Number of registered vehicles in India and India’s population.. Year Total vehicles (Lakhs) Two wheelers (Lakhs) Cars, Jeeps, Taxis (Lakhs) Buses (Lakhs) Goods vehicles (Lakhs) Others (Lakhs) Population (India) (Millions) 1961 6.65 0.88 3.1 0.57 1.68 0.42 439.23 1971 18.65 5.76 6.82 0.94 3.43 1.70 548.15 1981 53.19 26.18 11.60 1.62 5.54 8.97 683.32 1991 213.74 142.00 29.54 3.31 13.56 25.33 846.42 2001 549.91 385.56 70.58 6.34 29.48 57.95 1028.73 2011 1417.58 1018.65 191.23 16.04 70.64 121.02 1210.19. 16. In which year the decadal growth (%) in number of cars surpassed that of the two wheelers?. (1) 1981. (2) 2011. (3) 1991. (4) 2001. 17. What was the average decadal growth in the number of cars during 1961 - 2011?. (1) ~ 217%. (2) ~ 157%. (3) ~ 131%. (4) ~ 68%. 18. In the year 2001, out of total number of vehicles, the number of passenger vehicles (4 wheelers) accounted for:. (1) ~ 31%. (2) ~ 43%. (3) ~ 14%. (4) ~ 24%. 19. What was the per capita ownership of two wheelers in India in the year 2011?. (1) ~ 0.84%. (2) ~ 0.068%. (3) ~ 0.084%. (4) ~ 0.0084%. 20. The maximum decadal growth in population of India is registered in the period:. (1) 2001 - 2011. (2) 1981 - 1991. (3) 1961 - 1971. (4) 1991 - 2001. Click here to know the UGC NET Exam Previous Year Cut-Off Marks. Read the passage carefully and answer question numbers from 21 to 25.. 21. The traditional knowledge should be used through:. (1) Synergy between government and local interventions. (2) Modern technology. (3) Its dissemination. (4) Improvement in national circumstances. 22. Given below are the factors of vulnerability of poor people to climate change. Select the code that contains the correct answer.. (a) Their dependence on natural resources. (b) Geographical attributes. (c) Lack of financial resources. Code:. (1) (a), (b), (c) and (d). (2) (c) only. (3) (a), (b) and (c). (4) (b), (c) and (d). 23. Adaptation as a process enables societies to cope with:. (a) An uncertain future. (c) Negative impact of climate change. (d) Positive impact of climate change. Select the most appropriate answer from the following code:. (1) (b), (c) and (d). (2) (c) only. (3) (a), (b), (c) and (d). (4) (a) and (c). 24. To address the challenge of climate change, developing countries urgently require:. (3) Imposition of climate change tax. (4) Implementation of national adaptation policy at their level. 25. The main focus of the passage is on:. (2) Social dimensions of climate change. (3) Combining traditional knowledge with appropriate technology. (4) Co-ordination between regional and national efforts. 26. Which of the following come(s) within the ambit of the term ‘corruption’?. (a) Misuse of official position. (b) Deviation from rules, laws and norms. (c) Non-action when action is required. (d) Harm to public good.	Select the correct answer from the code given below:. (1) (a), (b) and (d). (2) (a), (b), (c) and (d). (3) (a) only. (4) (a) and (b) only. 27. Which of the following universities has received the Visitor’s Award for the best Central University in India in Feb. 2017?. (1) Tezpur University. (3) Jawaharlal Nehru University. (4) Banaras Hindu University. 28. What is the name for a webpage address?. (1) Protocol. (2) URL. (3) Domain. (4) Directory. 29. Occurrence of natural hazards is affected by:. (a) Land use changes. (b) Drainage and construction. (c) Ozone depletion. (d) Climate change. Choose the correct answer from the code given below:. (1) (a), (b) and (d). (2) (b), (c) and (d). (3) (a), (c) and (d). (4) (a), (b) and (c). 30. The data storage hierarchy consists of:. (1) Bits, bytes, records, fields, files and databases. (2) Bits, bytes, fields, files, records and databases. (3) Bytes, bits, fields, records, files and databases. (4) Bits, bytes, fields, records, files and databases. 31. What is the full form of USB as used in computer related activities?. (1) Universal Serial Bus. (2) United Serial Bus. (3) Ultra Security Block. (4) Universal Security Block. 32. Which of the following are the goals of higher education in India?. (a) Access. (b) Equity. (c) Quality and Excellence. (d) Relevance. (e) Value based education. (f) Compulsory and free education. Select the correct answer from the code given below:. (1) (a), (b), (c), (d) and (e). (2) (a), (b), (c), (d), (e) and (f). (3) (a), (b) and (e) only. (4) (a), (b), (e) and (f). 33. Which of the following represents billion characters?. (1) Kilobytes. (2) Gigabytes. (3) Terabytes. (4) Megabytes. 34. Who among the following can be removed by the President without Parliament’s resolution?. (1) Chief Election Commissioner. (2) Comptroller and Auditor - General. (3) Judge of a High Court. (4) Governor of a State. 35. Which of the following pollutants is the major cause of respiratory diseases?. (1) Carbon monoxide. (2) Volatile organic compounds. (3) Suspended fine particles. (4) Nitrogen oxides. 36. Which of the following domains is used for - profit businesses?. (1) .edu. (2) .com. (3) .org. (4) .net. 37. Which of the following pollutant gases is not produced both naturally and as a result of industrial activity?. (1) Methane. (2) Carbon dioxide. (3) Chlorofluoro carbons. (4) Nitrous oxide. 38. Which of the following has been ranked the best college in the country (2017) as per the National Institutional Ranking Framework (NIRF)?. (1) Fergusson College, Pune. (2) Maharaja’s College, Mysore. (3) Miranda House, Delhi. (4) St. Stephen’s College, Delhi. 39. Assertion (A): In urban areas, smog episodes occur frequently in winters.. Reason (R): In winters, a lot of biomass is burnt by people for heating purposes or to keep themselves warm.. Choose the correct answer from the code given below:. (1) (A) is true and (R) is false. (2) Both (A) and (R) are false. (3) Both (A) and (R) are true and (R) is the correct explanation of (A). (4) Both (A) and (R) are true but (R) is not the correct explanation of (A). 40. Among the following fuels of energy, which is the most environment friendly?. (1) CNG. (2) Hydrogen. (3) Ethanol. (4) Biogas. Click here to know the Eligibility Criteria for NTA UGC NET December 2019 Exam. 41. In which of the following arrangements a wider spectrum of ideas and issues may be made possible?. (1) Conference. (2) Symposium. (3) Research Article. (4) Workshop mode. 42. A researcher attempts to evaluate the effect of method of feeding on anxiety - proneness of children. Which method of research would be appropriate for this?. (1) Ex-post-facto method. (2) Survey method. (3) Case study method. (4) Experimental method. 43. Which of the following is susceptible to the issue of research ethics?. (1) Choice of sampling techniques. (2) Reporting of research findings. (3) Inaccurate application of statistical techniques. (4) Faulty research design. 44. Which one of the following is a key behaviour in effective teaching?. (1) Instructional variety. (2) Questioning. (3) Using student ideas and contribution. (4) Structuring. 45. From the list given below identify the learner characteristics which would facilitate teaching learning system to become effective. Choose the correct code to indicate your answer.. (a) Prior experience of learner. (b) Learner’s family lineage. (c) Aptitude of the learner. (d) Learner’s stage of development. (e) Learner’s food habits and hobbies. (f) Learner’s religious affiliation. Code:. (1) (a), (d) and (e). (2) (b), (c) and (f). (3) (a), (c) and (d). (4) (d), (e) and (f). 46. Which of the following set of statements best represents the nature and objective of teaching and learning?. (a) Teaching is like selling and learning is like buying.. (b) Teaching is a social act while learning is a personal act.. (c) Teaching implies learning whereas learning does not imply teaching.. (d) Teaching is a kind of delivery of knowledge while learning is like receiving it.. (e) Teaching is an interaction and is triadic in nature whereas learning is an active engagement in a subject domain.. Code:. (1) (a), (b) and (c). (2) (a), (b) and (d). (3) (a), (d) and (e). (4) (b), (c) and (e). 47. Which of the following research types focuses on ameliorating the prevailing situations?. (1) Action Research. (2) Experimental Research. (3) Fundamental Research. (4) Applied Research. 48. In finalizing a thesis writing format which of the following would form part of supplementary pages?. (1) Conclusions of the study. (2) Bibliography and Appendices. (3) List of tables and figures. 49. Assertion (A): All teaching implies learning.. Reason (R): Learning to be useful must be derived from teaching.. Choose the correct answer from the following:. (1) (A) is true, but (R) is false.. (2) (A) is false, but (R) is true.. (3) Both (A) and (R) are true and (R) is the correct explanation of (A).. (4) Both (A) and (R) are true but (R) is not the correct explanation of (A).. 50. On the basis of summative tests, a teacher is interpreting his/her students, performance in terms of their wellness life style evident in behaviour. This will be called:. (1) Norm - referenced testing. (2) Criterion - referenced testing. (3) Formative testing. (4) Continuous and comprehensive evaluation. Q. No Ans. No. Q. No Ans. No. 1 3 26 2 2 3 27 3 3 1 28 2 4 2 29 1 5 4 30 4 6 3 31 1 7 1 32 1 8 3 33 2 9 1 34 4 10 3 35 3 11 2 36 2 12 1 37 3 13 1 38 3 14 2 39 4 15 4 40 2 16 2 41 1 17 3 42 1 18 3 43 2 19 9 44 1 20 3 45 3 21 1 46 4 22 3 47 1 23 3 48 2 24 4 49 1 25 1 50 2. Practice makes the man perfect! The more you will practice, the more accuracy you will gain which will eventually lead you to a high score in the exam. Practice will help you in avoiding silly mistakes and making unnecessary guess works while attempting NTA UGC NET December 2019 Exam. Therefore, practicing previous year papers will help you in achieving accuracy and high score in NTA UGC NET December 2019 Exam.. Comment (0). ### Post Comment. 0 + 7 =. Post. Disclaimer: Comments will be moderated by Jagranjosh editorial team. Comments that are abusive, personal, incendiary or irrelevant will not be published. Please use a genuine email ID and provide your name, to avoid rejection.
https://statisticool.com/martialarts/maai.html	1,716,064,624,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971057494.65/warc/CC-MAIN-20240518183301-20240518213301-00156.warc.gz	466,635,030	4,974	# Mathematical Combat Distance 1/27/07 Intuitively, one understands that danger is inversely proportional to the distance from a threat. In the martial arts, this concept of "combative distance" is crucial. This informal article will outline one way of looking at combat distance. I've personally found it useful, and hope others can. Let's represent each Personi by a circle of radius Legi and concentric circle of radius Armi. The center of the circle is the center of a person; their center of gravity, core, tanden, tantien, etc. Each radius is the reach of the arm and leg respectively, and the extent of each reach is marked by the edge of the circle. Note that these distances can be measured for any person. Simply stand stationary, extend a limb, and have someone measure the distance the limb extends, using a measuring tape. What happens when two people, who are a distance, d, apart, approach each other? The outcome (keep in mind our assumptions, listed at the end of the article), hinges on two factors 1. How does Leg1 compare to Leg2? 2. How does Arm1 compare to Arm2? Consider If these people approach (cross your eyes to make them approach), Person1 is able to touch Person2's center first or able to "cover" a larger fraction of Person2. How can we mathematically express this? Note that when Person1 is able to do this, more of Person2 is overlapped by Person1 than vice versa. This overlap, expressed mathematically as a measure of area, is A = (1/2) * [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2 Therefore, the percentage overlap of Person2 is the overlap divided by Person2's area, or (((1/2) * [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2)) / (piLeg22) Let's label this number by LegDanger2, and note that if LegDangeri > LegDangerj, then Personi is in more trouble than Personj from leg attacks. Note, however, that if Person1's leg attack fails, then Person2's arm attack is able to get to Person1's center first, barely. This is why one must take into account both leg and arm attacks. The above formula for A is calculated using the Arm radii, and we get ArmDanger1 and ArmDanger2 respectively. Therefore, Person1's total danger is Danger1 = LegDanger1 + ArmDanger1 Written out in full, this is Danger1 = [(((1/2) [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2)) / (piLeg12)] + [(((1/2) [(-d+Arm1+Arm2)(d+Arm1-Arm2)(d-Arm1+Arm2)(d+Arm1+Arm2)]1/2)) / (pi*Arm12)] For a given time in an encounter, one can compare Danger1 and Danger2. If Danger1 > Danger2, then Person1 is, theoretically, in trouble. The discussion has been held to open-hand combat. Most weapons could be included in the Arm radius. However, a weapon like a gun would have a huge radius, hundreds of feet, and its circle would easily overlap any non-weapon carrying person. What happens with multiple opponents? Say Person1 is being attacked by Person2 and Person3 To assess danger, we'd compare Danger1 to Danger2 + Danger3. In this article, there are several assumptions that are needed so we can talk about such a simple model • model is 2-dimensional. One could consider doing the same analysis using spheres and volumes • each person can attack and defend 360 degrees • beside arm and leg reach, each person is equal in all other aspects • attacking or controlling the opponent's center is the goal, and is most devastating I think looking at overlapping circles of various radii is the simplest case of developing a model to help explain combat distance. It has the bonus of everything being measurable, and explains the common sense of why fighting against weapons and multiple attackers is very, very difficult.	948	3,684	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.28125	4	CC-MAIN-2024-22	latest	en	0.930797	# Mathematical Combat Distance. 1/27/07. Intuitively, one understands that danger is inversely proportional to the distance from a threat. In the martial arts, this concept of "combative distance" is crucial.. This informal article will outline one way of looking at combat distance. I've personally found it useful, and hope others can.. Let's represent each Personi by a circle of radius Legi and concentric circle of radius Armi. The center of the circle is the center of a person; their center of gravity, core, tanden, tantien, etc.. Each radius is the reach of the arm and leg respectively, and the extent of each reach is marked by the edge of the circle. Note that these distances can be measured for any person. Simply stand stationary, extend a limb, and have someone measure the distance the limb extends, using a measuring tape.. What happens when two people, who are a distance, d, apart, approach each other? The outcome (keep in mind our assumptions, listed at the end of the article), hinges on two factors. 1. How does Leg1 compare to Leg2?. 2. How does Arm1 compare to Arm2?. Consider. If these people approach (cross your eyes to make them approach), Person1 is able to touch Person2's center first or able to "cover" a larger fraction of Person2. How can we mathematically express this?. Note that when Person1 is able to do this, more of Person2 is overlapped by Person1 than vice versa. This overlap, expressed mathematically as a measure of area, is. A = (1/2) * [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2. Therefore, the percentage overlap of Person2 is the overlap divided by Person2's area, or. (((1/2) * [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2)) / (pi*Leg22).	Let's label this number by LegDanger2, and note that if LegDangeri > LegDangerj, then Personi is in more trouble than Personj from leg attacks.. Note, however, that if Person1's leg attack fails, then Person2's arm attack is able to get to Person1's center first, barely. This is why one must take into account both leg and arm attacks. The above formula for A is calculated using the Arm radii, and we get ArmDanger1 and ArmDanger2 respectively.. Therefore, Person1's total danger is. Danger1 = LegDanger1 + ArmDanger1. Written out in full, this is. Danger1 = [(((1/2) * [(-d+Leg1+Leg2)(d+Leg1-Leg2)(d-Leg1+Leg2)(d+Leg1+Leg2)]1/2)) / (piLeg12)] + [(((1/2) [(-d+Arm1+Arm2)(d+Arm1-Arm2)(d-Arm1+Arm2)(d+Arm1+Arm2)]1/2)) / (pi*Arm12)]. For a given time in an encounter, one can compare Danger1 and Danger2. If Danger1 > Danger2, then Person1 is, theoretically, in trouble.. The discussion has been held to open-hand combat. Most weapons could be included in the Arm radius. However, a weapon like a gun would have a huge radius, hundreds of feet, and its circle would easily overlap any non-weapon carrying person.. What happens with multiple opponents? Say Person1 is being attacked by Person2 and Person3. To assess danger, we'd compare Danger1 to Danger2 + Danger3.. In this article, there are several assumptions that are needed so we can talk about such a simple model. • model is 2-dimensional. One could consider doing the same analysis using spheres and volumes. • each person can attack and defend 360 degrees. • beside arm and leg reach, each person is equal in all other aspects. • attacking or controlling the opponent's center is the goal, and is most devastating. I think looking at overlapping circles of various radii is the simplest case of developing a model to help explain combat distance. It has the bonus of everything being measurable, and explains the common sense of why fighting against weapons and multiple attackers is very, very difficult.
https://civilblog.org/2014/04/10/volume-batching-of-concrete-worked-out-example/	1,656,857,984,000,000,000	text/html	crawl-data/CC-MAIN-2022-27/segments/1656104244535.68/warc/CC-MAIN-20220703134535-20220703164535-00670.warc.gz	225,950,845	17,674	# VOLUME BATCHING OF CONCRETE – WORKED OUT EXAMPLE In my previous post I discussed with you about volume batching of concrete and how it is done in field. Now let us solve an example, so that you become more confident in the field. In this example I have also tried o show you, how correction for bulking is applied and how correction of any surface moisture is applied. ## Data Given • Relative proportion of cement : sand : coarse aggregate is 1:2:4 • Water-cement ratio is 0.6 • Moisture content in fine aggregate (i.e. sand) is 6% by volume • Moisture content in coarse aggregate is 1.5% by volume • Bulking of fine aggregate is 20% ## Solution Step-1-Calculating Volume of Dry material Required Ratio of materials by volume =1:2:4 Volume of one bag of cement =35 litres Volume of dry fine aggregate required =352=70 litres Volume of dry coarse aggregate required =354=140 litres Water-cement ratio =0.60 Total water required =0.6050=30 litres Step-2-Applying Correction for Moisture Present in Aggregate Moisture percent in fine aggregate (F.A.) =6% Amount of moisture present in F.A. =70(6/100)=4.2 litres Moisture percent in coarse aggregate (C.A.) =1.5% Amount of moisture present in C.A. =140(1.5/100)=2.1 litres So extra amount of moisture present in C.A & F.A. =4.2+2.1=6.3 litres Net water to be added =Total water-extra amount of water present as moisture content=30-6.3 =23.7 litres Step-3-Applying Correction for Bulking of fine aggregate Bulking of fine aggregate =20% of total volume of F.A.=70(20/100) =14 litres Moist fine aggregate required =70+14=84 litres One Comment	428	1,604	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.125	4	CC-MAIN-2022-27	latest	en	0.879285	# VOLUME BATCHING OF CONCRETE – WORKED OUT EXAMPLE. In my previous post I discussed with you about volume batching of concrete and how it is done in field.. Now let us solve an example, so that you become more confident in the field. In this example I have also tried o show you, how correction for bulking is applied and how correction of any surface moisture is applied.. ## Data Given. • Relative proportion of cement : sand : coarse aggregate is 1:2:4. • Water-cement ratio is 0.6. • Moisture content in fine aggregate (i.e. sand) is 6% by volume.	• Moisture content in coarse aggregate is 1.5% by volume. • Bulking of fine aggregate is 20%. ## Solution. Step-1-Calculating Volume of Dry material Required Ratio of materials by volume =1:2:4 Volume of one bag of cement =35 litres Volume of dry fine aggregate required =352=70 litres Volume of dry coarse aggregate required =354=140 litres Water-cement ratio =0.60 Total water required =0.6050=30 litres Step-2-Applying Correction for Moisture Present in Aggregate Moisture percent in fine aggregate (F.A.) =6% Amount of moisture present in F.A. =70(6/100)=4.2 litres Moisture percent in coarse aggregate (C.A.) =1.5% Amount of moisture present in C.A. =140(1.5/100)=2.1 litres So extra amount of moisture present in C.A & F.A. =4.2+2.1=6.3 litres Net water to be added =Total water-extra amount of water present as moisture content=30-6.3 =23.7 litres Step-3-Applying Correction for Bulking of fine aggregate Bulking of fine aggregate =20% of total volume of F.A.=70(20/100) =14 litres Moist fine aggregate required =70+14=84 litres. One Comment.
http://gmatclub.com/forum/quant-question-answer-pls-122802.html	1,472,516,868,000,000,000	text/html	crawl-data/CC-MAIN-2016-36/segments/1471982967797.63/warc/CC-MAIN-20160823200927-00026-ip-10-153-172-175.ec2.internal.warc.gz	106,338,724	49,588	Find all School-related info fast with the new School-Specific MBA Forum It is currently 29 Aug 2016, 17:27 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # Events & Promotions ###### Events & Promotions in June Open Detailed Calendar Author Message TAGS: ### Hide Tags Intern Joined: 12 Oct 2011 Posts: 3 Followers: 1 Kudos [?]: 1 [0], given: 0 ### Show Tags 06 Nov 2011, 10:03 For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3x5x5. How many two-digit positive integers have length 6 ? A) None B) One C) Two D) Three E) Four Intern Status: Accepted - SMU, Singapore (Class of 2013) Joined: 08 Sep 2011 Posts: 27 Location: Singapore Concentration: Finance, General Management GMAT 1: 620 Q46 V29 GMAT 2: 710 Q50 V35 WE: Information Technology (Computer Software) Followers: 3 Kudos [?]: 13 [0], given: 0 ### Show Tags 06 Nov 2011, 10:10 rvind wrote: For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3x5x5. How many two-digit positive integers have length 6 ? A) None B) One C) Two D) Three E) Four 2x2x2x2x2x2=64 2x2x2x2x2x3=96 hence two such two-digit numbers Ans. C _________________ p!yu\$h http://gmatclub.com/forum/from-620-to-710-my-gmat-story-test-date-30-oct-122547.html#p995276 Manager Status: SC SC SC SC SC.... Concentrating on SC alone. Joined: 20 Dec 2010 Posts: 240 Location: India Concentration: General Management GMAT Date: 12-30-2011 Followers: 3 Kudos [?]: 58 [0], given: 47 ### Show Tags 06 Nov 2011, 10:12 It is two. C The minimal possible value for a length of 6 is by taking all the primes as 2. 2 * 2 * 2 * 2 * 2 * 2 = 64. The next alteration we do is to replace a number by 3. 2 * 2 * 2 * 2 * 2 * 3 = 96. So it is still possible. Another replacement of 3 will make it a 3 digit. _________________ D- Day December 30 2011. Hoping for the happiest new year celebrations ! Aiming for 700+ Kudo me if the post is worth it Similar topics Replies Last post Similar Topics: 1 quants question--Pls help solve? 1 17 Nov 2014, 10:38 4 Standard 700 level quantitaive questions with answers 2 16 Jan 2013, 11:19 54 436 Amazing GMAT Data Sufficiency Questions with Answers 12 29 Nov 2012, 07:29 PS1000 Questions and Answers 3 11 Mar 2011, 02:44 Converting Questions answered correctly to scaled score 2 09 Aug 2010, 17:11 Display posts from previous: Sort by	915	2,948	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2016-36	longest	en	0.82954	Find all School-related info fast with the new School-Specific MBA Forum. It is currently 29 Aug 2016, 17:27. ### GMAT Club Daily Prep. #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.. Customized. for You. we will pick new questions that match your level based on your Timer History. Track. every week, we’ll send you an estimated GMAT score based on your performance. Practice. Pays. we will pick new questions that match your level based on your Timer History. # Events & Promotions. ###### Events & Promotions in June. Open Detailed Calendar. Author Message. TAGS:. ### Hide Tags. Intern. Joined: 12 Oct 2011. Posts: 3. Followers: 1. Kudos [?]: 1 [0], given: 0. ### Show Tags. 06 Nov 2011, 10:03. For any positive integer n, the length of n is defined as the number of prime factors whose product is n. For example, the length of 75 is 3, since 75 = 3x5x5. How many two-digit positive integers have length 6 ?. A) None. B) One. C) Two. D) Three. E) Four. Intern. Status: Accepted - SMU, Singapore (Class of 2013). Joined: 08 Sep 2011. Posts: 27. Location: Singapore. Concentration: Finance, General Management. GMAT 1: 620 Q46 V29. GMAT 2: 710 Q50 V35. WE: Information Technology (Computer Software). Followers: 3. Kudos [?]: 13 [0], given: 0. ### Show Tags. 06 Nov 2011, 10:10. rvind wrote:. For any positive integer n, the length of n is defined as the number of prime factors whose product is n.	For example, the length of 75 is 3, since 75 = 3x5x5. How many two-digit positive integers have length 6 ?. A) None. B) One. C) Two. D) Three. E) Four. 2x2x2x2x2x2=64. 2x2x2x2x2x3=96. hence two such two-digit numbers. Ans. C. _________________. p!yu\$h. http://gmatclub.com/forum/from-620-to-710-my-gmat-story-test-date-30-oct-122547.html#p995276. Manager. Status: SC SC SC SC SC.... Concentrating on SC alone.. Joined: 20 Dec 2010. Posts: 240. Location: India. Concentration: General Management. GMAT Date: 12-30-2011. Followers: 3. Kudos [?]: 58 [0], given: 47. ### Show Tags. 06 Nov 2011, 10:12. It is two. C. The minimal possible value for a length of 6 is by taking all the primes as 2.. 2 * 2 * 2 * 2 * 2 * 2 = 64.. The next alteration we do is to replace a number by 3.. 2 * 2 * 2 * 2 * 2 * 3 = 96.. So it is still possible. Another replacement of 3 will make it a 3 digit.. _________________. D- Day December 30 2011. Hoping for the happiest new year celebrations !. Aiming for 700+. Kudo me if the post is worth it. Similar topics Replies Last post. Similar. Topics:. 1 quants question--Pls help solve? 1 17 Nov 2014, 10:38. 4 Standard 700 level quantitaive questions with answers 2 16 Jan 2013, 11:19. 54 436 Amazing GMAT Data Sufficiency Questions with Answers 12 29 Nov 2012, 07:29. PS1000 Questions and Answers 3 11 Mar 2011, 02:44. Converting Questions answered correctly to scaled score 2 09 Aug 2010, 17:11. Display posts from previous: Sort by.
https://www.gradesaver.com/textbooks/math/algebra/introductory-algebra-for-college-students-7th-edition/chapter-6-section-6-4-factoring-special-forms-exercise-set-page-454/33	1,537,367,383,000,000,000	text/html	crawl-data/CC-MAIN-2018-39/segments/1537267156252.31/warc/CC-MAIN-20180919141825-20180919161825-00222.warc.gz	750,376,072	13,897	## Introductory Algebra for College Students (7th Edition) $3x(x^2+9)$ Factor out $3x$ to obtain: $=3x(x^2+9)$ The binomial factor is a sum of two square, which is prime. Thus, the factored form of the given polynomial is : $=3x(x^2+9)$	79	237	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2018-39	longest	en	0.843604	## Introductory Algebra for College Students (7th Edition). $3x(x^2+9)$. Factor out $3x$ to obtain: $=3x(x^2+9)$ The binomial factor is a sum of two square, which is prime.	Thus, the factored form of the given polynomial is : $=3x(x^2+9)$.
https://socratic.org/questions/what-is-the-integral-of-1-1-x-2#636724	1,669,735,246,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446710698.62/warc/CC-MAIN-20221129132340-20221129162340-00412.warc.gz	562,600,972	6,211	What is the integral of 1/(1+x^2)? Jun 28, 2018 $\int \frac{1}{1 + {x}^{2}} \mathrm{dx} = {\tan}^{-} 1 x + C$ Explanation: color(blue)(int(du)/(1+u^2)=tan^-1u+C$\rightarrow$ Where $u$ is a function of $x$ $\textcolor{red}{\text{Proof:}}$ $\int \frac{\mathrm{du}}{1 + {u}^{2}}$ $u = \tan \theta$$\rightarrow$$\mathrm{du} = {\sec}^{2} \theta d \left(\theta\right)$ int(du)/(1+u^2)=int(sec^2thetad(theta))/(1+tan^2theta color(green)(sec^2theta=1+tan^2theta $\int \frac{{\sec}^{2} \theta d \left(\theta\right)}{1 + {\tan}^{2} \theta} = \int \frac{\left(\cancel{1 + {\tan}^{2} \theta}\right) d \left(\theta\right)}{\cancel{1 + {\tan}^{2} \theta}}$ $= \int d \left(\theta\right) = \theta$ Reverse the Substitution $u = \tan \theta$color(red)(rarr$\theta = {\tan}^{-} 1 u$ $\therefore \int \frac{\mathrm{du}}{1 + {u}^{2}} = {\tan}^{-} 1 u + C$ Simply by Substituting in this relation $\int \frac{\mathrm{dx}}{1 + {x}^{2}} = {\tan}^{-} 1 x + C$ Jun 28, 2018 $\arctan x + C$. Explanation: It is one of the Standard Integral : $\int \frac{1}{1 + {x}^{2}} = \arctan x + C$. Aliter : Let, $I = \int \frac{1}{1 + {x}^{2}} \mathrm{dx}$. Subst. x=tanu. :. dx=sec^2udu, &, u=arctanx. $\therefore I = \int \frac{1}{1 + {\tan}^{2} u} {\sec}^{2} u \mathrm{du}$, $= \int \frac{1}{\sec} ^ 2 u {\sec}^{2} u \mathrm{du}$, $= \int 1 \mathrm{du}$, $= u$. $\Rightarrow I = \arctan x + C$.	592	1,390	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 28, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.40625	4	CC-MAIN-2022-49	latest	en	0.370331	What is the integral of 1/(1+x^2)?. Jun 28, 2018. $\int \frac{1}{1 + {x}^{2}} \mathrm{dx} = {\tan}^{-} 1 x + C$. Explanation:. color(blue)(int(du)/(1+u^2)=tan^-1u+C$\rightarrow$ Where $u$ is a function of $x$. $\textcolor{red}{\text{Proof:}}$. $\int \frac{\mathrm{du}}{1 + {u}^{2}}$. $u = \tan \theta$$\rightarrow$$\mathrm{du} = {\sec}^{2} \theta d \left(\theta\right)$. int(du)/(1+u^2)=int(sec^2thetad(theta))/(1+tan^2theta. color(green)(sec^2theta=1+tan^2theta. $\int \frac{{\sec}^{2} \theta d \left(\theta\right)}{1 + {\tan}^{2} \theta} = \int \frac{\left(\cancel{1 + {\tan}^{2} \theta}\right) d \left(\theta\right)}{\cancel{1 + {\tan}^{2} \theta}}$. $= \int d \left(\theta\right) = \theta$. Reverse the Substitution. $u = \tan \theta$color(red)(rarr$\theta = {\tan}^{-} 1 u$. $\therefore \int \frac{\mathrm{du}}{1 + {u}^{2}} = {\tan}^{-} 1 u + C$. Simply by Substituting in this relation.	$\int \frac{\mathrm{dx}}{1 + {x}^{2}} = {\tan}^{-} 1 x + C$. Jun 28, 2018. $\arctan x + C$.. Explanation:. It is one of the Standard Integral : $\int \frac{1}{1 + {x}^{2}} = \arctan x + C$.. Aliter :. Let, $I = \int \frac{1}{1 + {x}^{2}} \mathrm{dx}$.. Subst. x=tanu. :. dx=sec^2udu, &, u=arctanx.. $\therefore I = \int \frac{1}{1 + {\tan}^{2} u} {\sec}^{2} u \mathrm{du}$,. $= \int \frac{1}{\sec} ^ 2 u {\sec}^{2} u \mathrm{du}$,. $= \int 1 \mathrm{du}$,. $= u$.. $\Rightarrow I = \arctan x + C$.
https://www.maplesoft.com/support/help/view.aspx?path=describe(deprecated)%2Fsumdata	1,721,855,524,000,000,000	text/html	crawl-data/CC-MAIN-2024-30/segments/1720763518454.54/warc/CC-MAIN-20240724202030-20240724232030-00655.warc.gz	771,987,602	21,381	describe(deprecated)/sumdata - Maple Help stats[describe] sumdata Sum powers of Statistical List Calling Sequence stats[describe, sumdata[which, origin]](data) describe[sumdata[which, origin, Nconstraint]](data) Parameters data - statistical list which - (optional, default=1) value that specifies the required power origin - (optional, default=0) the quantity about which the moment is computed. Description • Important: The stats package has been deprecated. Use the superseding package Statistics instead. • The function sumdata of the subpackage stats[describe, ...] computes sums of the various powers of the given data about any origin. • The sum of r-th power about an origin R is computed as follows: S_R:= sum( (X-R)^r ). • The function sumdata is closely related to the function moment. • The value of origin can be either a number, or the various descriptive statistic functions that can be specified via stats[describe, descriptive statistic function]. • Classes are assumed to be represented by the class mark, for example 10..12 has the value 11. Missing data are ignored. • The command with(stats[describe],sumdata) allows the use of the abbreviated form of this command. Examples Important: The stats package has been deprecated. Use the superseding package Statistics instead. > $\mathrm{with}\left(\mathrm{stats}\right):$ > $\mathrm{data}≔\left[1,3,5\right]$ ${\mathrm{data}}{≔}\left[{1}{,}{3}{,}{5}\right]$ (1) Sum of third power about 0: > $\mathrm{describe}\left[\mathrm{sumdata}\left[3\right]\right]\left(\mathrm{data}\right)={1}^{3}+{3}^{3}+{5}^{3}$ ${153}{=}{153}$ (2) Sum of fourth power about 1: > $\mathrm{describe}\left[\mathrm{sumdata}\left[4,1\right]\right]\left(\mathrm{data}\right)={\left(1-1\right)}^{4}+{\left(3-1\right)}^{4}+{\left(5-1\right)}^{4}$ ${272}{=}{272}$ (3) Sum of fifth power about the mean > $\mathrm{describe}\left[\mathrm{moment}\left[5,\mathrm{mean}\right]\right]\left(\mathrm{data}\right)={\left(1-3\right)}^{5}+{\left(3-3\right)}^{5}+{\left(5-3\right)}^{5}$ ${0}{=}{0}$ (4)	608	2,064	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 9, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5625	4	CC-MAIN-2024-30	latest	en	0.533456	describe(deprecated)/sumdata - Maple Help. stats[describe]. sumdata. Sum powers of Statistical List. Calling Sequence stats[describe, sumdata[which, origin]](data) describe[sumdata[which, origin, Nconstraint]](data). Parameters. data - statistical list which - (optional, default=1) value that specifies the required power origin - (optional, default=0) the quantity about which the moment is computed.. Description. • Important: The stats package has been deprecated. Use the superseding package Statistics instead.. • The function sumdata of the subpackage stats[describe, ...] computes sums of the various powers of the given data about any origin.. • The sum of r-th power about an origin R is computed as follows: S_R:= sum( (X-R)^r ).. • The function sumdata is closely related to the function moment.. • The value of origin can be either a number, or the various descriptive statistic functions that can be specified via stats[describe, descriptive statistic function].. • Classes are assumed to be represented by the class mark, for example 10..12 has the value 11. Missing data are ignored.. • The command with(stats[describe],sumdata) allows the use of the abbreviated form of this command.	Examples. Important: The stats package has been deprecated. Use the superseding package Statistics instead.. > $\mathrm{with}\left(\mathrm{stats}\right):$. > $\mathrm{data}≔\left[1,3,5\right]$. ${\mathrm{data}}{≔}\left[{1}{,}{3}{,}{5}\right]$ (1). Sum of third power about 0:. > $\mathrm{describe}\left[\mathrm{sumdata}\left[3\right]\right]\left(\mathrm{data}\right)={1}^{3}+{3}^{3}+{5}^{3}$. ${153}{=}{153}$ (2). Sum of fourth power about 1:. > $\mathrm{describe}\left[\mathrm{sumdata}\left[4,1\right]\right]\left(\mathrm{data}\right)={\left(1-1\right)}^{4}+{\left(3-1\right)}^{4}+{\left(5-1\right)}^{4}$. ${272}{=}{272}$ (3). Sum of fifth power about the mean. > $\mathrm{describe}\left[\mathrm{moment}\left[5,\mathrm{mean}\right]\right]\left(\mathrm{data}\right)={\left(1-3\right)}^{5}+{\left(3-3\right)}^{5}+{\left(5-3\right)}^{5}$. ${0}{=}{0}$ (4).
https://en.wikibooks.org/wiki/Physics_with_Calculus/Mechanics/Work_and_Energy	1,679,933,817,000,000,000	text/html	crawl-data/CC-MAIN-2023-14/segments/1679296948673.1/warc/CC-MAIN-20230327154814-20230327184814-00303.warc.gz	260,367,953	14,426	# Physics with Calculus/Mechanics/Work and Energy ## Work Work is a special name given to the (scalar) quantity ${\displaystyle W=\int {\vec {F}}\cdot d{\vec {x}}.}$ where ${\displaystyle W}$ is work, ${\displaystyle F}$ is force on the object and ${\displaystyle x}$ is displacement. Since the dot product is a projection, the work is the component of the force in the direction of the displacement times the displacement. If the force is constant and the object travels in a straight line, this reduces to ${\displaystyle W={\vec {F}}\cdot {\vec {x}}=Fx\cos \theta \ }$ where ${\displaystyle W}$ is work and ${\displaystyle F}$ is force on the object and ${\displaystyle x}$ is displacement. We say that W is the "work done by the force, F." Let us derive a useful relationship between work and kinetic energy. Say we have a total force ${\displaystyle {\vec {F}}}$ acting on an object. Then the work is ${\displaystyle \int {\vec {F}}\cdot d{\vec {r}}=\int m{\frac {d{\vec {v}}}{dt}}\cdot d{\vec {r}}=m\int {\frac {d{\vec {v}}}{dt}}\cdot {\vec {v}}dt=m\int {\vec {v}}\cdot d{\vec {v}}=\Delta (mv^{2}/2)=\Delta K}$ I simply used Newton's second law in the first step and a nice substitution in the integral. This states that the work done by the total force on an object is the change in kinetic energy of the object. In fact, this is sufficient to be the motivation for defining K to be ${\displaystyle mv^{2}/2}$. For example, if you hold an apple, then move the apple up a little bit then stop, what is happening? The potential energy of the apple changes, so someone is doing work even though there is no change in kinetic energy -- and by our theorem, this means that there was no work done -- how can that be? Gravity did (negative) work on the apple, but you also did (positive) work on the apple. That is, the total work done is zero, although the work you did is non zero. Always remember that it is the total force that changes kinetic energy. Another useful property of work is that it is linear in the force. That is, the work done by the sum of two forces is the sum of the work done by each force. Because of this, you can interpret the work as how much kinetic energy each force is giving to the object. In the apple example, at first the force from your hand is greater than the force of gravity, so the kinetic energy increases and the apple accelerates up. Then as you slow down, gravity does more work so the total work is negative and the apple decreases its kinetic energy and comes to a stop. In a very special case, the quantity of work does not depend on how you move a particle around, but only on the beginning and ending points. Such a field is called "conservative." It means that we can introduce a potential. Gravity is such a conservative force, amazingly, which is why we can talk about the "potential energy" of an object. It is just shorthand for saying the work it takes to move the object from somewhere (the reference point) to wherever we are talking about. Consequently, the change in kinetic energy equals the negative change in potential energy, which means that the total energy of the system is constant. This is in fact why such a force is called conservative -- it conserves mechanical energy! The converse, however, is not true. That is, if a system conserves mechanical energy, it is not necessarily a conservative force field. Dissipative forces, such as friction (it eats up energy) are sometimes called non-conservative forces. This is somewhat of a mistake because on the molecular level, the forces really are conservative. However, it is often nicer to just say that energy is not conserved in a given scenario, even though we know full well that it is disappearing into the motion of atoms, or heat. You will hear many people say that energy is not conserved in a given situation, but of course it is; energy is always conserved. It turns out that a force is conservative if and only if the force is "irrotational," or "curl-less" which has to do with vector calculus. It means that if you put a paddle wheel in, it won't spontaneously start to turn. It is an interesting fact that there are no non-conservative forces! To quantify everything, we have the work done by a nonconservative force is the change in the total energy of the body. By total energy, we mean potential energy plus kinetic energy, since the total force can be split into conservative (which appears in the potential energy term) and nonconservative (the new term). ## Power Power is the rate of doing work. To come up with a useful expression for this, consider a short amount of time ${\displaystyle \Delta t}$. How much work is done in that time? Well, by the definition of power, it is very nearly ${\displaystyle P\Delta t}$. By the definition of work, this is ${\displaystyle {\vec {F}}\cdot {\vec {\Delta x}}}$ where ${\displaystyle \Delta x}$ is the displacement which occurs in ${\displaystyle \Delta t}$. We have ${\displaystyle P\Delta t={\vec {F}}\cdot {\vec {\Delta x}}}$ So, ${\displaystyle P={\vec {F}}\cdot {\frac {\vec {\Delta x}}{\Delta t}}}$, which in the limit as ${\displaystyle \Delta t}$ goes to zero (which is when our "equations" become exact), ${\displaystyle P={\vec {F}}\cdot {\frac {d{\vec {x}}}{dt}}={\vec {F}}\cdot {\vec {v}}}$. Equivalently, we could have gotten the expression by simply differentiating our expression for work. No matter the derivation, because it simply does not matter that much; we have a useful expression for power. This means that if the force is acting perpendicular to the velocity, the speed does not change, because the work is zero so the change in kinetic energy is zero. But wait, how can that be, since a force necessarily accelerates something? It is accelerating it, it is changing the direction of travel -- acceleration means the derivative of the vector velocity, not the magnitude of velocity. In fact, this tells us that the component of force in the same direction as velocity is responsible for (and only for) changes in the magnitude of the velocity, and the component of force perpendicular to the velocity is responsible for (and only for) changes in the direction of the velocity. Just to quantify this a little bit, it can be shown that ${\displaystyle {\vec {a}}=\|\|v\|\|{\vec {T}}+{\frac {\|\|v\|\|^{2}}{\rho }}{\vec {N}}}$ where a is acceleration, v is the velocity, T is the unit tangent vector (tangent to the path of the particle and consequently parallel to the velocity vector), N is the unit normal vector (perpendicular to the tangent vector and in the direction of the derivative of the tangent vector, which you can picture by drawing two pretty close tangent vectors on a curve), and ${\displaystyle \rho }$ is the radius of curvature, which is essentially the radius of the circle which closest fits the path at the point (the radius of curvature of a circle is the radius of the circle, and the radius of curvature of a straight line is infinity). All this business is not really necessary for understanding physics, but if you understand it it will help you understand what is going on. Notice that the second term is the centripetal acceration -- this is in fact where we get the formula for it. Finally, just writing out the definition of power to look pretty, if the work is done at a changing rate, then ${\displaystyle P={\frac {dW}{dt}}}$ If the work is done at a constant rate, then this becomes ${\displaystyle P={\frac {W}{\Delta t}}={\frac {\Delta E}{\Delta t}}}$.	1,791	7,519	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 24, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.28125	4	CC-MAIN-2023-14	latest	en	0.890036	# Physics with Calculus/Mechanics/Work and Energy. ## Work. Work is a special name given to the (scalar) quantity. ${\displaystyle W=\int {\vec {F}}\cdot d{\vec {x}}.}$. where ${\displaystyle W}$ is work, ${\displaystyle F}$ is force on the object and ${\displaystyle x}$ is displacement. Since the dot product is a projection, the work is the component of the force in the direction of the displacement times the displacement. If the force is constant and the object travels in a straight line, this reduces to. ${\displaystyle W={\vec {F}}\cdot {\vec {x}}=Fx\cos \theta \ }$. where ${\displaystyle W}$ is work and ${\displaystyle F}$ is force on the object and ${\displaystyle x}$ is displacement.. We say that W is the "work done by the force, F." Let us derive a useful relationship between work and kinetic energy.. Say we have a total force ${\displaystyle {\vec {F}}}$ acting on an object. Then the work is. ${\displaystyle \int {\vec {F}}\cdot d{\vec {r}}=\int m{\frac {d{\vec {v}}}{dt}}\cdot d{\vec {r}}=m\int {\frac {d{\vec {v}}}{dt}}\cdot {\vec {v}}dt=m\int {\vec {v}}\cdot d{\vec {v}}=\Delta (mv^{2}/2)=\Delta K}$. I simply used Newton's second law in the first step and a nice substitution in the integral. This states that the work done by the total force on an object is the change in kinetic energy of the object. In fact, this is sufficient to be the motivation for defining K to be ${\displaystyle mv^{2}/2}$.. For example, if you hold an apple, then move the apple up a little bit then stop, what is happening? The potential energy of the apple changes, so someone is doing work even though there is no change in kinetic energy -- and by our theorem, this means that there was no work done -- how can that be? Gravity did (negative) work on the apple, but you also did (positive) work on the apple. That is, the total work done is zero, although the work you did is non zero. Always remember that it is the total force that changes kinetic energy.. Another useful property of work is that it is linear in the force. That is, the work done by the sum of two forces is the sum of the work done by each force. Because of this, you can interpret the work as how much kinetic energy each force is giving to the object. In the apple example, at first the force from your hand is greater than the force of gravity, so the kinetic energy increases and the apple accelerates up. Then as you slow down, gravity does more work so the total work is negative and the apple decreases its kinetic energy and comes to a stop.. In a very special case, the quantity of work does not depend on how you move a particle around, but only on the beginning and ending points. Such a field is called "conservative." It means that we can introduce a potential. Gravity is such a conservative force, amazingly, which is why we can talk about the "potential energy" of an object. It is just shorthand for saying the work it takes to move the object from somewhere (the reference point) to wherever we are talking about. Consequently, the change in kinetic energy equals the negative change in potential energy, which means that the total energy of the system is constant. This is in fact why such a force is called conservative -- it conserves mechanical energy! The converse, however, is not true. That is, if a system conserves mechanical energy, it is not necessarily a conservative force field.	Dissipative forces, such as friction (it eats up energy) are sometimes called non-conservative forces. This is somewhat of a mistake because on the molecular level, the forces really are conservative. However, it is often nicer to just say that energy is not conserved in a given scenario, even though we know full well that it is disappearing into the motion of atoms, or heat. You will hear many people say that energy is not conserved in a given situation, but of course it is; energy is always conserved.. It turns out that a force is conservative if and only if the force is "irrotational," or "curl-less" which has to do with vector calculus. It means that if you put a paddle wheel in, it won't spontaneously start to turn. It is an interesting fact that there are no non-conservative forces!. To quantify everything, we have the work done by a nonconservative force is the change in the total energy of the body. By total energy, we mean potential energy plus kinetic energy, since the total force can be split into conservative (which appears in the potential energy term) and nonconservative (the new term).. ## Power. Power is the rate of doing work. To come up with a useful expression for this, consider a short amount of time ${\displaystyle \Delta t}$. How much work is done in that time? Well, by the definition of power, it is very nearly ${\displaystyle P\Delta t}$. By the definition of work, this is ${\displaystyle {\vec {F}}\cdot {\vec {\Delta x}}}$ where ${\displaystyle \Delta x}$ is the displacement which occurs in ${\displaystyle \Delta t}$. We have. ${\displaystyle P\Delta t={\vec {F}}\cdot {\vec {\Delta x}}}$. So,. ${\displaystyle P={\vec {F}}\cdot {\frac {\vec {\Delta x}}{\Delta t}}}$,. which in the limit as ${\displaystyle \Delta t}$ goes to zero (which is when our "equations" become exact),. ${\displaystyle P={\vec {F}}\cdot {\frac {d{\vec {x}}}{dt}}={\vec {F}}\cdot {\vec {v}}}$.. Equivalently, we could have gotten the expression by simply differentiating our expression for work. No matter the derivation, because it simply does not matter that much; we have a useful expression for power.. This means that if the force is acting perpendicular to the velocity, the speed does not change, because the work is zero so the change in kinetic energy is zero. But wait, how can that be, since a force necessarily accelerates something? It is accelerating it, it is changing the direction of travel -- acceleration means the derivative of the vector velocity, not the magnitude of velocity. In fact, this tells us that the component of force in the same direction as velocity is responsible for (and only for) changes in the magnitude of the velocity, and the component of force perpendicular to the velocity is responsible for (and only for) changes in the direction of the velocity. Just to quantify this a little bit, it can be shown that. ${\displaystyle {\vec {a}}=\|\|v\|\|{\vec {T}}+{\frac {\|\|v\|\|^{2}}{\rho }}{\vec {N}}}$. where a is acceleration, v is the velocity, T is the unit tangent vector (tangent to the path of the particle and consequently parallel to the velocity vector), N is the unit normal vector (perpendicular to the tangent vector and in the direction of the derivative of the tangent vector, which you can picture by drawing two pretty close tangent vectors on a curve), and ${\displaystyle \rho }$ is the radius of curvature, which is essentially the radius of the circle which closest fits the path at the point (the radius of curvature of a circle is the radius of the circle, and the radius of curvature of a straight line is infinity). All this business is not really necessary for understanding physics, but if you understand it it will help you understand what is going on. Notice that the second term is the centripetal acceration -- this is in fact where we get the formula for it.. Finally, just writing out the definition of power to look pretty, if the work is done at a changing rate, then. ${\displaystyle P={\frac {dW}{dt}}}$. If the work is done at a constant rate, then this becomes. ${\displaystyle P={\frac {W}{\Delta t}}={\frac {\Delta E}{\Delta t}}}$.
http://www.talkstats.com/showthread.php/5654-Standard-error-why-sqrt-of-n	1,419,098,072,000,000,000	text/html	crawl-data/CC-MAIN-2014-52/segments/1418802770071.75/warc/CC-MAIN-20141217075250-00123-ip-10-231-17-201.ec2.internal.warc.gz	823,028,729	14,610	# Thread: Standard error - why sqrt of n? 1. ## Standard error - why sqrt of n? I happened to be browsing through some standard explanations of confidence intervals and came upon the formula for standard error... t x (stddev / sqrt of n) I couldn't quite grasp why we would use sqrt of the sample size. Yes, a stats 101 question, but my brain can't seem to go that far back. Can someone explain this in simple terms? Thank you. 2. Originally Posted by jawon I happened to be browsing through some standard explanations of confidence intervals and came upon the formula for standard error... t x (stddev / sqrt of n) I couldn't quite grasp why we would use sqrt of the sample size. Yes, a stats 101 question, but my brain can't seem to go that far back. Can someone explain this in simple terms? Thank you. stddev is the standard deviation of one observation from the population. But then you take average of n observations(sample mean), standard deviation sample mean will smaller than stddev (sample mean should vary less from sample-to-sample, when sample size is large we will get approximately same sample mean every time ) and it will be stddev / sqrt of n if you need more understandingon this, do a simulation study. 3. Xm, nice one to read > http://yusung.******.com/2008/09/s...deviation.html ****=blog spot !!! Since when that's banned? 4. is the ****** means blog spot ? 5. Ha, I thought the ban only worked for hyperlinks 6. Hmmm. Thanks for the replies. I think my question was not very clear. Let me try this... For the sample size, why do you square-root? I of course understand mathematically what the result of the formulas are, but I'm trying to get a more general explanation why you would square root. Sorry if this is a silly naive question... may not be worth any more of anyone's time! 7. Originally Posted by jawon Hmmm. Thanks for the replies. I think my question was not very clear. Let me try this... For the sample size, why do you square-root? ... I'm trying to get a more general explanation why you would square root. Let X1,...Xi,...,XN form a random sample from a population with variance Sigma^2. Let the sample mean be XBar=(X1+...+Xi+...+XN)/N. The variance of XBar is: (Sum of each variance of Xi + Sum of all covariances between Xi and Xj)/N^2. Because each pair of Xi and Xj are uncorrelated, each covariance is zero. So the variance of XBar is the sum of all the variances of Xi/N^2. Each component Xi of the sample has the same variance Sigma^2, the variance of XBar is Sigma^2/N. Thus, the standard error of XBar is Sigma/Sqrt[N]. How's that? 8. when you square root n, is n the sample mean or the sample of the problem. For example finding the standard error when the standard deviation is 18, sample mean is 45 and the sample is 100. What would you square root? 9. Originally Posted by tiffnnie01 when you square root n, is n the sample mean or the sample of the problem. For example finding the standard error when the standard deviation is 18, sample mean is 45 and the sample is 100. What would you square root? 18 divided by 10 ( sqrt of 100) (multiplied by t) is the standard error, I think. Tweet #### Posting Permissions • You may not post new threads • You may not post replies • You may not post attachments • You may not edit your posts	804	3,331	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.59375	4	CC-MAIN-2014-52	longest	en	0.94401	# Thread: Standard error - why sqrt of n?. 1. ## Standard error - why sqrt of n?. I happened to be browsing through some standard explanations of confidence intervals and came upon the formula for standard error.... t x (stddev / sqrt of n). I couldn't quite grasp why we would use sqrt of the sample size. Yes, a stats 101 question, but my brain can't seem to go that far back. Can someone explain this in simple terms? Thank you.. 2. Originally Posted by jawon. I happened to be browsing through some standard explanations of confidence intervals and came upon the formula for standard error.... t x (stddev / sqrt of n). I couldn't quite grasp why we would use sqrt of the sample size. Yes, a stats 101 question, but my brain can't seem to go that far back. Can someone explain this in simple terms? Thank you.. stddev is the standard deviation of one observation from the population.. But then you take average of n observations(sample mean), standard deviation sample mean will smaller than stddev. (sample mean should vary less from sample-to-sample, when sample size is large we will get approximately same sample mean every time ). and it will be. stddev / sqrt of n. if you need more understandingon this, do a simulation study.. 3. Xm, nice one to read. >. http://yusung.******.com/2008/09/s...deviation.html. ****=blog spot !!! Since when that's banned?. 4. is the ****** means blog spot ?. 5. Ha, I thought the ban only worked for hyperlinks. 6. Hmmm. Thanks for the replies. I think my question was not very clear. Let me try this.... For the sample size, why do you square-root?.	I of course understand mathematically what the result of the formulas are, but I'm trying to get a more general explanation why you would square root. Sorry if this is a silly naive question... may not be worth any more of anyone's time!. 7. Originally Posted by jawon. Hmmm. Thanks for the replies.. I think my question was not very clear. Let me try this.... For the sample size, why do you square-root?. ... I'm trying to get a more general explanation why you would square root.. Let X1,...Xi,...,XN form a random sample from a population with variance Sigma^2. Let the sample mean be XBar=(X1+...+Xi+...+XN)/N.. The variance of XBar is: (Sum of each variance of Xi + Sum of all covariances between Xi and Xj)/N^2.. Because each pair of Xi and Xj are uncorrelated, each covariance is zero.. So the variance of XBar is the sum of all the variances of Xi/N^2.. Each component Xi of the sample has the same variance Sigma^2,. the variance of XBar is Sigma^2/N.. Thus, the standard error of XBar is Sigma/Sqrt[N].. How's that?. 8. when you square root n, is n the sample mean or the sample of the problem. For example finding the standard error when the standard deviation is 18, sample mean is 45 and the sample is 100. What would you square root?. 9. Originally Posted by tiffnnie01. when you square root n, is n the sample mean or the sample of the problem. For example finding the standard error when the standard deviation is 18, sample mean is 45 and the sample is 100. What would you square root?. 18 divided by 10 ( sqrt of 100) (multiplied by t) is the standard error, I think.. 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https://nrich.maths.org/public/leg.php?code=-39&cl=1&cldcmpid=9692	1,540,043,054,000,000,000	text/html	crawl-data/CC-MAIN-2018-43/segments/1539583512750.15/warc/CC-MAIN-20181020121719-20181020143219-00156.warc.gz	748,754,711	10,000	# Search by Topic #### Resources tagged with Combinations similar to School Fair Necklaces: Filter by: Content type: Age range: Challenge level: ### There are 111 results Broad Topics > Decision Mathematics and Combinatorics > Combinations ### Home City ##### Age 7 to 11 Challenge Level: Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in. ### Calcunos ##### Age 7 to 11 Challenge Level: If we had 16 light bars which digital numbers could we make? How will you know you've found them all? ### Halloween Investigation ##### Age 7 to 11 Challenge Level: Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. 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He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away? ### Three Way Mix Up ##### Age 5 to 11 Challenge Level: Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it? ### Homes ##### Age 5 to 7 Challenge Level: There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find? ### New House ##### Age 7 to 11 Challenge Level: In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with? ### Red Express Train ##### Age 5 to 7 Challenge Level: The Red Express Train usually has five red carriages. 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How many different towers can you make? ### Cereal Packets ##### Age 7 to 11 Challenge Level: How can you put five cereal packets together to make different shapes if you must put them face-to-face?	2,480	10,305	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.875	4	CC-MAIN-2018-43	latest	en	0.92868	# Search by Topic. #### Resources tagged with Combinations similar to School Fair Necklaces:. Filter by: Content type:. Age range:. Challenge level:. ### There are 111 results. Broad Topics > Decision Mathematics and Combinatorics > Combinations. ### Home City. ##### Age 7 to 11 Challenge Level:. Use the clues to work out which cities Mohamed, Sheng, Tanya and Bharat live in.. ### Calcunos. ##### Age 7 to 11 Challenge Level:. If we had 16 light bars which digital numbers could we make? How will you know you've found them all?. ### Halloween Investigation. ##### Age 7 to 11 Challenge Level:. Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?. ### Two Egg Timers. ##### Age 7 to 11 Challenge Level:. You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?. ##### Age 7 to 11 Challenge Level:. Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?. ### Street Party. ##### Age 7 to 11 Challenge Level:. The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.. ### Polo Square. ##### Age 7 to 11 Challenge Level:. Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.. ### Newspapers. ##### Age 7 to 11 Challenge Level:. When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?. ### Prison Cells. ##### Age 7 to 11 Challenge Level:. There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?. ### Room Doubling. ##### Age 7 to 11 Challenge Level:. Investigate the different ways you could split up these rooms so that you have double the number.. ### Plates of Biscuits. ##### Age 7 to 11 Challenge Level:. Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?. ### Elf Suits. ##### Age 7 to 11 Challenge Level:. If these elves wear a different outfit every day for as many days as possible, how many days can their fun last?. ### Ice Cream. ##### Age 7 to 11 Challenge Level:. You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.. ### Team Scream. ##### Age 7 to 11 Challenge Level:. Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?. ### Bean Bags for Bernard's Bag. ##### Age 7 to 11 Challenge Level:. How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?. ### Octa Space. ##### Age 7 to 11 Challenge Level:. In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?. ### Zargon Glasses. ##### Age 7 to 11 Challenge Level:. Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?. ### Hubble, Bubble. ##### Age 7 to 11 Challenge Level:. Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?. ### Finding Fifteen. ##### Age 7 to 11 Challenge Level:. Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?. ### On Target. ##### Age 7 to 11 Challenge Level:. You have 5 darts and your target score is 44. How many different ways could you score 44?. ### 3 Rings. ##### Age 7 to 11 Challenge Level:. If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?. ### More and More Buckets. ##### Age 7 to 11 Challenge Level:. In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?. ### Button-up Some More. ##### Age 7 to 11 Challenge Level:. How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?. ### Wag Worms. ##### Age 7 to 11 Challenge Level:. When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.. ### It Figures. ##### Age 7 to 11 Challenge Level:. Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?.	### Calendar Cubes. ##### Age 7 to 11 Challenge Level:. Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.. ### Three by Three. ##### Age 5 to 11 Challenge Level:. Arrange 3 red, 3 blue and 3 yellow counters into a three-by-three square grid, so that there is only one of each colour in every row and every column. ### Chocs, Mints, Jellies. ##### Age 7 to 11 Challenge Level:. In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?. ### Delia's Routes. ##### Age 7 to 11 Challenge Level:. A little mouse called Delia lives in a hole in the bottom of a tree.....How many days will it be before Delia has to take the same route again?. ### Button-up. ##### Age 5 to 7 Challenge Level:. My coat has three buttons. How many ways can you find to do up all the buttons?. ### Tiles on a Patio. ##### Age 7 to 11 Challenge Level:. How many ways can you find of tiling the square patio, using square tiles of different sizes?. ### Chocoholics. ##### Age 7 to 11 Challenge Level:. George and Jim want to buy a chocolate bar. George needs 2p more and Jim need 50p more to buy it. How much is the chocolate bar?. ### Jumping Cricket. ##### Age 5 to 7 Challenge Level:. El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?. ### Six Is the Sum. ##### Age 7 to 11 Challenge Level:. What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?. ### Snakes. ##### Age 5 to 7 Challenge Level:. Explore the different snakes that can be made using 5 cubes.. ### Penta Primes. ##### Age 7 to 11 Challenge Level:. Using all ten cards from 0 to 9, rearrange them to make five prime numbers. Can you find any other ways of doing it?. ### Sweets in a Box. ##### Age 7 to 11 Challenge Level:. How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?. ### The Puzzling Sweet Shop. ##### Age 7 to 11 Challenge Level:. There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?. ### Penta Place. ##### Age 7 to 11 Challenge Level:. Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?. ##### Age 5 to 7 Challenge Level:. In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?. ### Sealed Solution. ##### Age 7 to 11 Challenge Level:. Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?. ### Briefcase Lock. ##### Age 5 to 7 Challenge Level:. My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?. ### The School Trip. ##### Age 5 to 7 Challenge Level:. Lorenzie was packing his bag for a school trip. He packed four shirts and three pairs of pants. "I will be able to have a different outfit each day", he said. How many days will Lorenzie be away?. ### Three Way Mix Up. ##### Age 5 to 11 Challenge Level:. Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?. ### Homes. ##### Age 5 to 7 Challenge Level:. There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?. ### New House. ##### Age 7 to 11 Challenge Level:. In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?. ### Red Express Train. ##### Age 5 to 7 Challenge Level:. The Red Express Train usually has five red carriages. How many ways can you find to add two blue carriages?. ### Train Carriages. ##### Age 5 to 11 Challenge Level:. Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?. ### 3 Blocks Towers. ##### Age 5 to 7 Challenge Level:. Take three differently coloured blocks - maybe red, yellow and blue. Make a tower using one of each colour. How many different towers can you make?. ### Cereal Packets. ##### Age 7 to 11 Challenge Level:. How can you put five cereal packets together to make different shapes if you must put them face-to-face?.
https://www.hackmath.net/en/calculator/fraction?input=1%2F4+divided+by+2	1,713,436,747,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296817206.28/warc/CC-MAIN-20240418093630-20240418123630-00606.warc.gz	731,654,975	7,585	# Fraction calculator This fraction calculator performs basic and advanced fraction operations, expressions with fractions combined with integers, decimals, and mixed numbers. It also shows detailed step-by-step information about the fraction calculation procedure. The calculator helps in finding value from multiple fractions operations. Solve problems with two, three, or more fractions and numbers in one expression. ## The result: ### (1/4) : 2 = 1/8 = 0.125 Spelled result in words is one eighth. ### How do we solve fractions step by step? 1. Divide: 1/4 : 2 = 1/4 · 1/2 = 1 · 1/4 · 2 = 1/8 Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 2/1 is 1/2) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling. In other words - one quarter divided by two is one eighth. #### Rules for expressions with fractions: Fractions - use a forward slash to divide the numerator by the denominator, i.e., for five-hundredths, enter 5/100. If you use mixed numbers, leave a space between the whole and fraction parts. Mixed numerals (mixed numbers or fractions) keep one space between the integer and fraction and use a forward slash to input fractions i.e., 1 2/3 . An example of a negative mixed fraction: -5 1/2. Because slash is both sign for fraction line and division, use a colon (:) as the operator of division fractions i.e., 1/2 : 1/3. Decimals (decimal numbers) enter with a decimal point . and they are automatically converted to fractions - i.e. 1.45. ### Math Symbols SymbolSymbol nameSymbol MeaningExample -minus signsubtraction 1 1/2 - 2/3 asteriskmultiplication 2/3 3/4 ×times signmultiplication 2/3 × 5/6 :division signdivision 1/2 : 3 /division slashdivision 1/3 / 5 :coloncomplex fraction 1/2 : 1/3 ^caretexponentiation / power 1/4^3 ()parenthesescalculate expression inside first-3/5 - (-1/4) The calculator follows well-known rules for the order of operations. The most common mnemonics for remembering this order of operations are: PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. BEDMAS - Brackets, Exponents, Division, Multiplication, Addition, Subtraction BODMAS - Brackets, Of or Order, Division, Multiplication, Addition, Subtraction. GEMDAS - Grouping Symbols - brackets (){}, Exponents, Multiplication, Division, Addition, Subtraction. MDAS - Multiplication and Division have the same precedence over Addition and Subtraction. The MDAS rule is the order of operations part of the PEMDAS rule. Be careful; always do multiplication and division before addition and subtraction. Some operators (+ and -) and (* and /) have the same priority and must be evaluated from left to right.	728	2,954	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.5	4	CC-MAIN-2024-18	latest	en	0.823695	# Fraction calculator. This fraction calculator performs basic and advanced fraction operations, expressions with fractions combined with integers, decimals, and mixed numbers. It also shows detailed step-by-step information about the fraction calculation procedure. The calculator helps in finding value from multiple fractions operations. Solve problems with two, three, or more fractions and numbers in one expression.. ## The result:. ### (1/4) : 2 = 1/8 = 0.125. Spelled result in words is one eighth.. ### How do we solve fractions step by step?. 1. Divide: 1/4 : 2 = 1/4 · 1/2 = 1 · 1/4 · 2 = 1/8. Dividing two fractions is the same as multiplying the first fraction by the reciprocal value of the second fraction. The first sub-step is to find the reciprocal (reverse the numerator and denominator, reciprocal of 2/1 is 1/2) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators. In the following intermediate step, it cannot further simplify the fraction result by canceling.. In other words - one quarter divided by two is one eighth.. #### Rules for expressions with fractions:. Fractions - use a forward slash to divide the numerator by the denominator, i.e., for five-hundredths, enter 5/100. If you use mixed numbers, leave a space between the whole and fraction parts.. Mixed numerals (mixed numbers or fractions) keep one space between the integer and. fraction and use a forward slash to input fractions i.e., 1 2/3 . An example of a negative mixed fraction: -5 1/2.. Because slash is both sign for fraction line and division, use a colon (:) as the operator of division fractions i.e., 1/2 : 1/3.	Decimals (decimal numbers) enter with a decimal point . and they are automatically converted to fractions - i.e. 1.45.. ### Math Symbols. SymbolSymbol nameSymbol MeaningExample. -minus signsubtraction 1 1/2 - 2/3. asteriskmultiplication 2/3 3/4. ×times signmultiplication 2/3 × 5/6. :division signdivision 1/2 : 3. /division slashdivision 1/3 / 5. :coloncomplex fraction 1/2 : 1/3. ^caretexponentiation / power 1/4^3. ()parenthesescalculate expression inside first-3/5 - (-1/4). The calculator follows well-known rules for the order of operations. The most common mnemonics for remembering this order of operations are:. PEMDAS - Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.. BEDMAS - Brackets, Exponents, Division, Multiplication, Addition, Subtraction. BODMAS - Brackets, Of or Order, Division, Multiplication, Addition, Subtraction.. GEMDAS - Grouping Symbols - brackets (){}, Exponents, Multiplication, Division, Addition, Subtraction.. MDAS - Multiplication and Division have the same precedence over Addition and Subtraction. The MDAS rule is the order of operations part of the PEMDAS rule.. Be careful; always do multiplication and division before addition and subtraction. Some operators (+ and -) and (* and /) have the same priority and must be evaluated from left to right.
http://www.permutationpuzzles.org/rubik/mball/rainbow.html	1,719,057,189,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198862396.85/warc/CC-MAIN-20240622112740-20240622142740-00554.warc.gz	47,458,495	3,459	# Rainbow masterball 1. Notation for the basic moves: Let L be one of the longitudinal lines going from the north pole to the south pole. Let f1 denote the longitudinal rotation by 180 degrees along L. (The f stands for "flip".) Going left-to-right (i.e., counterclockwise from above), let the other "flips" be denoted f2, ..., f8, resp.. Let r1 denote the rotation of the north polar cap by 45 degrees right-to-left (i.e., clockwise from above). Positive exponents will be used to apply this more than once: for example, let r13 denote the rotation of the north polar cap by 135(=3x45) degrees right-to-left. Let r2 denote the rotation of the north-of-the-equatoral belt by 45 degrees right-to-left, r3 the rotation of the south-of-the-equatoral belt by 45 degrees right-to-left, and r4 the rotation of the south polar cap by 45 degrees right-to-left. Each of these moves has an "inverse" move going in the reverse direction which we denote by putting a superscript of -1 on it. For example, r1-1 denotes the rotation of the north polar cap by 45 degrees left-to-right (i.e., counterclockwise from above). Notice that each f1,...,f8 is equal to its inverse move. If you want to make several moves in sequence we simply multiplify these symbols together left-to-right: to move f1 then r3 twice then the inverse of r4 you could simply write f1r32r4-1. (Note that the order is in general important but in this case r3r4=r4r3.) We call the length of a move the smallest number of these generators or their inverses (r1,...,r4, f1,...,f8,r1-1,...,f8-1) required to make the move. For example, f1r32r4-1 is length 4 but r4r32r4-1 is length 2. Question: What is the longest move of the masterball? Here is a small catalog of moves for the masterball: some masterball moves . A 2-cycle will swap exactly 2 facets : 2-cycle position (~27K) The shortest move I know for an honest 2-cycle is very long (discovered by using GAP). If you know of a short one, please let me know. Close to this is Andrew Southern's product of two 2-cycles (from the beachball pattern): f1r3r4f2f4r1 r4-1f4r44f4 r4r1-1 f4r44 f2r3-1r4-1f1 Here are two pictures of this (from different orientations): Let's call such a product of two 2-cycles (where one of the transpositions only affects facets along the same column) a column double swap . In other words, a column double swap will swap two facets in different rows and* swap two facets in the same column (which would not be noticed if that column had already been "solved"). Such moves are very useful to know. Along with "fishing" (see the documentation above), one only needs to know some column double swaps to solve the puzzle. Andrew Southern's pictures of a column double swap . Here's another column double swap: f1r1f4* r1-1r4f4* r4-1f1 Both of these moves were discovered by Andrew Southern. 2. ## Some pictures Here are some pictures of some MAPLE 3d plots created by this package: Some pictures of these basic moves: r1 (~12K) r2 (~12K) r3 (~12K) r4 (~18K) f1 (~21K) f2 (~21K) f3 (~21K) f4 (~21K) ### Some pretty patterns These are all based on ideas of Andrew Southern. 1. quadrantized pattern (side view) (~39K), quadrantized pattern (front and back) . In the above notation, this pattern can be reached from the beachball pattern by the moves (r3r4)4f1 (r3r4)2f2* (r3r4)2f3 2. banded pattern r3-4r1-4f1r32 r12f2r32r12f3 3. checkered pattern (top view), checkered pattern (side), checkered pattern (front and back) move1=r1r3f1r3-1 r1-1f7f2, move2=r1r3f1r3-1 r1-1f1, move3=r1r3f3 r1-1r3-1f4f5, move4=r1r3f2 r1-1r3-1f2, move5=f3f4f5f6 f3f4f5* f6r1r2r3r4 (swaps columns 1,2), move6=f4f5f6f7f4f5 f6f7r1r2 r3r4 (swaps columns 2,3) checkered=move1move2move3move4move5move6 Back to puzzle page Created 1997. Last modified 8-26-2007.	1,179	3,874	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.859375	4	CC-MAIN-2024-26	latest	en	0.864749	# Rainbow masterball. 1. Notation for the basic moves: Let L be one of the longitudinal lines going from the north pole to the south pole. Let f1 denote the longitudinal rotation by 180 degrees along L. (The f stands for "flip".) Going left-to-right (i.e., counterclockwise from above), let the other "flips" be denoted f2, ..., f8, resp.. Let r1 denote the rotation of the north polar cap by 45 degrees right-to-left (i.e., clockwise from above). Positive exponents will be used to apply this more than once: for example, let r13 denote the rotation of the north polar cap by 135(=3x45) degrees right-to-left. Let r2 denote the rotation of the north-of-the-equatoral belt by 45 degrees right-to-left, r3 the rotation of the south-of-the-equatoral belt by 45 degrees right-to-left, and r4 the rotation of the south polar cap by 45 degrees right-to-left. Each of these moves has an "inverse" move going in the reverse direction which we denote by putting a superscript of -1 on it. For example, r1-1 denotes the rotation of the north polar cap by 45 degrees left-to-right (i.e., counterclockwise from above). Notice that each f1,...,f8 is equal to its inverse move. If you want to make several moves in sequence we simply multiplify these symbols together left-to-right: to move f1 then r3 twice then the inverse of r4 you could simply write f1r32r4-1. (Note that the order is in general important but in this case r3r4=r4r3.). We call the length of a move the smallest number of these generators or their inverses (r1,...,r4, f1,...,f8,r1-1,...,f8-1) required to make the move. For example, f1r32r4-1 is length 4 but r4r32r4-1 is length 2.. Question: What is the longest move of the masterball?. Here is a small catalog of moves for the masterball: some masterball moves .. A 2-cycle will swap exactly 2 facets : 2-cycle position (~27K) The shortest move I know for an honest 2-cycle is very long (discovered by using GAP). If you know of a short one, please let me know.. Close to this is Andrew Southern's product of two 2-cycles (from the beachball pattern):. f1r3r4f2f4r1 r4-1f4r44f4 r4r1-1 f4r44 f2r3-1r4-1f1. Here are two pictures of this (from different orientations):. Let's call such a product of two 2-cycles (where one of the transpositions only affects facets along the same column) a column double swap . In other words, a column double swap will swap two facets in different rows and* swap two facets in the same column (which would not be noticed if that column had already been "solved"). Such moves are very useful to know. Along with "fishing" (see the documentation above), one only needs to know some column double swaps to solve the puzzle.. Andrew Southern's pictures of a column double swap . Here's another column double swap:. f1r1f4* r1-1r4f4* r4-1*f1. Both of these moves were discovered by Andrew Southern.. 2.	## Some pictures. Here are some pictures of some MAPLE 3d plots created by this package:. Some pictures of these basic moves: r1 (~12K) r2 (~12K) r3 (~12K) r4 (~18K) f1 (~21K) f2 (~21K) f3 (~21K) f4 (~21K). ### Some pretty patterns. These are all based on ideas of Andrew Southern.. 1. quadrantized pattern (side view) (~39K), quadrantized pattern (front and back) . In the above notation, this pattern can be reached from the beachball pattern by the moves. (r3r4)4f1* (r3r4)2f2* (r3r4)2f3. 2. banded pattern. r3-4r1-4f1r32 r12f2r32r12f3. 3. checkered pattern (top view), checkered pattern (side), checkered pattern (front and back). move1=r1r3f1r3-1 r1-1f7f2,. move2=r1r3f1r3-1 r1-1f1,. move3=r1r3f3 r1-1r3-1f4f5,. move4=r1r3f2 r1-1r3-1f2,. move5=f3f4f5f6 f3f4f5* f6r1r2r3r4 (swaps columns 1,2),. move6=f4f5f6f7f4f5 f6f7r1r2 r3r4 (swaps columns 2,3). checkered=move1move2move3move4move5move6. Back to puzzle page. Created 1997. Last modified 8-26-2007.
https://nrich.maths.org/public/topic.php?code=-333&cl=1&cldcmpid=2353	1,571,306,983,000,000,000	text/html	crawl-data/CC-MAIN-2019-43/segments/1570986673538.21/warc/CC-MAIN-20191017095726-20191017123226-00263.warc.gz	636,964,571	9,801	# Search by Topic #### Resources tagged with Investigations similar to Here to There 1 2 3: Filter by: Content type: Age range: Challenge level: ### Homes ##### Age 5 to 7 Challenge Level: There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find? ### Bean Bags for Bernard's Bag ##### Age 7 to 11 Challenge Level: How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this? ### The Pied Piper of Hamelin ##### Age 7 to 11 Challenge Level: This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether! ### Polo Square ##### Age 7 to 11 Challenge Level: Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total. ### Cubes Here and There ##### Age 7 to 11 Challenge Level: How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green? ##### Age 7 to 11 Challenge Level: I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take? ##### Age 7 to 11 Challenge Level: Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon? ### Birthday Cake Candles ##### Age 7 to 11 Challenge Level: This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things. ### Newspapers ##### Age 7 to 11 Challenge Level: When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different? ### Street Sequences ##### Age 5 to 11 Challenge Level: Investigate what happens when you add house numbers along a street in different ways. ### Halloween Investigation ##### Age 7 to 11 Challenge Level: Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make? ### Ice Cream ##### Age 7 to 11 Challenge Level: You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream. ### Train Carriages ##### Age 5 to 11 Challenge Level: Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done? ### Street Party ##### Age 7 to 11 Challenge Level: The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks. ### Room Doubling ##### Age 7 to 11 Challenge Level: Investigate the different ways you could split up these rooms so that you have double the number. ### 3 Rings ##### Age 7 to 11 Challenge Level: If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities? ### Doplication ##### Age 7 to 11 Challenge Level: We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes? ### Building with Rods ##### Age 7 to 11 Challenge Level: In how many ways can you stack these rods, following the rules? ### Crossing the Town Square ##### Age 7 to 11 Challenge Level: This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares. ### It Figures ##### Age 7 to 11 Challenge Level: Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all? ### Worms ##### Age 7 to 11 Challenge Level: Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make? ### Magic Constants ##### Age 7 to 11 Challenge Level: In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square? ### Calcunos ##### Age 7 to 11 Challenge Level: If we had 16 light bars which digital numbers could we make? How will you know you've found them all? ### My New Patio ##### Age 7 to 11 Challenge Level: What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes? ### Let's Investigate Triangles ##### Age 5 to 7 Challenge Level: Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make? ##### Age 7 to 11 Challenge Level: Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square. ### New House ##### Age 7 to 11 Challenge Level: In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with? ### Sweets in a Box ##### Age 7 to 11 Challenge Level: How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction? ### Sometimes We Lose Things ##### Age 7 to 11 Challenge Level: Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table. ### Stairs ##### Age 5 to 11 Challenge Level: This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high. ### Move a Match ##### Age 7 to 11 Challenge Level: How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement? ### Caterpillars ##### Age 5 to 7 Challenge Level: These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square? ### Two on Five ##### Age 5 to 11 Challenge Level: Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table? ### Abundant Numbers ##### Age 7 to 11 Challenge Level: 48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers? ### Tiling ##### Age 7 to 11 Challenge Level: An investigation that gives you the opportunity to make and justify predictions. ### It Was 2010! ##### Age 5 to 11 Challenge Level: If the answer's 2010, what could the question be? ### Number Squares ##### Age 5 to 11 Challenge Level: Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be? ### Little Boxes ##### Age 7 to 11 Challenge Level: How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six? ##### Age 7 to 11 Challenge Level: What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules. ### Sorting the Numbers ##### Age 5 to 11 Challenge Level: Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws? ### Teddy Town ##### Age 5 to 14 Challenge Level: There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules? ### Five Coins ##### Age 7 to 11 Challenge Level: Ben has five coins in his pocket. How much money might he have? ### Sending Cards ##### Age 7 to 11 Challenge Level: This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six? ### Tiles on a Patio ##### Age 7 to 11 Challenge Level: How many ways can you find of tiling the square patio, using square tiles of different sizes? ### Balance of Halves ##### Age 7 to 11 Challenge Level: Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance? ### Sticks and Triangles ##### Age 7 to 11 Challenge Level: Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles? ### Plants ##### Age 5 to 11 Challenge Level: Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this? ### Calendar Patterns ##### Age 7 to 11 Challenge Level: In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers? ### Magazines ##### Age 7 to 11 Challenge Level: Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages. ### Polygonals ##### Age 7 to 11 Challenge Level: Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.	2,336	10,140	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.71875	4	CC-MAIN-2019-43	latest	en	0.927843	# Search by Topic. #### Resources tagged with Investigations similar to Here to There 1 2 3:. Filter by: Content type:. Age range:. Challenge level:. ### Homes. ##### Age 5 to 7 Challenge Level:. There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?. ### Bean Bags for Bernard's Bag. ##### Age 7 to 11 Challenge Level:. How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?. ### The Pied Piper of Hamelin. ##### Age 7 to 11 Challenge Level:. This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!. ### Polo Square. ##### Age 7 to 11 Challenge Level:. Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.. ### Cubes Here and There. ##### Age 7 to 11 Challenge Level:. How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?. ##### Age 7 to 11 Challenge Level:. I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?. ##### Age 7 to 11 Challenge Level:. Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?. ### Birthday Cake Candles. ##### Age 7 to 11 Challenge Level:. This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.. ### Newspapers. ##### Age 7 to 11 Challenge Level:. When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?. ### Street Sequences. ##### Age 5 to 11 Challenge Level:. Investigate what happens when you add house numbers along a street in different ways.. ### Halloween Investigation. ##### Age 7 to 11 Challenge Level:. Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?. ### Ice Cream. ##### Age 7 to 11 Challenge Level:. You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.. ### Train Carriages. ##### Age 5 to 11 Challenge Level:. Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?. ### Street Party. ##### Age 7 to 11 Challenge Level:. The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.. ### Room Doubling. ##### Age 7 to 11 Challenge Level:. Investigate the different ways you could split up these rooms so that you have double the number.. ### 3 Rings. ##### Age 7 to 11 Challenge Level:. If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?. ### Doplication. ##### Age 7 to 11 Challenge Level:. We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?. ### Building with Rods. ##### Age 7 to 11 Challenge Level:. In how many ways can you stack these rods, following the rules?. ### Crossing the Town Square. ##### Age 7 to 11 Challenge Level:. This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.. ### It Figures. ##### Age 7 to 11 Challenge Level:. Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?. ### Worms. ##### Age 7 to 11 Challenge Level:. Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?. ### Magic Constants. ##### Age 7 to 11 Challenge Level:. In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?. ### Calcunos. ##### Age 7 to 11 Challenge Level:. If we had 16 light bars which digital numbers could we make? How will you know you've found them all?. ### My New Patio. ##### Age 7 to 11 Challenge Level:. What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?. ### Let's Investigate Triangles. ##### Age 5 to 7 Challenge Level:.	Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?. ##### Age 7 to 11 Challenge Level:. Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.. ### New House. ##### Age 7 to 11 Challenge Level:. In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?. ### Sweets in a Box. ##### Age 7 to 11 Challenge Level:. How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?. ### Sometimes We Lose Things. ##### Age 7 to 11 Challenge Level:. Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.. ### Stairs. ##### Age 5 to 11 Challenge Level:. This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.. ### Move a Match. ##### Age 7 to 11 Challenge Level:. How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?. ### Caterpillars. ##### Age 5 to 7 Challenge Level:. These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?. ### Two on Five. ##### Age 5 to 11 Challenge Level:. Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?. ### Abundant Numbers. ##### Age 7 to 11 Challenge Level:. 48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?. ### Tiling. ##### Age 7 to 11 Challenge Level:. An investigation that gives you the opportunity to make and justify predictions.. ### It Was 2010!. ##### Age 5 to 11 Challenge Level:. If the answer's 2010, what could the question be?. ### Number Squares. ##### Age 5 to 11 Challenge Level:. Start with four numbers at the corners of a square and put the total of two corners in the middle of that side. Keep going... Can you estimate what the size of the last four numbers will be?. ### Little Boxes. ##### Age 7 to 11 Challenge Level:. How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?. ##### Age 7 to 11 Challenge Level:. What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.. ### Sorting the Numbers. ##### Age 5 to 11 Challenge Level:. Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?. ### Teddy Town. ##### Age 5 to 14 Challenge Level:. There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?. ### Five Coins. ##### Age 7 to 11 Challenge Level:. Ben has five coins in his pocket. How much money might he have?. ### Sending Cards. ##### Age 7 to 11 Challenge Level:. This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?. ### Tiles on a Patio. ##### Age 7 to 11 Challenge Level:. How many ways can you find of tiling the square patio, using square tiles of different sizes?. ### Balance of Halves. ##### Age 7 to 11 Challenge Level:. Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?. ### Sticks and Triangles. ##### Age 7 to 11 Challenge Level:. Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?. ### Plants. ##### Age 5 to 11 Challenge Level:. Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?. ### Calendar Patterns. ##### Age 7 to 11 Challenge Level:. In this section from a calendar, put a square box around the 1st, 2nd, 8th and 9th. Add all the pairs of numbers. What do you notice about the answers?. ### Magazines. ##### Age 7 to 11 Challenge Level:. Let's suppose that you are going to have a magazine which has 16 pages of A5 size. Can you find some different ways to make these pages? Investigate the pattern for each if you number the pages.. ### Polygonals. ##### Age 7 to 11 Challenge Level:. Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
https://gradeup.co/harmonic-progressions-i-e384eef5-9042-11e6-b2df-84026b72af21	1,620,807,504,000,000,000	text/html	crawl-data/CC-MAIN-2021-21/segments/1620243991685.16/warc/CC-MAIN-20210512070028-20210512100028-00412.warc.gz	312,900,092	39,564	# Harmonic Progressions Updated : October 12th, 2016 Share via \| ### Harmonic Progression If inverse of a sequence follows rule of an A.P. then it is said to be in harmonic progression. e.g. 1,1/2,1/3, 1/4, 1/5 ............... 1/10, 1/7, 1/4, 1, – 1/2, ........... In general 1/a, 1/a+d, 1/a+2d, .................. Note: Three convenient numbers in H.P. are 1/a–d, 1/a, 1/a+d Four convenient numbers in H.P. are 1/a–3d, 1/a–d, 1/a+d, 1/a+3d Five convenient numbers in H.P. are 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d Example 1: Find the 4th and 8th term of the series 6, 4, 3, …… Solution: Consider 1/6, /14, 1/3, ...... ∞ Here T2 – T1 = T3 – T2 = 1/12 ⇒ 1/6, 1/4, 1/3 is an A.P. 4th term of this A.P. = 1/6 + 3 × 1/12 = 1/6 + 1/4 = 5/12, And the 8th term = 1/6 + 7 × 1/12 = 9/12. Hence the 8th term of the H.P. = 12/9 = 4/3 and the 4th term = 12/5. Example 2: If a, b, c are in H.P., show that a/b+c, b/c+a, c/a+b are also in H.P. Solution: Given that a, b, c are in H.P. ⇒ 1/a, 1/b, 1/c are in A.P. ⇒ a+b+c/a, a+b+c/b, a+b+c/c are in A.P. ⇒ 1 + b+c/a, 1 + c+a/b, 1 + a+b/c are in A.P. ⇒ b+c/a, c+a/b, a+b/c are in A.P. ⇒ a/b+c, b/c+a, c/a+b are in H.P. #### Some Important Results • 1 + 2 + 3 +…+ n = n/2(n + 1) (sum of first n natural numbers). • 12 + 22 + 32 +…+ n2 = n(n+1)(2n+1)/6 (sum of the squares of first n natural numbers). • 13 + 23 + 33 +…+ n3 = n2(n+1)2/4 = (1 + 2 + 3 +…+ n)2 (sum of the cubes of first n natural numbers). • (1 – x)–1 = 1 + x + x2 + x3 +… –1 < x < 1. • (1 – x)–2 = 1 + 2x + 3x2 +… –1 < x < 1. Example 1: Find the nth term and the sum of n terms of the series 1.2.4 + 2.3.5 + 3.4.6 +… Solution: rth term of the series = r(r+1).(r+3)=r3 + 4r2 + 3r So sum of n terms = Σnr=1 r3 + 4Σnr=1 r2 + 3Σnr=1 r = (n(n+1)/2)2 + 4 n(n+1)(2n+1)/6 + 3n(n+1)/2 = n(n+1)/12 {3n2 + 19n + 26}. Example 2: Find the sum of the series 1.n + 2(n–1) + 3.(n–2) +…+ n.1. Solution: The rth term of the series is tr = (1 + (r – 1).1)(n + (r–1)(–1)) = r(n – r + 1) = r(n + 1) – r2 ⇒ Sn= Σnr=1 tr Σnr=1 (n+1)r – Σnr=1 r2 = (n+1) n.(n+1)/2 – n(n+1)(2n+1)/6 = n.(n+1)/2 [n + 1 – 2n+1/3] = n(n+1)(n–2)/6. Posted by: Sharing is caring📗📔📗📕 Member since Apr 2016 0 Thanks Recieved	1,085	2,219	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.75	5	CC-MAIN-2021-21	latest	en	0.795456	# Harmonic Progressions. Updated : October 12th, 2016. Share via \|. ### Harmonic Progression. If inverse of a sequence follows rule of an A.P. then it is said to be in harmonic progression.. e.g. 1,1/2,1/3, 1/4, 1/5 ................ 1/10, 1/7, 1/4, 1, – 1/2, ............ In general 1/a, 1/a+d, 1/a+2d, ................... Note:. Three convenient numbers in H.P. are. 1/a–d, 1/a, 1/a+d. Four convenient numbers in H.P. are. 1/a–3d, 1/a–d, 1/a+d, 1/a+3d. Five convenient numbers in H.P. are. 1/a–2d, 1/a–d, 1/a, 1/a+d, 1/a+2d. Example 1:. Find the 4th and 8th term of the series 6, 4, 3, ……. Solution:. Consider 1/6, /14, 1/3, ...... ∞. Here T2 – T1 = T3 – T2 = 1/12 ⇒ 1/6, 1/4, 1/3 is an A.P.. 4th term of this A.P. = 1/6 + 3 × 1/12 = 1/6 + 1/4 = 5/12,. And the 8th term = 1/6 + 7 × 1/12 = 9/12.. Hence the 8th term of the H.P. = 12/9 = 4/3 and the 4th term = 12/5.. Example 2:. If a, b, c are in H.P., show that a/b+c, b/c+a, c/a+b are also in H.P.	Solution:. Given that a, b, c are in H.P.. ⇒ 1/a, 1/b, 1/c are in A.P.. ⇒ a+b+c/a, a+b+c/b, a+b+c/c are in A.P.. ⇒ 1 + b+c/a, 1 + c+a/b, 1 + a+b/c are in A.P.. ⇒ b+c/a, c+a/b, a+b/c are in A.P.. ⇒ a/b+c, b/c+a, c/a+b are in H.P.. #### Some Important Results. • 1 + 2 + 3 +…+ n = n/2(n + 1) (sum of first n natural numbers).. • 12 + 22 + 32 +…+ n2 = n(n+1)(2n+1)/6 (sum of the squares of first n natural numbers).. • 13 + 23 + 33 +…+ n3 = n2(n+1)2/4 = (1 + 2 + 3 +…+ n)2 (sum of the cubes of first n natural numbers).. • (1 – x)–1 = 1 + x + x2 + x3 +… –1 < x < 1.. • (1 – x)–2 = 1 + 2x + 3x2 +… –1 < x < 1.. Example 1:. Find the nth term and the sum of n terms of the series 1.2.4 + 2.3.5 + 3.4.6 +…. Solution:. rth term of the series = r(r+1).(r+3)=r3 + 4r2 + 3r. So sum of n terms = Σnr=1 r3 + 4Σnr=1 r2 + 3Σnr=1 r. = (n(n+1)/2)2 + 4 n(n+1)(2n+1)/6 + 3n(n+1)/2 = n(n+1)/12 {3n2 + 19n + 26}.. Example 2:. Find the sum of the series 1.n + 2(n–1) + 3.(n–2) +…+ n.1.. Solution:. The rth term of the series is. tr = (1 + (r – 1).1)(n + (r–1)(–1)). = r(n – r + 1) = r(n + 1) – r2. ⇒ Sn= Σnr=1 tr Σnr=1 (n+1)r – Σnr=1 r2 = (n+1) n.(n+1)/2 – n(n+1)(2n+1)/6. = n.(n+1)/2 [n + 1 – 2n+1/3] = n(n+1)(n–2)/6.. Posted by:. Sharing is caring📗📔📗📕. Member since Apr 2016. 0 Thanks Recieved.
https://www.scribd.com/document/335484889/Intersecting-Line	1,542,722,867,000,000,000	text/html	crawl-data/CC-MAIN-2018-47/segments/1542039746398.20/warc/CC-MAIN-20181120130743-20181120152743-00149.warc.gz	989,016,155	36,733	You are on page 1of 3 # Introduction to Computer Programming(CSE1001 ) Lab Test (Set- E) Problem Statement: To test if two straight lines are co-linear, parallel or intersecting. If they are intersecting lines then we need to find their intersection point, and if lines are co-linear then find the length of overlapping portion. Description: The general equation of a line is usually written as y=mx + b where m is the slope of the line and b is the intercept. Slope is how much the vertical (y) coordinate changes for each unit of change in the horizontal direction (x). Intercept is where the line crosses the y-axis, i.e. the value of y when x=0. If we know two points on a line (x1,y1) and (x2,y2), we can determine m and b as follows. From its definition, change in y divided by change in x = m = (y2-y1)/(x2-x1). And since b=y-mx for any point on the line, we can compute b1=y1-mx1 and b2=y2-mx2 With two line segments, we can derive an equation for each, say y=m1x+b1 and y=m2x+b2. The x and y values at the intersection point must satisfy both, so y = m1x+b1=m2x+b2. solving for x, we get m1x+b1=m2x+b2 => (m1-m2)x=b2-b1 => x=(b2-b1)/(m1-m2). Putting x in y=m1x+b1 we can get y=m1*((b2-b1)/(m1-m2)) + b1 Notice that when m1 equals m2, the lines are parallel and they will never intersect. (unless b1 also equals b2 in which case the lines are co-linear). .   Sample Runs: Run1: Input 2 points on line1 (x1.y2) and two points on line2 (u1.int x2.v1) and (u2.y) using the above method described.v2) [2 points] Find slopes m1 and m2 using a method findSlope().int y1.}  [4 points] check if lines are co-linear ◦ check if lines are overlapping ▪ find co-ordinates of two end points of the overlapping portion ▪ find length of the overlapping portion using method findLength() public static double findLength(int x1.int x2.} [2 points] check if lines are parallel [4 points] otherwise lines are intersecting. Header of the method can be public static double findSlope(int x1. Header of the method can be public static double findB(int x1.y1). ◦ find intersecting point (x..double m){.int y1.(x2..}  [2 points] Find b1 and b2 using a method findB()..co-linear lines intersecting lines parallel lines Algorithm Development: In this assignment you need to do the following things:   [1 points] Input two points on line1 (x1.int y1.int y2){..y2) 4 2 2 0 .y1) and (x2..int y2){. (x2.v2) 1 2 2 3 Lines are parallel [In this assignment you need to modify and complete the IntersectingLines.y2) 1 1 2 2 Input 2 points on line2 (u1.(x2.y2) 0 0 1 1 Input 2 points on line2 (u1.v1).y1).v1).v2) 0 4 4 0 Lines are intersecting and intersection point is (3.v1).y1).v2) 2 2 3 3 Lines are co-linear and length of overlapping portion is 0.Input 2 points on line2 (u1.(u2.0 Run3: Input 2 points on line1 (x1.1) Run2: Input 2 points on line1 (x1.(u2.] .(u2.java file provided to you.	908	2,879	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.34375	4	CC-MAIN-2018-47	latest	en	0.880632	You are on page 1of 3. # Introduction to Computer Programming(CSE1001. ). Lab Test (Set- E). Problem Statement: To test if two straight lines are co-linear, parallel or intersecting. If they. are intersecting lines then we need to find their intersection point, and if lines are co-linear then. find the length of overlapping portion.. Description:. The general equation of a line is usually written as y=mx + b where m is the slope of the line and. b is the intercept. Slope is how much the vertical (y) coordinate changes for each unit of change. in the horizontal direction (x). Intercept is where the line crosses the y-axis, i.e. the value of y. when x=0.. If we know two points on a line (x1,y1) and (x2,y2), we can determine m and b as follows. From. its definition, change in y divided by change in x =. m = (y2-y1)/(x2-x1).	And since b=y-mx for any point on the line, we can compute. b1=y1-mx1 and b2=y2-mx2. With two line segments, we can derive an equation for each, say. y=m1x+b1 and y=m2x+b2.. The x and y values at the intersection point must satisfy both, so. y = m1x+b1=m2x+b2.. solving for x, we get m1x+b1=m2x+b2. => (m1-m2)x=b2-b1. => x=(b2-b1)/(m1-m2).. Putting x in y=m1x+b1 we can get. y=m1*((b2-b1)/(m1-m2)) + b1. Notice that when m1 equals m2, the lines are parallel and they will never intersect. (unless b1. also equals b2 in which case the lines are co-linear).. .   Sample Runs: Run1: Input 2 points on line1 (x1.y2) and two points on line2 (u1.int x2.v1) and (u2.y) using the above method described.v2) [2 points] Find slopes m1 and m2 using a method findSlope().int y1.}  [4 points] check if lines are co-linear ◦ check if lines are overlapping ▪ find co-ordinates of two end points of the overlapping portion ▪ find length of the overlapping portion using method findLength() public static double findLength(int x1.int x2.} [2 points] check if lines are parallel [4 points] otherwise lines are intersecting. Header of the method can be public static double findSlope(int x1. Header of the method can be public static double findB(int x1.y1). ◦ find intersecting point (x..double m){.int y1.(x2..}  [2 points] Find b1 and b2 using a method findB()..co-linear lines intersecting lines parallel lines Algorithm Development: In this assignment you need to do the following things:   [1 points] Input two points on line1 (x1.int y1.int y2){..y2) 4 2 2 0 .y1) and (x2..int y2){.. (x2.v2) 1 2 2 3 Lines are parallel [In this assignment you need to modify and complete the IntersectingLines.y2) 1 1 2 2 Input 2 points on line2 (u1.(x2.y2) 0 0 1 1 Input 2 points on line2 (u1.v1).y1).v1).v2) 0 4 4 0 Lines are intersecting and intersection point is (3.v1).y1).v2) 2 2 3 3 Lines are co-linear and length of overlapping portion is 0.Input 2 points on line2 (u1.(u2.0 Run3: Input 2 points on line1 (x1.1) Run2: Input 2 points on line1 (x1.(u2.] .(u2.java file provided to you.
https://fr.maplesoft.com/support/help/Maple/view.aspx?path=Student/MultivariateCalculus/GetNormal	1,675,757,174,000,000,000	text/html	crawl-data/CC-MAIN-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00038.warc.gz	293,381,312	29,067	GetNormal - Maple Help # Online Help ###### All Products Maple MapleSim Student[MultivariateCalculus] GetNormal a normal vector for a plane Calling Sequence GetNormal(p) Parameters p - Plane ; Plane defined in Student[MultivariateCalculus]. Returns • Vector ; Normal vector for the plane. Description • The GetNormal command obtains a normal vector for a plane. This vector is by definition orthogonal to the plane. • The normal vector is not necessarily simplified. Examples > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right):$ > $\mathrm{p1}≔\mathrm{Plane}\left(\left[2,3,-3\right],⟨5,3,1⟩\right):$ > $\mathrm{GetNormal}\left(\mathrm{p1}\right)$ $\left[\begin{array}{c}{5}\\ {3}\\ {1}\end{array}\right]$ (1) > $\mathrm{GetPlot}\left(\mathrm{Plane}\left(\left[2,3,-3\right],⟨5,3,1⟩\right)\right)$ The normal vector can be obtained from a plane defined by any Plane constructor. > $\mathrm{p2}≔\mathrm{Plane}\left(\left[3,6,1\right],\left[4,1,-5\right],\left[3,2,0\right],'\mathrm{variables}'=\left[u,v,x\right]\right):$ > $\mathrm{GetRepresentation}\left(\mathrm{p2}\right)$ ${-}{19}{}{u}{+}{v}{-}{4}{}{x}{=}{-55}$ (2) > $\mathrm{GetNormal}\left(\mathrm{p2}\right)$ $\left[\begin{array}{c}{-19}\\ {1}\\ {-4}\end{array}\right]$ (3) > $\mathrm{GetPlot}\left(\mathrm{Plane}\left(\left[3,6,1\right],\left[4,1,-5\right],\left[3,2,0\right],'\mathrm{variables}'=\left[u,v,x\right]\right)\right)$ Compatibility • The Student[MultivariateCalculus][GetNormal] command was introduced in Maple 17. • For more information on Maple 17 changes, see Updates in Maple 17.	533	1,636	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 11, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5	4	CC-MAIN-2023-06	latest	en	0.395246	GetNormal - Maple Help. # Online Help. ###### All Products Maple MapleSim. Student[MultivariateCalculus]. GetNormal. a normal vector for a plane. Calling Sequence GetNormal(p). Parameters. p - Plane ; Plane defined in Student[MultivariateCalculus].. Returns. • Vector ; Normal vector for the plane.. Description. • The GetNormal command obtains a normal vector for a plane. This vector is by definition orthogonal to the plane.. • The normal vector is not necessarily simplified.. Examples.	> $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right):$. > $\mathrm{p1}≔\mathrm{Plane}\left(\left[2,3,-3\right],⟨5,3,1⟩\right):$. > $\mathrm{GetNormal}\left(\mathrm{p1}\right)$. $\left[\begin{array}{c}{5}\\ {3}\\ {1}\end{array}\right]$ (1). > $\mathrm{GetPlot}\left(\mathrm{Plane}\left(\left[2,3,-3\right],⟨5,3,1⟩\right)\right)$. The normal vector can be obtained from a plane defined by any Plane constructor.. > $\mathrm{p2}≔\mathrm{Plane}\left(\left[3,6,1\right],\left[4,1,-5\right],\left[3,2,0\right],'\mathrm{variables}'=\left[u,v,x\right]\right):$. > $\mathrm{GetRepresentation}\left(\mathrm{p2}\right)$. ${-}{19}{}{u}{+}{v}{-}{4}{}{x}{=}{-55}$ (2). > $\mathrm{GetNormal}\left(\mathrm{p2}\right)$. $\left[\begin{array}{c}{-19}\\ {1}\\ {-4}\end{array}\right]$ (3). > $\mathrm{GetPlot}\left(\mathrm{Plane}\left(\left[3,6,1\right],\left[4,1,-5\right],\left[3,2,0\right],'\mathrm{variables}'=\left[u,v,x\right]\right)\right)$. Compatibility. • The Student[MultivariateCalculus][GetNormal] command was introduced in Maple 17.. • For more information on Maple 17 changes, see Updates in Maple 17.
https://pballew.blogspot.com/2020/01/so-you-though-you-knew-everything-about.html	1,716,644,276,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971058829.24/warc/CC-MAIN-20240525131056-20240525161056-00665.warc.gz	389,721,329	25,780	## Sunday 19 January 2020 ### So You Thought You Knew Everything About Equilateral Triangles, Expanded Version Catching up on some reading lately I came across an interesting book on Academia called Mysteries of the Equilateral Triangle. The only mystery is why he chose to use that word in the title, but it does include a lot of obscure historical and mathematical tidbits about this basic geometric standard. I will share a few I like and add a couple of things I've come across in other places to share as well. The first is in the historical section where he includes an ancient geometric motif from a temple in Chennai, India. It shows three interlocking triangles, but notice they are interlocked in a Brunnian link, like the famous Borromean Rings. While the three can not be separated, no two are individually linked. Along the way he enumerates lots of interesting things about history and property of equilateral triangles that you may well teach, Viviani's theorem, and one of my favorites, Morley's theorem Then he mentions two common recreational challenges. The harder, in my mind, Find the side length and area of the largest square that can be inscribed in an equilateral triangle with unit side lengths. The easier, perhaps, find the side length and area of an the largest equilateral triangle that can be inscribed in a unit square. The two are interestingly tied together. The Figure above showed up on the wonderful Futility Closet blog a while back. It shows that two simple maximization problems are curiously related. Problem one, shown in the equilateral triangle at top is the solution to what is the largest square that can be inscribed in an equilateral triangle with unit side lengths. Problem two, shown in the bottom square is the solution to what is the largest equilateral triangle that can be inscribed in a square with unit side lengths. The coincidence? The length of the side of the square in the triangle is $2 \sqrt3 - 3$ units; and the area of the equilateral triangle in the bottom square, is $2 \sqrt3 - 3$ square units. Greg Ross, the mastermind behind the Futility Closet blog credits John Conway with discovering the proof of this relationship. Digressing a moment from equilateral triangles, another couple of relationships I came across that, somehow, lead me back to the same topic in a roundabout way. I'm strolling through my twitter feed, and Colin Beveridge AKA @icecolbeveridge posted one of his always entertaining "Math Ninja" blogs and it was about Ailles Rectangle. He just showed that it was a great memory device to figure out the trigonometric values of 15o and 75o. It is easy to construct and a nice way to verify directly the sum and addition formulas. I had seen the Ailles (pronounced like the beverage, by coincidence) rectangle several years before, (it's been around even before I was a teacher, but seems not very well known) and had to do a little research to figure out why Colin's looked different. I found an article by Jack S. Calcut at Oberlin College. He gave the original Ailles rectangle from the 1971 article, and sure enough it was different. (Chris Maslanka pointed out that the segment in the upper left should be $\sqrt 3 - 1$ Where Colin had a 30-60-90 triangle inscribed in the rectangle, Ailles had used an isosceles right triangle. Both however, contain the three essential triangles necessary to demonstrate, what I believe is it's great power as a classroom demonstration, they contain ALL of the right triangles that exist with rational angles and each side length containing at most one square root. There is a 30-60-90, a 45-45-90, and a 15-75-90 triangle. That's it, that's all of them, there are no others. And that seems to be impressive as heck to high school students. "Here they are, memorize this image and you have the whole set!" And all those Pythagorean triples you know how to create.... None of them have rational angles in degree measure or as multiples of $\pi$. Since the theme of the day is coincidences, I noticed that the diagrams of Ailles contained another somewhat well known historical result.; If you look remove the 15-75-90 triangle at the top, you are left with a trapezoid that is used in the proof of the Pythagorean Theorem by President James Garfield of the US . Garfield was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Ohio Senate in 1859. He was in congress, not president,when he did the proof which was published in the New England Journal of Education in 1876. As I was studying Conway's curiosity, I realized that there was one more coincidence between the diagram and Ailles rectangle. Using Colin's illustration, if you reflect the 30-60-90 triangle about its longest leg and extend the other lines it will look like this. Constructing the Equilateral Triangle at the bottom side of the figure and we have Conway's figure rotated 180 degrees. So Garfield's Pythagorean proof, with an additional triangle becomes the Ailles Rectangle showing all the rational right triangles with sides with a single quadratic root; and reflecting part of that gives us the square that demonstrates a curious equality between two classic maximization problems. I imagine you could assemble the whole thing out of tiles available for the elementary school. Another little know truth the book reveals is that if you find the Euler Line in a triangle with a 60 degree angle, the Euler line will cut off an equilateral triangle. Ok, how about creating a parabolic arc with equilateral triangles? Simply done by erecting equilateral triangles along a straight line with progressive side lengths of the odd numbers. Another novelty to let your students discover, take an equilateral triangle and circumscribe it in a rectangle. (there are several ways to do that, so they should do a couple each. Now find the area of the three new triangles created inside the rectangle but outside the equilateral triangle. Make an observation. One they may pick up after some thought, if we call the three area A, B, and C, then there will always be one that is the sum of the other two. Two of my favorite equilateral triangle ideas are the arithmetic triangle and Sierpinski's gasket. And of course the second can be arrived by coloring in all the odd numbers in one color and the evens in another. Ok, How about a mathematical quadrilateral you never (probably) heard of, the equlic quadrilateral. A quadrilateral ABCD such that AD and BC are of equal length, and angles A and B have a sum of 120 degrees. The two lower images show two relations of this shape with Equilateral triangles. I an equilatera PCD is erected outside the quadrilateral, then PAB is also equilateral. The bottom shows the midpoints of diagonals P, and Q, and the midpoint of CD will always form an equilateral triangle at all. Toss "equalic quadrilateral" out at the next math dept meeting and rule the crew. Two non-associated pieces not from the book, I'm writing this on November 7th and just printed up my post for On This Day in Math for November 8th, and it includes a note about the Death of Gino Fano on that date in 1952. Fano's relation to the equilateral Triangle, of course, is his famous illustration of a finite geometry in which every line had three points and every point lies on three lines. One last hat tip to John D Cook who pointed out on Linkedin a connection between the Fermat Primes and the Sierpinski Gasket. If you've forgotten, the only Fermat numbers now known to be prime are 3, 5, 17, 257, and 65537. You may also know that these numbers are related to what regular polygons are constructable by straight edge and compass. Only numbers of the form $2^k F$ are constructable by classic geometry methods, where F is a product of distinct Fermat Primes . So that means there can only be $2^5=32$ values for F if we allow the empty set to be represented by 1, If you print out those values in binary, and he did, you get a perfect Gasket. John aligned his on the left margin so his gasket is a right triangle, but it's the same beast. So go check out this book, especially if you are a student of geometry, a parent of a geometry student, or a teacher of geometry, or just like me, and you love looking at beautiful math. And check out John Cook's blogsite as well. Oh, and you might drop back by this blog some time just to see that I'm staying busy and out of trouble. One application of equilateral triangles is the use of the Warren Truss. The Warren Truss was designed in 1848 by James Warren and Willoughby Theobald Monzani. This truss consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length. One example of the use is in the new Bridge across the Tennessee River between Paducah and Ledbetter Kentucky on Hwy US 60. How about some other ideas related to equilateral triangles and circles: One blog I follow regularly is Antonio Gutierrez's gogeometry. If you teach/study/like plane geometry he should be one of your regular references. Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle. I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss. I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site. So on we go... 1) draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices. The sum of the lengths of the two shorter segments will equal the third. The problem, and solution is here 2) OK: Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle. That proof is here. 3) And now one with the circle on the inside. Again, from any point on the circle construct segments to the three vertices of the equilateral triangle. Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself. Or you can go to the site here. Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three. With that same diagram, we should point out an interesting and almost trivial relationship that I think is often overlooked. If you draw a median from either side to the opposite vertex, the encircle will trident that median into three equal parts. The proof for any student in elementary geometry involves only two relationships. The first is that the incenter divides each of the medians in a ratio of 2:1. Now use the fact that the distance from the incenter to the encircle is equal on both sides. And let's add one beautiful proof about infinite series that is approachable to clever students at the geometry level. Analysis by elementary geometry. Try it with your students. I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it. The problem, is that, except for mathematicians who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)2 + (1/4)3+ ... +(1/4)n is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do. The amazing thing in my experience, is that a question about analysis can be illustrated with simple plane geometry. I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual" Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) 2. Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)3... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area. And from a slightly different approach, making equilateral triangles from matchsticks. I posed this question on the appropriate day of the year on my "On This Day in Math " blog. If you build an equilateral triangle with nine matchsticks on each side, then subdivide into additional equilateral triangles, there will be a total of 235 triangles of several different sizes. The image shows the subdivision of a equilateral triangle with three matchsticks on a side. Can you find the thirteen triangles in it? Students will see that there is obviously a single triangle when there is only one matchstick per side. At 2 per side, they get four small triangles and one larger, or four triangles in all. Above I've given away the number of triangles for lengths of 3 on a side, and 9 on a side. Can students determine the relationship for the number of triangles with n toothpicks on a side?	3,290	14,565	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.953125	4	CC-MAIN-2024-22	latest	en	0.910364	## Sunday 19 January 2020. ### So You Thought You Knew Everything About Equilateral Triangles, Expanded Version. Catching up on some reading lately I came across an interesting book on Academia called Mysteries of the Equilateral Triangle. The only mystery is why he chose to use that word in the title, but it does include a lot of obscure historical and mathematical tidbits about this basic geometric standard. I will share a few I like and add a couple of things I've come across in other places to share as well.. The first is in the historical section where he includes an ancient geometric motif from a temple in Chennai, India. It shows three interlocking triangles, but notice they are interlocked in a Brunnian link, like the famous Borromean Rings. While the three can not be separated, no two are individually linked.. Along the way he enumerates lots of interesting things about history and property of equilateral triangles that you may well teach, Viviani's theorem, and one of my favorites, Morley's theorem. Then he mentions two common recreational challenges. The harder, in my mind, Find the side length and area of the largest square that can be inscribed in an equilateral triangle with unit side lengths. The easier, perhaps, find the side length and area of an the largest equilateral triangle that can be inscribed in a unit square.. The two are interestingly tied together.. The Figure above showed up on the wonderful Futility Closet blog a while back. It shows that two simple maximization problems are curiously related. Problem one, shown in the equilateral triangle at top is the solution to what is the largest square that can be inscribed in an equilateral triangle with unit side lengths. Problem two, shown in the bottom square is the solution to what is the largest equilateral triangle that can be inscribed in a square with unit side lengths. The coincidence? The length of the side of the square in the triangle is $2 \sqrt3 - 3$ units; and the area of the equilateral triangle in the bottom square, is $2 \sqrt3 - 3$ square units. Greg Ross, the mastermind behind the Futility Closet blog credits John Conway with discovering the proof of this relationship.. Digressing a moment from equilateral triangles, another couple of relationships I came across that, somehow, lead me back to the same topic in a roundabout way.. I'm strolling through my twitter feed, and Colin Beveridge AKA @icecolbeveridge posted one of his always entertaining "Math Ninja" blogs and it was about Ailles Rectangle. He just showed that it was a great memory device to figure out the trigonometric values of 15o and 75o. It is easy to construct and a nice way to verify directly the sum and addition formulas.. I had seen the Ailles (pronounced like the beverage, by coincidence) rectangle several years before, (it's been around even before I was a teacher, but seems not very well known) and had to do a little research to figure out why Colin's looked different.. I found an article by Jack S. Calcut at Oberlin College. He gave the original Ailles rectangle from the 1971 article, and sure enough it was different. (Chris Maslanka pointed out that the segment in the upper left should be $\sqrt 3 - 1$. Where Colin had a 30-60-90 triangle inscribed in the rectangle, Ailles had used an isosceles right triangle. Both however, contain the three essential triangles necessary to demonstrate, what I believe is it's great power as a classroom demonstration, they contain ALL of the right triangles that exist with rational angles and each side length containing at most one square root. There is a 30-60-90, a 45-45-90, and a 15-75-90 triangle. That's it, that's all of them, there are no others. And that seems to be impressive as heck to high school students. "Here they are, memorize this image and you have the whole set!" And all those Pythagorean triples you know how to create.... None of them have rational angles in degree measure or as multiples of $\pi$.. Since the theme of the day is coincidences, I noticed that the diagrams of Ailles contained another somewhat well known historical result.; If you look remove the 15-75-90 triangle at the top, you are left with a trapezoid that is used in the proof of the Pythagorean Theorem by President James Garfield of the US . Garfield was a professor of mathematics at Hiram College in Ohio for several years before being elected to the Ohio Senate in 1859. He was in congress, not president,when he did the proof which was published in the New England Journal of Education in 1876.. As I was studying Conway's curiosity, I realized that there was one more coincidence between the diagram and Ailles rectangle. Using Colin's illustration, if you reflect the 30-60-90 triangle about its longest leg and extend the other lines it will look like this.. Constructing the Equilateral Triangle at the bottom side of the figure and we have Conway's figure rotated 180 degrees. So Garfield's Pythagorean proof, with an additional triangle becomes the Ailles Rectangle showing all the rational right triangles with sides with a single quadratic root; and reflecting part of that gives us the square that demonstrates a curious equality between two classic maximization problems. I imagine you could assemble the whole thing out of tiles available for the elementary school.. Another little know truth the book reveals is that if you find the Euler Line in a triangle with a 60 degree angle, the Euler line will cut off an equilateral triangle.. Ok, how about creating a parabolic arc with equilateral triangles? Simply done by erecting equilateral triangles along a straight line with progressive side lengths of the odd numbers.. Another novelty to let your students discover, take an equilateral triangle and circumscribe it in a rectangle. (there are several ways to do that, so they should do a couple each. Now find the area of the three new triangles created inside the rectangle but outside the equilateral triangle. Make an observation. One they may pick up after some thought, if we call the three area A, B, and C, then there will always be one that is the sum of the other two.. Two of my favorite equilateral triangle ideas are the arithmetic triangle and Sierpinski's gasket.. And of course the second can be arrived by coloring in all the odd numbers in one color and the evens in another.. Ok, How about a mathematical quadrilateral you never (probably) heard of, the equlic quadrilateral. A quadrilateral ABCD such that AD and BC are of equal length, and angles A and B have a sum of 120 degrees.. The two lower images show two relations of this shape with Equilateral triangles. I an equilatera PCD is erected outside the quadrilateral, then PAB is also equilateral. The bottom shows the midpoints of diagonals P, and Q, and the midpoint of CD will always form an equilateral triangle at all. Toss "equalic quadrilateral" out at the next math dept meeting and rule the crew.. Two non-associated pieces not from the book, I'm writing this on November 7th and just printed up my post for On This Day in Math for November 8th, and it includes a note about the Death of Gino Fano on that date in 1952. Fano's relation to the equilateral Triangle, of course, is his famous illustration of a finite geometry in which every line had three points and every point lies on three lines.. One last hat tip to John D Cook who pointed out on Linkedin a connection between the Fermat Primes and the Sierpinski Gasket. If you've forgotten, the only Fermat numbers now known to be prime are 3, 5, 17, 257, and 65537. You may also know that these numbers are related to what regular polygons are constructable by straight edge and compass. Only numbers of the form $2^k F$ are constructable by classic geometry methods, where F is a product of distinct Fermat Primes .. So that means there can only be $2^5=32$ values for F if we allow the empty set to be represented by 1, If you print out those values in binary, and he did, you get a perfect Gasket. John aligned his on the left margin so his gasket is a right triangle, but it's the same beast.	So go check out this book, especially if you are a student of geometry, a parent of a geometry student, or a teacher of geometry, or just like me, and you love looking at beautiful math. And check out John Cook's blogsite as well.. Oh, and you might drop back by this blog some time just to see that I'm staying busy and out of trouble.. One application of equilateral triangles is the use of the Warren Truss. The Warren Truss was designed in 1848 by James Warren and Willoughby Theobald Monzani. This truss consists of longitudinal members joined only by angled cross-members, forming alternately inverted equilateral triangle-shaped spaces along its length. One example of the use is in the new Bridge across the Tennessee River between Paducah and Ledbetter Kentucky on Hwy US 60.. How about some other ideas related to equilateral triangles and circles:. One blog I follow regularly is Antonio Gutierrez's gogeometry. If you teach/study/like plane geometry he should be one of your regular references.. Recently among his posts have been a couple with a related theme, circles inscribed or circumscribed about an equilateral triangle. I'm listing these because they are each a wonderful relationship, and together give these otherwise somewhat mundane seeming triangles a luster students?teachers/others might miss.. I will post the problems, but not the proofs, which (if you can't/won't work them out yourself you can find at the links provided to Antonio's site.. So on we go.... 1) draw a circle and inscribe an equilateral triangle. Now pick any point on the circumference and construct segments from this point to the three vertices. The sum of the lengths of the two shorter segments will equal the third. The problem, and solution is here. 2) OK:. Same triangle, same circle, but now we sum the square of the three distances ...????? and they sum to twice the square of a side of the equilateral triangle. That proof is here.. 3) And now one with the circle on the inside. Again, from any point on the circle construct segments to the three vertices of the equilateral triangle. Again the sum of the squares is related to a side length, but I'll let you chase that down for yourself. Or you can go to the site here.. Addendum: John Golden sent a comment with a link to a GeoGebra sketch showing all three.. With that same diagram, we should point out an interesting and almost trivial relationship that I think is often overlooked. If you draw a median from either side to the opposite vertex, the encircle will trident that median into three equal parts. The proof for any student in elementary geometry involves only two relationships. The first is that the incenter divides each of the medians in a ratio of 2:1. Now use the fact that the distance from the incenter to the encircle is equal on both sides.. And let's add one beautiful proof about infinite series that is approachable to clever students at the geometry level. Analysis by elementary geometry. Try it with your students.. I have stated previously how much I like "napkin" techniques that give quick calculations or estimates of a problem. I also like things like visual displays which essentially prove some mathematical idea. The one at the top of the page is from the cover of Roger Nelsen's "Proofs without words II.." which is way too expensive for a paperback, but I will probably break down and buy it.. The problem, is that, except for mathematicians who already know the proof, it seldom convinces "without words". Most of my high school students will not look at this image and be able to explain easily and clearly why it shows that the limit as n goes to infinity of 1/4 + (1/4)2 + (1/4)3+ ... +(1/4)n is equal to 1/3. If I'm wrong, not generally, but in your particular case, then stay with me and read what's written, and tell me if you see it the way I do. The amazing thing in my experience, is that a question about analysis can be illustrated with simple plane geometry.. I think they will be able to see that the triangle is divided into thirds... by the colors, 1/3 purple, 1/3 orange, 1/3 white. But I don't know if they can see the series of powers of 1/4 going off to infinity. That's why they have high school teachers... and so here are some words to help make it more "visual". Look at the largest white triangle... can you see it is 1/4 of the largest (outside) triangle?... Ok, now look at the line across the top of the biggest white triangle, it connects the midpoints of two legs of an equilateral triangle, so the triangle above this medial segment, the one with multiple smaller triangles in it, is exactly congruent to the Biggest white triangle and is 1/4 of the total area of the outside triangle also. This upper triangle is a scale model of the original outside triangle,with all the same colors in the same positions and the white triangle in it is 1/4 of the area of this upper replica. So the second largest white triangle is 1/4 of the area of the Largest white triangle.... its area is 1/16 or (1/4) 2. Now the line above the second smallest white triangle is a medial segment of the upper triangle, and so the triangle above it, which is also a scale model of the original biggest triangle, is also 1/16 of the total area... and the third smallest white triangle, is 1/4 of that, so its area is 1/4 of 1/16 or (1/4)3... OK, now you see it, and as you move out each white triangle is 1/4 of the previous one... and sure enough, all the white triangles add up to 1/3 of the total area.. And from a slightly different approach, making equilateral triangles from matchsticks. I posed this question on the appropriate day of the year on my "On This Day in Math " blog.. If you build an equilateral triangle with nine matchsticks on each side, then subdivide into additional equilateral triangles, there will be a total of 235 triangles of several different sizes. The image shows the subdivision of a equilateral triangle with three matchsticks on a side. Can you find the thirteen triangles in it?. Students will see that there is obviously a single triangle when there is only one matchstick per side. At 2 per side, they get four small triangles and one larger, or four triangles in all. Above I've given away the number of triangles for lengths of 3 on a side, and 9 on a side. Can students determine the relationship for the number of triangles with n toothpicks on a side?.
https://www.physicsforums.com/threads/geometric-sets-and-tangent-subspaces-mcinnerney-example-3.858102/	1,508,196,714,000,000,000	text/html	crawl-data/CC-MAIN-2017-43/segments/1508187820466.2/warc/CC-MAIN-20171016214209-20171016234209-00284.warc.gz	1,220,740,497	18,038	# Geometric Sets and Tangent Subspaces - McInnerney, Example 3 1. Feb 18, 2016 ### Math Amateur I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of $T_p ( \mathbb{R}^n )$ ... I need help with a basic aspect of Example 3.3.7 ... Example 3.3.7 reads as follows: Question 1 In the above text we read: " ... ... Let $p = ( x_0, y_0, z_0 ) \in S$ i.e. $2 x_0 - 3 y_0 - z_0 = 0$. ... ... " ... BUT ... as I read the example ... ... we have that $2 x_0 - 3 y_0 - z_0 = 0$ is the equation of $c(t)$ at $( x_0, y_0, z_0 )$ ... AND ... again as I see it ... this is not all of $S$ as $c$ maps $I$ into $S$ ... thus a general point $p = ( x_0, y_0, z_0 ) \in S$ may not satisfy the equation as it may not be in the range of $c$ ... Can some please clarify my issue with the example... ? (I hope I have made my question clear ..) =========================================================== * EDIT * After some reflection I now feel that all points in $S = \phi (U) = ( u, v, 2u - 3v )$ satisfy the equation $2x - 3y - z = 0$ ... so a particular point $p = ( x_0, y_0, z_0)$ obviously satisfies $2x_0 - 3y_0 - z_0 = 0$ ... is that right ... ? Please let me know if my edit is correct ... ========================================================== Question 2 In the above text we read: " ... ... This discussion shows that for all $p \in S$, $T_p (S) = \{ (a, b, c)_p \ \ \| \ \ 2a - 3b - c = 0 \} \subset T_p ( \mathbb{R}^3 )$ ... ... ... ... " Now, it seems that vectors at $p = (x_0, y_0, z_0)$ that have components $a, b, c$ respectively which obey the equation, $2a - 3b - c =0$ are (I think???) in $S$ ... ... so this would mean that $T_p(S)$ is a subset of $S$ ... Is that correct ...? Question 3 Does anyone know of any books with a simple approach to tangent spaces replete with a number of worked/computational exercises ...? Hope someone can help with the above questions ... Peter I have made a simple diagram of my understanding of the mappings involved ... as follows ... ... Is the above diagram a correct representation of the mappings involved? To help to give some of the context and some explanation of the theory and notation relevant to the above I am providing McInerney's introduction to Section 3.3 as follows: #### Attached Files: File size: 167.3 KB Views: 146 File size: 96.3 KB Views: 143 File size: 130.2 KB Views: 134 File size: 113 KB Views: 130 • ###### McInerney - 2 - Section 3.3 - PART 2 - Page 80 .png File size: 68.6 KB Views: 120 Last edited: Feb 18, 2016 2. Feb 18, 2016 ### andrewkirk Yes, it is correct. No. A vector in the tangent space $T_pS$ is not in the space $S$ itself, because the former holds both location and direction information, whereas the latter holds only location information. It's like the difference between 'going through Paris in a Northerly direction at 50 km/h' and just 'Paris'. You can think of the former as a vector of six components, with the first three giving location and the second three giving velocity (direction and magnitude), and the latter as a vector of only three components, giving just location. 3. Feb 18, 2016 ### Math Amateur Thanks for the help Andrew ... but just a clarification ... We know that $S = \phi (U) = (u, v, 2u - 3v)$ So points in $S$ satisfy $2x - 3y - z$... this is the equation that vectors in $T_P (S)$ must satisfy ... so surely all the points of the vector $< a, b, c >_p$ lie in $S$ ... I understand that the points of $S$ do not have a direction and that points are conceptually different from vectors in that vectors have magnitude and direction and points do not ... but can't we say that the points of the vector $< a, b, c >_p$ lie in $S$ ... ? Peter 4. Feb 18, 2016 ### andrewkirk Remember that the vector in $T_pS$ is a 6-tuple $<a,b,c,x,y,z>$ (a '6-vector') with the first three components indicating mag and direction and the second three indicating location. What we can say in this case, and only because the plane $S$ passes through the origin, is that both $<x,y,z>$ and $<a,b,c>$, regarded as '3-vectors', must satisfy the equation. So you can think of each of those two as being in $S$, but neither is the 6-vector, because each contains only half of the information content of the 6-vector. An element of $T_pS$ is a 6-vector, which in this case we can think of two 3-vectors, while an element of $S$ is just a 3-vector. Note also that while $<x,y,z>$ and $<a,b,c>$, interpreted as points in Euclidean 3-space, lie in the same plane S, they do not necessarily point in the same direction. Consider the vector $<-2,3,-13>_{<3,2,0>}$ which is the vector at point $<3,2,0>$ pointing in direction $<-2,3,-13>$, and has 6-tuple representation $<-2,3,-13,3,2,0>$. The direction in which the 3-vector is pointing is (if I've done my calcs right) perpendicular to the location 3-vector whose tail is at the origin and head is at $<3,2,0>$. But both 3-vectors lie in the plane $S$. Another way to visualize vectors in $T_pS$ is as signposts showing the direction and distance to another city. Consider a signpost in Glasgow pointing along a Roman road ('cos they're straight) to Edinburgh that says 'Edinburgh 100km'. The direction of the signpost and the distance 100km is the $<a,b,c>$ info of the 6-vector. The location of the signpost (eg its latitude and longitude) is the $<x,y,z>$ info of the 6-vector. 5. Feb 18, 2016 ### Math Amateur Thanks for the help, Andrew ... appreciate it Still reflecting on what you have said ... Do you know a good and preferably basic reference that treats tangent spaces at undergrad level ... and has some worked examples ,,, Peter Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook	1,701	5,922	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.8125	4	CC-MAIN-2017-43	longest	en	0.885346	# Geometric Sets and Tangent Subspaces - McInnerney, Example 3. 1. Feb 18, 2016. ### Math Amateur. I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic .... I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of $T_p ( \mathbb{R}^n )$ .... I need help with a basic aspect of Example 3.3.7 .... Example 3.3.7 reads as follows:. Question 1. In the above text we read:. " ... ... Let $p = ( x_0, y_0, z_0 ) \in S$ i.e. $2 x_0 - 3 y_0 - z_0 = 0$. ... ... ". ... BUT ... as I read the example ... ... we have that $2 x_0 - 3 y_0 - z_0 = 0$ is the equation of $c(t)$ at $( x_0, y_0, z_0 )$ ... AND ... again as I see it ... this is not all of $S$ as $c$ maps $I$ into $S$ ... thus a general point $p = ( x_0, y_0, z_0 ) \in S$ may not satisfy the equation as it may not be in the range of $c$ .... Can some please clarify my issue with the example... ? (I hope I have made my question clear ..). ===========================================================. * EDIT *. After some reflection I now feel that all points in $S = \phi (U) = ( u, v, 2u - 3v )$ satisfy the equation $2x - 3y - z = 0$ ... so a particular point $p = ( x_0, y_0, z_0)$ obviously satisfies $2x_0 - 3y_0 - z_0 = 0$ ... is that right ... ?. Please let me know if my edit is correct .... ==========================================================. Question 2. In the above text we read:. " ... ... This discussion shows that for all $p \in S$,. $T_p (S) = \{ (a, b, c)_p \ \ \| \ \ 2a - 3b - c = 0 \} \subset T_p ( \mathbb{R}^3 )$. ... ... ... ... ". Now, it seems that vectors at $p = (x_0, y_0, z_0)$ that have components $a, b, c$ respectively which obey the equation, $2a - 3b - c =0$ are (I think???) in $S$ ... ... so this would mean that $T_p(S)$ is a subset of $S$ ... Is that correct ...?. Question 3. Does anyone know of any books with a simple approach to tangent spaces replete with a number of worked/computational exercises ...?. Hope someone can help with the above questions .... Peter. I have made a simple diagram of my understanding of the mappings involved ... as follows ... .... Is the above diagram a correct representation of the mappings involved?. To help to give some of the context and some explanation of the theory and notation relevant to the above I am providing McInerney's introduction to Section 3.3 as follows:. #### Attached Files:. File size:. 167.3 KB. Views:. 146. File size:. 96.3 KB. Views:. 143. File size:. 130.2 KB. Views:. 134. File size:.	113 KB. Views:. 130. • ###### McInerney - 2 - Section 3.3 - PART 2 - Page 80 .png. File size:. 68.6 KB. Views:. 120. Last edited: Feb 18, 2016. 2. Feb 18, 2016. ### andrewkirk. Yes, it is correct.. No. A vector in the tangent space $T_pS$ is not in the space $S$ itself, because the former holds both location and direction information, whereas the latter holds only location information. It's like the difference between 'going through Paris in a Northerly direction at 50 km/h' and just 'Paris'.. You can think of the former as a vector of six components, with the first three giving location and the second three giving velocity (direction and magnitude), and the latter as a vector of only three components, giving just location.. 3. Feb 18, 2016. ### Math Amateur. Thanks for the help Andrew ... but just a clarification .... We know that $S = \phi (U) = (u, v, 2u - 3v)$. So points in $S$ satisfy $2x - 3y - z$... this is the equation that vectors in $T_P (S)$ must satisfy ... so surely all the points of the vector $< a, b, c >_p$ lie in $S$ .... I understand that the points of $S$ do not have a direction and that points are conceptually different from vectors in that vectors have magnitude and direction and points do not ... but can't we say that the points of the vector $< a, b, c >_p$ lie in $S$ ... ?. Peter. 4. Feb 18, 2016. ### andrewkirk. Remember that the vector in $T_pS$ is a 6-tuple $<a,b,c,x,y,z>$ (a '6-vector') with the first three components indicating mag and direction and the second three indicating location. What we can say in this case, and only because the plane $S$ passes through the origin, is that both $<x,y,z>$ and $<a,b,c>$, regarded as '3-vectors', must satisfy the equation. So you can think of each of those two as being in $S$, but neither is the 6-vector, because each contains only half of the information content of the 6-vector.. An element of $T_pS$ is a 6-vector, which in this case we can think of two 3-vectors, while an element of $S$ is just a 3-vector.. Note also that while $<x,y,z>$ and $<a,b,c>$, interpreted as points in Euclidean 3-space, lie in the same plane S, they do not necessarily point in the same direction. Consider the vector $<-2,3,-13>_{<3,2,0>}$ which is the vector at point $<3,2,0>$ pointing in direction $<-2,3,-13>$, and has 6-tuple representation $<-2,3,-13,3,2,0>$. The direction in which the 3-vector is pointing is (if I've done my calcs right) perpendicular to the location 3-vector whose tail is at the origin and head is at $<3,2,0>$. But both 3-vectors lie in the plane $S$.. Another way to visualize vectors in $T_pS$ is as signposts showing the direction and distance to another city. Consider a signpost in Glasgow pointing along a Roman road ('cos they're straight) to Edinburgh that says 'Edinburgh 100km'. The direction of the signpost and the distance 100km is the $<a,b,c>$ info of the 6-vector. The location of the signpost (eg its latitude and longitude) is the $<x,y,z>$ info of the 6-vector.. 5. Feb 18, 2016. ### Math Amateur. Thanks for the help, Andrew ... appreciate it. Still reflecting on what you have said .... Do you know a good and preferably basic reference that treats tangent spaces at undergrad level ... and has some worked examples ,,,. Peter. Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook.
http://science-site.net/logcalcII.htm	1,582,518,869,000,000,000	text/html	crawl-data/CC-MAIN-2020-10/segments/1581875145897.19/warc/CC-MAIN-20200224040929-20200224070929-00237.warc.gz	127,451,055	5,609	Tracking ID UA-46582519-1 THE SCIENCE SITE Understanding Logarithms II a.) Question by Jacinta: "What I am confused about is how 3dB corresponds to an increase in power of 2:1. I understand that 0.301 is log 2 but this value is on the RHS of the equal sign (ie why isn't it 1:2) ? I have had a problem to calculate the factor increase if there is a 60dB difference between 2 sounds. so following the steps above: 10 log(P2/P1)=60 log (P2/P1) = 60/10 log (P2/P1) = 6 I don't understand why in your answer above the result from an increase in 10 dB becomes 10^1 (though I understand steps before that). Just translating this result to my problem I would assume that my answer would be 10^6, but wouldn't this be an answer of 1000000? How does log(p2/P1) start to be referred to as 10^ (or at least that is how it appears to me)." Answer to question: Let's begin with the definition of a logarithm. If we take the expression x = 10^n, then log x = n. So if we set x = 2, then log 2 = 0.301. The definition of a decibel is dependent upon what the variable may be. For voltage it 20 log V2/V1, while for power it is 10 log P2/P1. Therefore dB = 10 log P2/P1, and it is on the RHS of the equal sign. In order to calculate P2/P1, you need to reverse the equation. For dB = 60, dB = 10 log P2/P1 = 60, and therefore log (P2/P1) = 6. Therefore there are six decades of power change =1,000,000. This is the great advantage of the use of logarithms when the numbers become quite large, and this is exactly what happens in electronics and electromagnetism. It also happens in mathematics when we deal with ratios. Note that the difference between 1,000,012 (the number you might want to seek) and log 1,000,000 is a small fraction of one percent. So othe log of 1,000,000 is usually sufficient. Iff you need the log equivalents of very high degrees of accuracies to many decimal places, then you can take the time to go to your log table. The base number doesn't need to be 10, but that is the way we count. It can be any number, such as 2. There are various base numbers, such as 2. The base can have a reference number. Ex: Base 10 ref one milliwatt refers to the ratio with respect to one milliwatt. b.) Solving the problem of calculating a relative increase in power of one dB: For an increase in power of one dB: 10 log(P2/P!) = 1.0 dB, or log (P2/P1) = 0.10. Therefore the increase in power is 10^0.1. We can utilize a calculator to find the answer, but we want to do it without a calculator or table of logarithms, using the process of known values and linear interpolations. The first step is to locate the known value on the log (N) vs N graph nearest the point log (N) = 0.1. We see that the value of log (1) = 0, which is 0 dB. This point is less than 1.0 dB, and the nearest point above it is log (1.414) = 0.1505, which is 1.505 dB. Therefore, N will lie between 1 and 1.414 (the square root of 2), and log (N) lies between 0 and 0.1505. We can now utilize linear interpolation to find a better value and increase the accuracy of the estimate. For linear interpolation, the estimated value of N lies along a straight line between the two points 1 and 1.414. We need to decrease the location of the estimate point from 1.505 dB to 1.00 dB, which is a change of The total spread of the values of log (N) is delta N = (0.1505 - 0) = 0.1505, and this corresponds to a spread of 1.505 dB. Since we want the value of one dB, we have to increase from 0 dB by a factor of 1.0 dB. This is a ratio of 1.0/1.515 = 0.66 The total spread of the values of N is (1.414 -1.0) = 0.414. Therefore, the estimated value of N lies 0.66 x 0.414 = 0.27327 above the 0 dB point, and N = 1.0 + 0.27327 = 1.27327 = P2/P1 (since N is a ratio of power level increase). The actual value is N = 1.25992105 = P2/P1 for a one dB increase in power, and therefore, the power increases by 26% for every dB, as compared to the estimated value of 27.3% . The db calculation error is (1.27327 - 1.25992105) = 0.013349 dB or 1.33%, which is considered to be quite good. Notice that (P2/P1)^3 = 1.25992105^3 = 2, which corresponds to 3 dB for twice the power level. b.) Solving the problem of calculating the absolute power level of one dBm: For a signal level of one dBm, the absolute power level is referenced to one miliwatt(zero dBm), so value of the absolute power is P(dBm) = 1.0 x10^-3 x1.25992105 = 1.26 x 10^-3 = 1.26 mW. Compare this exact value to the estimated value in (a.) of 1.27 mw This is a difference of only 10 microwatts as compared to the exact value of 1.26 microwatts. If you found this process to be somewhat difficult, you can try to visualize the estimation process using the graph below. We have shown that there is a 26% change in power for every one dB increment, and that a 3 dB increment represents a doubling of the power. The curved line is a plot of log(n) vs (n). The estimate point is the circle, and the actual value is the square. The x,y reference points are (1,0) and (1.414,1.5).	1,462	5,000	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.1875	4	CC-MAIN-2020-10	latest	en	0.951566	Tracking ID UA-46582519-1. THE SCIENCE SITE. Understanding Logarithms II. a.) Question by Jacinta:. "What I am confused about is how 3dB corresponds to an increase in power of 2:1. I understand that 0.301 is log 2 but this value is on the RHS of the equal sign (ie why isn't it 1:2) ? I have had a problem to calculate the factor increase if there is a 60dB difference between 2 sounds. so following the steps above:. 10 log(P2/P1)=60. log (P2/P1) = 60/10. log (P2/P1) = 6. I don't understand why in your answer above the result from an increase in 10 dB becomes 10^1 (though I understand steps before that). Just translating this result to my problem I would assume that my answer would be 10^6, but wouldn't this be an answer of 1000000? How does log(p2/P1) start to be referred to as 10^ (or at least that is how it appears to me).". Answer to question: Let's begin with the definition of a logarithm. If we take the expression. x = 10^n, then log x = n. So if we set x = 2, then log 2 = 0.301.. The definition of a decibel is dependent upon what the variable may be. For voltage it 20 log V2/V1, while for power it is 10 log P2/P1. Therefore dB = 10 log P2/P1, and it is on the RHS of the equal sign.. In order to calculate P2/P1, you need to reverse the equation. For dB = 60,. dB = 10 log P2/P1 = 60, and therefore log (P2/P1) = 6. Therefore there are six decades of. power change =1,000,000. This is the great advantage of the use of logarithms when the. numbers become quite large, and this is exactly what happens in electronics and electromagnetism.. It also happens in mathematics when we deal with ratios. Note that the difference between 1,000,012 (the number you might want to seek) and log 1,000,000 is a small fraction of one percent. So othe log of 1,000,000 is usually sufficient. Iff you need the log equivalents of very high degrees of accuracies to many decimal places, then you can take the time to go to your log table.. The base number doesn't need to be 10, but that is the way we count. It can be any number, such as 2. There are various base numbers, such as 2.	The base can have a reference number. Ex: Base 10 ref one milliwatt refers to the ratio with respect to one milliwatt.. b.) Solving the problem of calculating a relative increase in power of one dB:. For an increase in power of one dB: 10 log(P2/P!) = 1.0 dB, or log (P2/P1) = 0.10. Therefore the increase in power is 10^0.1. We can utilize a calculator to find the answer, but we want to do it without a calculator or table of logarithms, using the process of known values and linear interpolations. The first step is to locate the known value on the log (N) vs N graph nearest the point log (N) = 0.1. We see that the value of log (1) = 0, which is 0 dB. This point is less than 1.0 dB, and the nearest point above it is log (1.414) = 0.1505, which is 1.505 dB. Therefore, N will lie between 1 and 1.414 (the square root of 2), and log (N) lies between 0 and 0.1505.. We can now utilize linear interpolation to find a better value and increase the accuracy of the estimate. For linear interpolation, the estimated value of N lies along a straight line between the two points 1 and 1.414. We need to decrease the location of the estimate point from 1.505 dB to 1.00 dB, which is a change of. The total spread of the values of log (N) is delta N = (0.1505 - 0) = 0.1505, and this corresponds to a spread of 1.505 dB. Since we want the value of one dB, we have to increase from 0 dB by a factor of 1.0 dB. This is a ratio of 1.0/1.515 = 0.66. The total spread of the values of N is (1.414 -1.0) = 0.414.. Therefore, the estimated value of N lies. 0.66 x 0.414 = 0.27327 above the 0 dB point, and. N = 1.0 + 0.27327 = 1.27327 = P2/P1 (since N is a ratio of power level increase).. The actual value is N = 1.25992105 = P2/P1 for a one dB increase in power, and therefore, the power increases by 26% for every dB, as compared to the estimated value of 27.3% . The db calculation error is (1.27327 - 1.25992105) = 0.013349 dB or 1.33%, which is considered to be quite good. Notice that (P2/P1)^3 = 1.25992105^3 = 2, which corresponds to 3 dB for twice the power level.. b.) Solving the problem of calculating the absolute power level of one dBm:. For a signal level of one dBm, the absolute power level is referenced to one miliwatt(zero dBm), so value of the absolute power is. P(dBm) = 1.0 x10^-3 x1.25992105 = 1.26 x 10^-3 = 1.26 mW.. Compare this exact value to the estimated value in (a.) of 1.27 mw This is a difference of only 10 microwatts as compared to the exact value of 1.26 microwatts.. If you found this process to be somewhat difficult, you can try to visualize the estimation process using the graph below. We have shown that there is a 26% change in power for every one dB increment, and that a 3 dB increment represents a doubling of the power.. The curved line is a plot of log(n) vs (n). The estimate point is the circle, and the actual value is the square. The x,y reference points are (1,0) and (1.414,1.5).
https://gmat.magoosh.com/forum/5684-three-houses-are-being-sold-through-a	1,637,970,683,000,000,000	text/html	crawl-data/CC-MAIN-2021-49/segments/1637964358074.14/warc/CC-MAIN-20211126224056-20211127014056-00361.warc.gz	350,435,568	19,357	Source: Official Guide for GMAT Review 2016 Data Sufficiency ; #15 3 # Three houses are being sold through a Three houses are being sold through a real estate agent. What is the asking price for the house with the second-largest asking price? ### 1 Explanation 1 John Robertson Given: Mike draws quick diagrams usually, and I found that helps w/o sacrificing a lot of time: House 1 House 2 House 3 P1 P2 P3 -> (using greater/less than or equal to signs, can't do it on my mac) P1 < P2 < P3 . I think you can just assign House 2 as the mid-priced house and so on without making any math mistakes, so I did. 1) P3- P1 = \$130,000. NS for a few reasons. P3 and P1 could be a ton of #s that subtract to \$130,000, also tells you nothing about P2. Cross off A & D 2) P3-P2=\$85,000. NS. introduces P2, but P3 and P2 could be a ton of different test variables. Cross of B 1&2) P3 - P1 =\$130,000 and P3-P2= \$85,000. I think its possible to see, "ok two equations, I could solve for P2 bc there's probably a combination that works out to that." But, that's wrong. Strictly speaking, 3 variables, 2 equations, can't sub in and solve. NS. I think the OG provided a few different numbers that would satisfy Eq 1 and Eq 2 in this case. ANS: E Jan 15, 2017 • Comment Cydney Seigerman, Magoosh Tutor In the prompt, we're asked about the price of the house with the second-largest asking price, which, as you mentioned, we can call the mid-priced house or P2. Nice work analyzing the two statements and concluding that even when they are combined, the information is not sufficient to answer the question :)	442	1,610	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2021-49	longest	en	0.949612	Source: Official Guide for GMAT Review 2016 Data Sufficiency ; #15. 3. # Three houses are being sold through a. Three houses are being sold through a real estate agent. What is the asking price for the house with the second-largest asking price?. ### 1 Explanation. 1. John Robertson. Given:. Mike draws quick diagrams usually, and I found that helps w/o sacrificing a lot of time:. House 1 House 2 House 3. P1 P2 P3. -> (using greater/less than or equal to signs, can't do it on my mac) P1 < P2 < P3 . I think you can just assign House 2 as the mid-priced house and so on without making any math mistakes, so I did.. 1) P3- P1 = \$130,000. NS for a few reasons. P3 and P1 could be a ton of #s that subtract to \$130,000, also tells you nothing about P2.	Cross off A & D. 2) P3-P2=\$85,000. NS. introduces P2, but P3 and P2 could be a ton of different test variables. Cross of B. 1&2) P3 - P1 =\$130,000 and P3-P2= \$85,000. I think its possible to see, "ok two equations, I could solve for P2 bc there's probably a combination that works out to that." But, that's wrong. Strictly speaking, 3 variables, 2 equations, can't sub in and solve. NS. I think the OG provided a few different numbers that would satisfy Eq 1 and Eq 2 in this case.. ANS: E. Jan 15, 2017 • Comment. Cydney Seigerman, Magoosh Tutor. In the prompt, we're asked about the price of the house with the second-largest asking price, which, as you mentioned, we can call the mid-priced house or P2. Nice work analyzing the two statements and concluding that even when they are combined, the information is not sufficient to answer the question :).
https://www.physicsforums.com/threads/where-between-two-charges-does-voltage-0.796078/	1,656,861,981,000,000,000	text/html	crawl-data/CC-MAIN-2022-27/segments/1656104244535.68/warc/CC-MAIN-20220703134535-20220703164535-00539.warc.gz	1,006,768,307	17,697	# Where between two charges does voltage = 0? ## Homework Statement If Q1 in the above figure is twice Q2 and both are positive, where can a point of zero potential be found? V = kq/r ## The Attempt at a Solution I know that eventually I'll have to set it up so that kq/r = kq/r, but my problem is, how do you know where this point will be? How do you know whether it will be between the two points or outside the two points? Thanks! rcgldr Homework Helper If it was outside the points horizontally, then you have the sum of two forces pointing in the same direction, so it can't be outside the points. You might want to use the terms k Q1/R1 and k Q2/(8-R1). If both charges are positive, it couldn't be between the two points either. Although there would be a point somewhere between the two charges where the force on a positively charged object would be zero, the potential would be less at some vertical distance above that point, so the potential wouldn't be zero anywhere except at an infinite distance away from both point charges. The question was mis-worded and corrected below, so that one of the charges is negative and the other positive. In this case there would be a point where the potential is zero. Last edited: If it was outside the points horizontally, then you have the sum of two forces pointing in the same direction, so it can't be outside the points. But I thought voltage was a scalar? rcgldr Homework Helper But I thought voltage was a scalar? Voltage is defined as the difference in potential per unit charge between two points, say A and B. The voltage from A to B is the negative of the voltage from B to A, so in that sense it has a direction. Link to article, look at the lower half of the page where infinity is used as a reference point: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potpoi.html As corrected in my previous post, there's isn't any point within finite distance of two positive point charges where the potential is zero (it can be considered to be zero at infinity). Last edited: SORRY for the confusion, I accidentally copied and pasted the wrong question. The actual problem is, If Q1 is negative and twice Q2, which is positive, where will the potential be zero? rcgldr Homework Helper Now it makes sense to use: k q1/(r) + k q2 / (8-r) = 0. Last edited: BvU	546	2,333	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.9375	4	CC-MAIN-2022-27	latest	en	0.968708	# Where between two charges does voltage = 0?. ## Homework Statement. If Q1 in the above figure is twice Q2 and both are positive, where can a point of zero potential be found?. V = kq/r. ## The Attempt at a Solution. I know that eventually I'll have to set it up so that kq/r = kq/r, but my problem is, how do you know where this point will be? How do you know whether it will be between the two points or outside the two points? Thanks!. rcgldr. Homework Helper. If it was outside the points horizontally, then you have the sum of two forces pointing in the same direction, so it can't be outside the points. You might want to use the terms k Q1/R1 and k Q2/(8-R1).. If both charges are positive, it couldn't be between the two points either. Although there would be a point somewhere between the two charges where the force on a positively charged object would be zero, the potential would be less at some vertical distance above that point, so the potential wouldn't be zero anywhere except at an infinite distance away from both point charges.. The question was mis-worded and corrected below, so that one of the charges is negative and the other positive. In this case there would be a point where the potential is zero.. Last edited:. If it was outside the points horizontally, then you have the sum of two forces pointing in the same direction, so it can't be outside the points.. But I thought voltage was a scalar?. rcgldr.	Homework Helper. But I thought voltage was a scalar?. Voltage is defined as the difference in potential per unit charge between two points, say A and B. The voltage from A to B is the negative of the voltage from B to A, so in that sense it has a direction. Link to article, look at the lower half of the page where infinity is used as a reference point:. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/potpoi.html. As corrected in my previous post, there's isn't any point within finite distance of two positive point charges where the potential is zero (it can be considered to be zero at infinity).. Last edited:. SORRY for the confusion, I accidentally copied and pasted the wrong question. The actual problem is,. If Q1 is negative and twice Q2, which is positive, where will the potential be zero?. rcgldr. Homework Helper. Now it makes sense to use: k q1/(r) + k q2 / (8-r) = 0.. Last edited:. BvU.
https://www.mathworksheets4kids.com/proportions.php	1,532,084,336,000,000,000	text/html	crawl-data/CC-MAIN-2018-30/segments/1531676591578.1/warc/CC-MAIN-20180720100114-20180720120114-00628.warc.gz	939,655,688	12,331	See All Math Topics # Proportions Worksheets This series of proportion worksheets are prepared specifically for learners of grades 6 and 7. Exercises like finding proportions using a pair of ratios, determining proportions in function tables, creating a proportion with a given set of numbers and solving word problems are included here. Students will also learn to identify if the coordinates on a graph share a proportional relationship. Identify the proportion: A pair of ratios To ascertain whether a pair of ratios forms a proportion or not, cross multiply and simplify the fractions. If the cross products so arrived at are found to be equal, the ratios form a proportion. Identify the proportion: Function Tables Determine the ratios between the x and y values for each table. If all ratios obtained across the table are equal, then the values are proportional. Use the answer keys to check your responses. Form a proportion These three-level worksheets require students to create a proportion with the numbers provided. Form two equivalent sets of ratios from a range of 4, 5, and 6 numbers. Three levels of difficulty with 5 worksheets each Identify the Proportion: Graph Observe the coordinates on each graph to determine if they are proportional. The coordinates are in proportion if the graph is a straight line and passes through the origin. Plot the graph and identify the proportion Plot the x and y coordinates on the graphs provided. Draw a graph through the points to ascertain whether x and y values are in proportional relationship. Constant of proportionality Worksheets Access this assortment of proportion worksheets to find the constant of proportionality for the graphs, linear equations, tables and a lot more! (25 Worksheets) Solving proportions Worksheets This batch of proportion worksheets includes multi-level exercises like solving the proportions involving integers, decimals, algebraic expressions and a lot more! (30 Worksheets) Unit Rate Worksheets with Word Problems Employ this series of worksheets to determine the unit rate of a graph, expressing phrases in rates and unit rates, solve well-researched word problems and much more! (25 Worksheets) Proportional graphs - Word problems Examine your comprehension on proportion with this set of print-ready worksheets. Observe the proportional graphs based on real-life scenarios and answer the word problems given below. You may also be interested in:	469	2,463	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.125	4	CC-MAIN-2018-30	longest	en	0.863477	See All Math Topics. # Proportions Worksheets. This series of proportion worksheets are prepared specifically for learners of grades 6 and 7. Exercises like finding proportions using a pair of ratios, determining proportions in function tables, creating a proportion with a given set of numbers and solving word problems are included here. Students will also learn to identify if the coordinates on a graph share a proportional relationship.. Identify the proportion: A pair of ratios. To ascertain whether a pair of ratios forms a proportion or not, cross multiply and simplify the fractions. If the cross products so arrived at are found to be equal, the ratios form a proportion.. Identify the proportion: Function Tables. Determine the ratios between the x and y values for each table. If all ratios obtained across the table are equal, then the values are proportional. Use the answer keys to check your responses.. Form a proportion. These three-level worksheets require students to create a proportion with the numbers provided. Form two equivalent sets of ratios from a range of 4, 5, and 6 numbers.. Three levels of difficulty with 5 worksheets each. Identify the Proportion: Graph. Observe the coordinates on each graph to determine if they are proportional. The coordinates are in proportion if the graph is a straight line and passes through the origin.	Plot the graph and identify the proportion. Plot the x and y coordinates on the graphs provided. Draw a graph through the points to ascertain whether x and y values are in proportional relationship.. Constant of proportionality Worksheets. Access this assortment of proportion worksheets to find the constant of proportionality for the graphs, linear equations, tables and a lot more!. (25 Worksheets). Solving proportions Worksheets. This batch of proportion worksheets includes multi-level exercises like solving the proportions involving integers, decimals, algebraic expressions and a lot more!. (30 Worksheets). Unit Rate Worksheets with Word Problems. Employ this series of worksheets to determine the unit rate of a graph, expressing phrases in rates and unit rates, solve well-researched word problems and much more!. (25 Worksheets). Proportional graphs - Word problems. Examine your comprehension on proportion with this set of print-ready worksheets. Observe the proportional graphs based on real-life scenarios and answer the word problems given below.. You may also be interested in:.
http://www.math.grin.edu/~rebelsky/Courses/CS153/2003S/Labs/procedures.html	1,513,491,649,000,000,000	text/html	crawl-data/CC-MAIN-2017-51/segments/1512948593526.80/warc/CC-MAIN-20171217054825-20171217080825-00755.warc.gz	415,164,529	5,597	# User-Defined Procedures in Scheme Summary: This lab provides practice with simple user-defined procedures. You may want to glance over the corresponding reading before beginning this lab. ## Exercises ### Exercise 0: Preparation Start DrScheme and make sure that you're in full Scheme mode. ### Exercise 1: Values as Fractions Copy the `frac` procedure from the reading on procedures and make sure that it works as advertised. ```;;; Procedure: ;;; frac ;;; Parameters: ;;; val, a number ;;; Purpose: ;;; Express val as a fraction. ;;; Procedues: ;;; rat, a rational number. ;;; Preconditions: ;;; val cannot be complex. ;;; Postconditions: ;;; rat is exact. ;;; rat is approximately equal to val (within some unknown level ;;; of accuracy). ;;; rat is the ratio of two integers. (define frac (lambda (val) (/ (inexact->exact (numerator val)) (inexact->exact (denominator val))))) ``` ### Exercise 2: Adding 2 Write a Scheme procedure `(addtwo a)` that returns the sum a+2. ### Exercise 3: Converting Feet to Meters a. Define a Scheme procedure, `(feet->meters ft)` that takes one argument, a real number representing a length measured in feet, and returns the number that represents the same length as measured in meters. Note that one foot is equal to exactly 761/2500 meters. b. Use this procedure to determine the number of meters in one mile (5280 feet). ### Exercise 4: A Quadratic Polynomial a. Define a procedure, `(poly1 x)`, that corresponds to the polynomial 5x2 - 8x + 2. b. Test your procedure on the values 0, 1, 2, 3, 4. ### Exercise 5: Quadratic Roots a. Write a procedure `(quadratic-root a b c)` that finds one root of a quadratic equation ax2 + bx + c = 0 using the quadratic formula. Use it to find a root of the above equation. (I don't care which root you find.) b. Test your procedure by computing ```(quadratic-root 1 -5 6) (quadratic-root 2 -10 12) (quadratic-root 1 4 4). ``` c. Use algebra to check these answers. d. What are `(quadratic-root 1 0 1)` and `(quadratic-root 1 0 2)`? ### Exercise 6: Swapping List Elements Write a procedure, `(swap-first-two lst)`, that, given a list as an argument, creates a new list that interchanges the first two elements of the original list, leaving the rest of the list unchanged. Thus, ```> (swap-first-two (list 'a 'b 'c 'd 'e)) (b a c d e) ``` In this problem, assume that the list given to `swap-first-two` has at least two elements; do not worry about the possibility that `swap-first-two` might be applied to numbers, symbols, empty lists, or lists with only one element. ### Exercise 7: Spherical Calculations The volume of a sphere of radius r is 4/3 times pi times r3. The circumference of a sphere of radius r is 2 times pi times r. a. Write a procedure named `(sphere-volume r)` that takes as its argument the radius of a sphere (in, say, centimeters) and returns its volume (in, say, cubic centimeters). b. Write a procedure named `(sphere-circ->radius circ)`, which converts the circumference of a sphere to its radius. c. Use these procedures to compute the volume of a standard softball, which has a circumference of 12 inches. d. Use these procedures to compute the volume of a Chicago-style softball, which has a circumference of 16 inches. e. Compute the volumes of each kind of softball using centimeters instead of inches. ### Exercise 8: `snoc` Define a procedure `snoc` (```cons` backwards'') that takes two arguments, of which the second should be a list. `snoc` should return a list just like its second argument, except that the first argument has been added at the right end: ```> (snoc 'alpha (list 'beta 'gamma 'delta)) (beta gamma delta alpha) > (snoc 1 (list 2 3 4 5 6)) (2 3 4 5 6 1) > (snoc 'first null) (first) ``` Hint: There are at least two ways to define this procedure. One uses calls to `reverse` and `cons`; the other uses calls to `append` and `list`. ### Exercise 9: Rotate Write a procedure, `(rotate lst)`, that, given a nonempty list of elements (e.g., `(a b c)`), creates a new list with the original first element moved to the end . For example, ```> (rotate (list 'a 'b 'c)) (b c a) > (rotate (list 1 2)) (2 1) > (rotate (rotate '(first second third fourth))) (third fourth first second) ``` ### Exercise 10: Scoring Figure Skating In a figure-skating competition, judges have observed the competitors' performances and awarded three separate scores to each competitor: one for accuracy, one for style, and one for the difficulty of the chosen routine. Each score is in the range from 0 to 10. The rules of the competition specify that a competitor's three scores are to be combined into a weighted average, in which accuracy counts three times as much as difficulty and style counts twice as much as difficulty. The overall result should be a single number in the range from 0 to 10. a. Write a comment in which you describe the nature and purpose of a procedure that takes three arguments -- a competitor's accuracy, style, and difficulty scores -- and returns their weighted average. b. Define the procedure that you have described. c. Test your procedure, looking for cases in which the weighted average is computed incorrectly. (If you find any, make corrections in your definition.) If you find that you have extra time, you might want to attempt the following tasks: • Write `snoc` (exercise 8) in two different ways. (You should have already written it one way; find another way.) • Implement `(round-to-n-places val p)`, which rounds val to p places after the decimal point. ## History Tuesday, 5 September 2000 [Samuel A. Rebelsky] Wednesday, 31 January 2001 [Samuel A. Rebelsky] • Reformatted for CSC151 2001S. • Added a new problem 1 (that refers to the `frac` procedure from the reading). • Fixed my bad programming habit of using quote to create lists. Sunday, 4 February 2001 [Samuel A. Rebelsky] Tuesday, 10 September 2002 [Samuel A. Rebelsky] • Updated reference links to point to the Glimmer Scheme Reference. • Updated the spherical volume problem to (1) correct information about the size of softballs; (2) add a question on Chicago-style softballs in honor of my students who played softball in Chicago. • Minor updates to text. • Added for those who finish early section. Friday, 13 September 2002 [Samuel A. Rebelsky] Friday, 24 January 2003 [Samuel A. Rebelsky] Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes. This document was generated by Siteweaver on Tue May 6 09:19:58 2003. The source to the document was last modified on Fri Jan 24 09:20:30 2003. This document may be found at `http://www.cs.grinnell.edu/~rebelsky/Courses/CS153/2003S/Labs/procedures.html`. You may wish to validate this document's HTML ; ; Check with Bobby Samuel A. Rebelsky, rebelsky@grinnell.edu	1,763	7,042	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.9375	4	CC-MAIN-2017-51	latest	en	0.816352	# User-Defined Procedures in Scheme. Summary: This lab provides practice with simple user-defined procedures.. You may want to glance over the corresponding reading before beginning this lab.. ## Exercises. ### Exercise 0: Preparation. Start DrScheme and make sure that you're in full Scheme mode.. ### Exercise 1: Values as Fractions. Copy the `frac` procedure from the reading on procedures and make sure that it works as advertised.. ```;;; Procedure:. ;;; frac. ;;; Parameters:. ;;; val, a number. ;;; Purpose:. ;;; Express val as a fraction.. ;;; Procedues:. ;;; rat, a rational number.. ;;; Preconditions:. ;;; val cannot be complex.. ;;; Postconditions:. ;;; rat is exact.. ;;; rat is approximately equal to val (within some unknown level. ;;; of accuracy).. ;;; rat is the ratio of two integers.. (define frac. (lambda (val). (/ (inexact->exact (numerator val)). (inexact->exact (denominator val))))). ```. ### Exercise 2: Adding 2. Write a Scheme procedure `(addtwo a)` that returns the sum a+2.. ### Exercise 3: Converting Feet to Meters. a. Define a Scheme procedure, `(feet->meters ft)` that takes one argument, a real number representing a length measured in feet, and returns the number that represents the same length as measured in meters. Note that one foot is equal to exactly 761/2500 meters.. b. Use this procedure to determine the number of meters in one mile (5280 feet).. ### Exercise 4: A Quadratic Polynomial. a. Define a procedure, `(poly1 x)`, that corresponds to the polynomial 5x2 - 8x + 2.. b. Test your procedure on the values 0, 1, 2, 3, 4.. ### Exercise 5: Quadratic Roots. a. Write a procedure `(quadratic-root a b c)` that finds one root of a quadratic equation. ax2 + bx + c = 0 using the quadratic formula. Use it to find a root of the above equation. (I don't care which root you find.). b. Test your procedure by computing. ```(quadratic-root 1 -5 6). (quadratic-root 2 -10 12). (quadratic-root 1 4 4).. ```. c. Use algebra to check these answers.. d. What are `(quadratic-root 1 0 1)` and `(quadratic-root 1 0 2)`?. ### Exercise 6: Swapping List Elements. Write a procedure, `(swap-first-two lst)`, that, given a list as an argument, creates a new list that interchanges the first two elements of the original list, leaving the rest of the list unchanged. Thus,. ```> (swap-first-two (list 'a 'b 'c 'd 'e)). (b a c d e). ```. In this problem, assume that the list given to `swap-first-two` has at least two elements; do not worry about the possibility that `swap-first-two` might be applied to numbers, symbols, empty lists, or lists with only one element.. ### Exercise 7: Spherical Calculations. The volume of a sphere of radius r is 4/3 times pi times r3.. The circumference of a sphere of radius r is 2 times pi times r.. a. Write a procedure named `(sphere-volume r)` that takes as its argument the radius of a sphere (in, say, centimeters) and returns its volume (in, say, cubic centimeters).. b. Write a procedure named `(sphere-circ->radius circ)`, which converts the circumference of a sphere to its radius.. c.	Use these procedures to compute the volume of a standard softball, which has a circumference of 12 inches.. d. Use these procedures to compute the volume of a Chicago-style softball, which has a circumference of 16 inches.. e. Compute the volumes of each kind of softball using centimeters instead of inches.. ### Exercise 8: `snoc`. Define a procedure `snoc` (```cons` backwards'') that takes two arguments, of which the second should be a list. `snoc` should return a list just like its second argument, except that the first argument has been added at the right end:. ```> (snoc 'alpha (list 'beta 'gamma 'delta)). (beta gamma delta alpha). > (snoc 1 (list 2 3 4 5 6)). (2 3 4 5 6 1). > (snoc 'first null). (first). ```. Hint: There are at least two ways to define this procedure. One uses calls to `reverse` and `cons`; the other uses calls to `append` and `list`.. ### Exercise 9: Rotate. Write a procedure, `(rotate lst)`, that, given a nonempty list of elements (e.g., `(a b c)`), creates a new list with the original first element moved to the end .. For example,. ```> (rotate (list 'a 'b 'c)). (b c a). > (rotate (list 1 2)). (2 1). > (rotate (rotate '(first second third fourth))). (third fourth first second). ```. ### Exercise 10: Scoring Figure Skating. In a figure-skating competition, judges have observed the competitors' performances and awarded three separate scores to each competitor: one for accuracy, one for style, and one for the difficulty of the chosen routine. Each score is in the range from 0 to 10. The rules of the competition specify that a competitor's three scores are to be combined into a weighted average, in which accuracy counts three times as much as difficulty and style counts twice as much as difficulty. The overall result should be a single number in the range from 0 to 10.. a. Write a comment in which you describe the nature and purpose of a procedure that takes three arguments -- a competitor's accuracy, style, and difficulty scores -- and returns their weighted average.. b. Define the procedure that you have described.. c. Test your procedure, looking for cases in which the weighted average is computed incorrectly. (If you find any, make corrections in your definition.). If you find that you have extra time, you might want to attempt the following tasks:. • Write `snoc` (exercise 8) in two different ways. (You should have already written it one way; find another way.). • Implement `(round-to-n-places val p)`, which rounds val to p places after the decimal point.. ## History. Tuesday, 5 September 2000 [Samuel A. Rebelsky]. Wednesday, 31 January 2001 [Samuel A. Rebelsky]. • Reformatted for CSC151 2001S.. • Added a new problem 1 (that refers to the `frac` procedure from the reading).. • Fixed my bad programming habit of using quote to create lists.. Sunday, 4 February 2001 [Samuel A. Rebelsky]. Tuesday, 10 September 2002 [Samuel A. Rebelsky]. • Updated reference links to point to the Glimmer Scheme Reference.. • Updated the spherical volume problem to (1) correct information about the size of softballs; (2) add a question on Chicago-style softballs in honor of my students who played softball in Chicago.. • Minor updates to text.. • Added for those who finish early section.. Friday, 13 September 2002 [Samuel A. Rebelsky]. Friday, 24 January 2003 [Samuel A. Rebelsky]. Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes.. This document was generated by Siteweaver on Tue May 6 09:19:58 2003.. The source to the document was last modified on Fri Jan 24 09:20:30 2003.. This document may be found at `http://www.cs.grinnell.edu/~rebelsky/Courses/CS153/2003S/Labs/procedures.html`.. You may wish to validate this document's HTML ; ; Check with Bobby. Samuel A. Rebelsky, rebelsky@grinnell.edu.
https://whatisconvert.com/275-milligrams-in-metric-tons	1,585,740,297,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585370505730.14/warc/CC-MAIN-20200401100029-20200401130029-00047.warc.gz	798,616,866	6,840	# What is 275 Milligrams in Metric Tons? ## Convert 275 Milligrams to Metric Tons To calculate 275 Milligrams to the corresponding value in Metric Tons, multiply the quantity in Milligrams by 1.0E-9 (conversion factor). In this case we should multiply 275 Milligrams by 1.0E-9 to get the equivalent result in Metric Tons: 275 Milligrams x 1.0E-9 = 2.75E-7 Metric Tons 275 Milligrams is equivalent to 2.75E-7 Metric Tons. ## How to convert from Milligrams to Metric Tons The conversion factor from Milligrams to Metric Tons is 1.0E-9. To find out how many Milligrams in Metric Tons, multiply by the conversion factor or use the Mass converter above. Two hundred seventy-five Milligrams is equivalent to zero point zero zero zero zero zero zero two seven five Metric Tons. ## Definition of Milligram The milligram (abbreviation: mg) is a unit of mass, equal to 1/000 of a gram, and 1/10000000 of a kilogram (also written 1E-6 kg). ## Definition of Metric Ton The tonne (SI unit symbol: t), commonly referred to as the metric ton in the United States, is a non-SI metric unit of mass equal to 1,000 kilograms; or one megagram (Mg); it is equivalent to approximately 2,204.6 pounds, 1.10 short tons (US) or 0.984 long tons (imperial). Although not part of the SI per se, the tonne is "accepted for use with" SI units and prefixes by the International Committee for Weights and Measures. ### Using the Milligrams to Metric Tons converter you can get answers to questions like the following: • How many Metric Tons are in 275 Milligrams? • 275 Milligrams is equal to how many Metric Tons? • How to convert 275 Milligrams to Metric Tons? • How many is 275 Milligrams in Metric Tons? • What is 275 Milligrams in Metric Tons? • How much is 275 Milligrams in Metric Tons? • How many tonne are in 275 mg? • 275 mg is equal to how many tonne? • How to convert 275 mg to tonne? • How many is 275 mg in tonne? • What is 275 mg in tonne? • How much is 275 mg in tonne?	538	1,965	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.578125	4	CC-MAIN-2020-16	latest	en	0.859536	# What is 275 Milligrams in Metric Tons?. ## Convert 275 Milligrams to Metric Tons. To calculate 275 Milligrams to the corresponding value in Metric Tons, multiply the quantity in Milligrams by 1.0E-9 (conversion factor). In this case we should multiply 275 Milligrams by 1.0E-9 to get the equivalent result in Metric Tons:. 275 Milligrams x 1.0E-9 = 2.75E-7 Metric Tons. 275 Milligrams is equivalent to 2.75E-7 Metric Tons.. ## How to convert from Milligrams to Metric Tons. The conversion factor from Milligrams to Metric Tons is 1.0E-9. To find out how many Milligrams in Metric Tons, multiply by the conversion factor or use the Mass converter above. Two hundred seventy-five Milligrams is equivalent to zero point zero zero zero zero zero zero two seven five Metric Tons.. ## Definition of Milligram. The milligram (abbreviation: mg) is a unit of mass, equal to 1/000 of a gram, and 1/10000000 of a kilogram (also written 1E-6 kg).. ## Definition of Metric Ton. The tonne (SI unit symbol: t), commonly referred to as the metric ton in the United States, is a non-SI metric unit of mass equal to 1,000 kilograms; or one megagram (Mg); it is equivalent to approximately 2,204.6 pounds, 1.10 short tons (US) or 0.984 long tons (imperial). Although not part of the SI per se, the tonne is "accepted for use with" SI units and prefixes by the International Committee for Weights and Measures.. ### Using the Milligrams to Metric Tons converter you can get answers to questions like the following:.	• How many Metric Tons are in 275 Milligrams?. • 275 Milligrams is equal to how many Metric Tons?. • How to convert 275 Milligrams to Metric Tons?. • How many is 275 Milligrams in Metric Tons?. • What is 275 Milligrams in Metric Tons?. • How much is 275 Milligrams in Metric Tons?. • How many tonne are in 275 mg?. • 275 mg is equal to how many tonne?. • How to convert 275 mg to tonne?. • How many is 275 mg in tonne?. • What is 275 mg in tonne?. • How much is 275 mg in tonne?.
http://mathsvideos.net/2015/09/17/1-618-as-a-continued-fraction/	1,529,705,974,000,000,000	text/html	crawl-data/CC-MAIN-2018-26/segments/1529267864822.44/warc/CC-MAIN-20180622220911-20180623000911-00548.warc.gz	205,589,859	10,280	# 1.618 as a continued fraction The value 1.618 is an approximation to the golden ratio, a number which is found extensively in nature. As it’s a very interesting number, let’s find out what it would look like as a continued fraction… $1.618\\ \\ =1+\frac { 618 }{ 1000 } \\ \\ =1+\frac { 1 }{ \frac { 1000 }{ 618 } } \\ \\ =1+\frac { 1 }{ \frac { 618 }{ 618 } +\frac { 382 }{ 618 } } \\ \\ =1+\frac { 1 }{ 1+\frac { 382 }{ 618 } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 618 }{ 382 } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 382 }{ 382 } +\frac { 236 }{ 382 } } }$ $\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 236 }{ 382 } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 382 }{ 236 } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 236 }{ 236 } +\frac { 146 }{ 236 } } } }$ $\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 146 }{ 236 } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 236 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 146 }{ 146 } +\frac { 90 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 90 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 146 }{ 90 } } } } } }$ $\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 90 }{ 90 } +\frac { 56 }{ 90 } } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+... } } } } }$	770	1,565	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 4, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.640625	4	CC-MAIN-2018-26	longest	en	0.123081	# 1.618 as a continued fraction. The value 1.618 is an approximation to the golden ratio, a number which is found extensively in nature. As it’s a very interesting number, let’s find out what it would look like as a continued fraction…. $1.618\\ \\ =1+\frac { 618 }{ 1000 } \\ \\ =1+\frac { 1 }{ \frac { 1000 }{ 618 } } \\ \\ =1+\frac { 1 }{ \frac { 618 }{ 618 } +\frac { 382 }{ 618 } } \\ \\ =1+\frac { 1 }{ 1+\frac { 382 }{ 618 } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 618 }{ 382 } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 382 }{ 382 } +\frac { 236 }{ 382 } } }$.	$\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 236 }{ 382 } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 382 }{ 236 } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 236 }{ 236 } +\frac { 146 }{ 236 } } } }$. $\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 146 }{ 236 } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 236 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 146 }{ 146 } +\frac { 90 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 90 }{ 146 } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 146 }{ 90 } } } } } }$. $\\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ \frac { 90 }{ 90 } +\frac { 56 }{ 90 } } } } } } \\ \\ =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+... } } } } }$.
https://www.physicsforums.com/threads/computing-integration-of-bessel-function.590385/	1,537,580,053,000,000,000	text/html	crawl-data/CC-MAIN-2018-39/segments/1537267158001.44/warc/CC-MAIN-20180922005340-20180922025740-00340.warc.gz	839,189,880	13,964	# Computing Integration of Bessel Function 1. Mar 26, 2012 ### sugaku I tried to compute this exact solution, but faced difficulty if the value of $η$ approaching to $ζ$. Let say the value of $ζ$ is fix at 0.5 and the collocation points for η is from 0 to 1. $$θ(η,ζ)=e^{-ε\frac{η}{2}} \left\{ e^{-η}+\left(1-\frac{ε^2}{4}\right)^{1/2} η \int_η^ζ e^{-τ}\frac{I_1 \left\{\left[ \left(1-\frac{ε^2}{4}\right)\left(τ^2-η^2\right) \right]^{1/2} \right\}} {\left(τ^2-η^2\right)} \right\} U(ζ-η)$$ These are the values that is suppose to appear, but only when η=0.5 θ=0.295778, i don't manage to get that value, others is ok. I used trapz command in matlab to calculate the area. η=0.0 θ=1.000000 η=0.1 θ=0.915287 η=0.2 θ=0.831763 η=0.3 θ=0.749758 η=0.4 θ=0.669587 η=0.5 θ=0.295778 η=0.6 θ=0.000000 η=0.7 θ=0.000000 η=0.8 θ=0.000000 η=0.9 θ=0.000000 η=1.0 θ=0.000000 I do suspect that the integration of Bessel function is not simply become 0 when η=0.5 (approach to singularity to that point). I do appreciate if someone could give some advice. Here I attach the matlab program that I wrote. Thank you in advance format short %analytic solution tic ita=0:0.1:1; m=11; ep=0.1; zeta=0.5; area=zeros(1,m); %kira integration dahulu for i=1:m if ita(i)<=zeta tau=linspace(ita(i)+0.000001,0.5,100000); %argument for bessel function a=(1-(ep^2)/4);b=(tau.^2-ita(i)^2); Z=(ab).^(1/2); %Modified bessel function func=@(tau) (exp(-tau).besseli(1,Z))./sqrt(b); area(i)=trapz(tau,func(tau)); else area(i)=0; end end Theta=zeros(1,m); for i=1:m if ita(i)<=zeta Theta(i)=exp((-ita(i)./2)ep)(exp(-ita(i))+sqrt(a)ita(i).area(i)); else Theta(i)=0; end end plot(ita,Theta); axis([0 2.2 0 1]); tableresult(:,1)=ita'; tableresult(:,2)=Theta'; disp(' x Analytic') disp('') disp(tableresult); toc	725	1,790	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.703125	4	CC-MAIN-2018-39	latest	en	0.627361	# Computing Integration of Bessel Function. 1. Mar 26, 2012. ### sugaku. I tried to compute this exact solution, but faced difficulty if the value of $η$ approaching to $ζ$. Let say the value of $ζ$ is fix at 0.5 and the collocation points for η is from 0 to 1.. $$θ(η,ζ)=e^{-ε\frac{η}{2}} \left\{ e^{-η}+\left(1-\frac{ε^2}{4}\right)^{1/2} η \int_η^ζ e^{-τ}\frac{I_1 \left\{\left[ \left(1-\frac{ε^2}{4}\right)\left(τ^2-η^2\right) \right]^{1/2} \right\}} {\left(τ^2-η^2\right)} \right\} U(ζ-η)$$. These are the values that is suppose to appear, but only when η=0.5 θ=0.295778, i don't manage to get that value, others is ok. I used trapz command in matlab to calculate the area.. η=0.0 θ=1.000000. η=0.1 θ=0.915287. η=0.2 θ=0.831763. η=0.3 θ=0.749758. η=0.4 θ=0.669587. η=0.5 θ=0.295778. η=0.6 θ=0.000000. η=0.7 θ=0.000000. η=0.8 θ=0.000000. η=0.9 θ=0.000000. η=1.0 θ=0.000000. I do suspect that the integration of Bessel function is not simply become 0 when η=0.5 (approach to singularity to that point). I do appreciate if someone could give some advice.. Here I attach the matlab program that I wrote. Thank you in advance. format short. %analytic solution. tic. ita=0:0.1:1; m=11;. ep=0.1;. zeta=0.5;. area=zeros(1,m);. %kira integration dahulu.	for i=1:m. if ita(i)<=zeta. tau=linspace(ita(i)+0.000001,0.5,100000);. %argument for bessel function. a=(1-(ep^2)/4);b=(tau.^2-ita(i)^2);. Z=(ab).^(1/2);. %Modified bessel function. func=@(tau) (exp(-tau).besseli(1,Z))./sqrt(b);. area(i)=trapz(tau,func(tau));. else. area(i)=0;. end. end. Theta=zeros(1,m);. for i=1:m. if ita(i)<=zeta. Theta(i)=exp((-ita(i)./2)ep)(exp(-ita(i))+sqrt(a)ita(i).area(i));. else. Theta(i)=0;. end. end. plot(ita,Theta);. axis([0 2.2 0 1]);. tableresult(:,1)=ita';. tableresult(:,2)=Theta';. disp(' x Analytic'). disp(''). disp(tableresult);. toc.
https://www.toppr.com/guides/maths-formulas/cos-2x-formula/	1,718,226,921,000,000,000	text/html	crawl-data/CC-MAIN-2024-26/segments/1718198861261.53/warc/CC-MAIN-20240612203157-20240612233157-00246.warc.gz	967,781,245	27,654	# Cos 2X Formula The word ‘trigonometry’ being driven from the Greek words’ ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. The subject originally thought and part of the scope of development to solve geometric problems involving triangles. We know about the trigonometric ratios of acute angles as the ratio of the sides of a right-angle triangle. In this Chapter, we will generalize the concept and Cos 2X formula of one such trigonometric ratios namely cos 2X with other trigonometric ratios. Let us start with our learning! ## Cos 2X Formula What is a Cos 2X? The trigonometric ratios of an angle in a right triangle define the relationship between the angle and the length of its sides. Cosine 2X or Cos 2X is also, one such trigonometrical formula, also known as double angle formula, as it has a double angle in it. Because of this, it is being driven by the expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions. Let us start with the cos two thetas or cos 2X or cosine of double angle formula. ### Derivation of Cos 2X Formula $$\cos \left ( X+Y \right )$$ = $$\cos X \cos Y – \sin X \sin Y$$ Let us equate, X and Y, i.e. X = Y So, the above formula for cos 2X, becomes $$\cos 2X = \cos \left ( X+X \right ) = \cos X \cos X – \sin X \sin X$$ $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ Hence, the first cos 2X formula follows, as $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ And for this reason, we know this formula as double the angle formula, because we are doubling the angle. ### Other Formulae of cos 2X $$\cos 2X = 1 – 2 \sin ^{2}X$$ To derive this, we need to start from the earlier derivation. As we already know that, $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ $$\cos 2X = \left ( 1-\sin ^{2}X \right ) – \sin ^{2}X [Since, \cos^{2}X = \left ( 1-\sin ^{2}X \right )]$$ $$\cos 2X = 1 – \sin ^{2}X – \sin ^{2}X$$ So, $$\cos 2X = 1 – \left (\sin ^{2}X + \sin ^{2}X \right)$$ $$Hence, \cos 2X = 1 – 2 \sin ^{2}X$$ $$\cos 2X = 2 \cos ^{2}X – 1$$ To derive this, we need to start from the earlier derivation. As we already know that, $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ $$\cos 2X = \cos ^{2}X – \left ( 1-\cos ^{2}X \right ) [Since, \sin^{2}X = \left ( 1-\cos ^{2}X \right )$$ $$\cos 2X = \cos ^{2}X – 1 + \cos ^{2}X$$ $$\cos 2X = \left ( \cos ^{2}X + \cos ^{2}X \right ) – 1$$ $$Hence, \cos 2X = 2\cos ^{2}X – 1$$ $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X}$$ To derive this, we need to start from the earlier derivation. As we already know that, $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$ $$\cos 2X = \frac{\cos ^{2}X – \sin ^{2}X}{1 }$$ $$\cos 2X = \frac{\cos ^{2}X – \sin ^{2}X}{\cos ^{2}X + \sin ^{2}X} [Since, cos ^{2}X + \sin ^{2}X = 1]$$ Dividing both numerator and denominator by $$\cos ^{2}$$X, we get $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X} [ Since, \tan X = \frac{\sin X}{\cos X}]$$ Hence, $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X}$$ ## Solved Examples Now that we have seen the formula of Cos 2X, let us try some examples to deepen our understanding. Q: Prove that, $$\cos 3X = 4 \cos ^{3}X – 3 \cos X,$$ Ans: We have, $$\cos 3X = \cos \left ( 2X + X \right )$$ $$\cos 3X = \cos 2X \cos X – \sin 2X \sin X$$ $$\cos 3X = \left ( 2\cos ^{2}X – 1 \right)\cos X – 2 \sin X\cos X\sin X$$ $$\cos 3X = \left ( 2\cos ^{2}X – 1 \right)\cos X – 2 \cos X\left ( 1-\cos ^{2}X \right )$$ $$\cos 3X = 2\cos ^{3}X – \cos X – 2 \cos X + 2 \cos ^{3}X$$ $$\cos 3X = 4\cos ^{3}X – 3 \cos X$$ [Hence, proved] Share with friends ## Customize your course in 30 seconds ##### Which class are you in? 5th 6th 7th 8th 9th 10th 11th 12th Get ready for all-new Live Classes! Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes. Ashhar Firdausi IIT Roorkee Biology Dr. Nazma Shaik VTU Chemistry Gaurav Tiwari APJAKTU Physics Get Started ## Browse ### One response to “Equation Formula” 1. KUCKOO B says: I get a different answer for first example. I got Q1 as 20.5 median 23 and Q3 26 ## Browse ##### Maths Formulas Watch lectures, practise questions and take tests on the go. No thanks.	1,563	4,227	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.875	5	CC-MAIN-2024-26	latest	en	0.879728	# Cos 2X Formula. The word ‘trigonometry’ being driven from the Greek words’ ‘trigon’ and ‘metron’ and it means ‘measuring the sides of a triangle’. The subject originally thought and part of the scope of development to solve geometric problems involving triangles. We know about the trigonometric ratios of acute angles as the ratio of the sides of a right-angle triangle. In this Chapter, we will generalize the concept and Cos 2X formula of one such trigonometric ratios namely cos 2X with other trigonometric ratios. Let us start with our learning!. ## Cos 2X Formula. What is a Cos 2X?. The trigonometric ratios of an angle in a right triangle define the relationship between the angle and the length of its sides. Cosine 2X or Cos 2X is also, one such trigonometrical formula, also known as double angle formula, as it has a double angle in it.. Because of this, it is being driven by the expressions for trigonometric functions of the sum and difference of two numbers (angles) and related expressions.. Let us start with the cos two thetas or cos 2X or cosine of double angle formula.. ### Derivation of Cos 2X Formula. $$\cos \left ( X+Y \right )$$ = $$\cos X \cos Y – \sin X \sin Y$$. Let us equate, X and Y, i.e. X = Y. So, the above formula for cos 2X, becomes. $$\cos 2X = \cos \left ( X+X \right ) = \cos X \cos X – \sin X \sin X$$. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. Hence, the first cos 2X formula follows, as. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. And for this reason, we know this formula as double the angle formula, because we are doubling the angle.. ### Other Formulae of cos 2X. $$\cos 2X = 1 – 2 \sin ^{2}X$$. To derive this, we need to start from the earlier derivation. As we already know that,. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. $$\cos 2X = \left ( 1-\sin ^{2}X \right ) – \sin ^{2}X [Since, \cos^{2}X = \left ( 1-\sin ^{2}X \right )]$$. $$\cos 2X = 1 – \sin ^{2}X – \sin ^{2}X$$. So,. $$\cos 2X = 1 – \left (\sin ^{2}X + \sin ^{2}X \right)$$. $$Hence, \cos 2X = 1 – 2 \sin ^{2}X$$. $$\cos 2X = 2 \cos ^{2}X – 1$$. To derive this, we need to start from the earlier derivation. As we already know that,. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. $$\cos 2X = \cos ^{2}X – \left ( 1-\cos ^{2}X \right ) [Since, \sin^{2}X = \left ( 1-\cos ^{2}X \right )$$. $$\cos 2X = \cos ^{2}X – 1 + \cos ^{2}X$$. $$\cos 2X = \left ( \cos ^{2}X + \cos ^{2}X \right ) – 1$$. $$Hence, \cos 2X = 2\cos ^{2}X – 1$$. $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X}$$. To derive this, we need to start from the earlier derivation. As we already know that,. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. $$\cos 2X = \cos ^{2}X – \sin ^{2}X$$. $$\cos 2X = \frac{\cos ^{2}X – \sin ^{2}X}{1 }$$. $$\cos 2X = \frac{\cos ^{2}X – \sin ^{2}X}{\cos ^{2}X + \sin ^{2}X} [Since, cos ^{2}X + \sin ^{2}X = 1]$$. Dividing both numerator and denominator by $$\cos ^{2}$$X, we get. $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X} [ Since, \tan X = \frac{\sin X}{\cos X}]$$. Hence, $$\cos 2X = \frac{1-\tan ^{2}X}{1+\tan ^{2}X}$$. ## Solved Examples.	Now that we have seen the formula of Cos 2X, let us try some examples to deepen our understanding.. Q: Prove that,. $$\cos 3X = 4 \cos ^{3}X – 3 \cos X,$$. Ans: We have,. $$\cos 3X = \cos \left ( 2X + X \right )$$. $$\cos 3X = \cos 2X \cos X – \sin 2X \sin X$$. $$\cos 3X = \left ( 2\cos ^{2}X – 1 \right)\cos X – 2 \sin X\cos X\sin X$$. $$\cos 3X = \left ( 2\cos ^{2}X – 1 \right)\cos X – 2 \cos X\left ( 1-\cos ^{2}X \right )$$. $$\cos 3X = 2\cos ^{3}X – \cos X – 2 \cos X + 2 \cos ^{3}X$$. $$\cos 3X = 4\cos ^{3}X – 3 \cos X$$. [Hence, proved]. Share with friends. ## Customize your course in 30 seconds. ##### Which class are you in?. 5th. 6th. 7th. 8th. 9th. 10th. 11th. 12th. Get ready for all-new Live Classes!. Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.. Ashhar Firdausi. IIT Roorkee. Biology. Dr. Nazma Shaik. VTU. Chemistry. Gaurav Tiwari. APJAKTU. Physics. Get Started. ## Browse. ### One response to “Equation Formula”. 1. KUCKOO B says:. I get a different answer for first example.. I got Q1 as 20.5. median 23 and. Q3 26. ## Browse. ##### Maths Formulas. Watch lectures, practise questions and take tests on the go.. No thanks.
https://courses.lumenlearning.com/calculus2/chapter/summary-of-arc-length-of-a-curve-and-surface-area/	1,726,467,488,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651676.3/warc/CC-MAIN-20240916044225-20240916074225-00697.warc.gz	162,493,800	17,614	Summary of Arc Length of a Curve and Surface Area Essential Concepts • The arc length of a curve can be calculated using a definite integral. • The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of $y.$ • The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution. • The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals. Key Equations • Arc Length of a Function of $x$ $\text{Arc Length}={\displaystyle\int }_{a}^{b}\sqrt{1+{\left[{f}^{\prime }(x)\right]}^{2}}dx$ • Arc Length of a Function of $y$ $\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy$ • Surface Area of a Function of $x$ $\text{Surface Area}={\displaystyle\int }_{a}^{b}(2\pi f(x)\sqrt{1+{({f}^{\prime }(x))}^{2}})dx$ Glossary arc length the arc length of a curve can be thought of as the distance a person would travel along the path of the curve frustum a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base surface area the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces	402	1,523	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.28125	4	CC-MAIN-2024-38	latest	en	0.841799	Summary of Arc Length of a Curve and Surface Area. Essential Concepts. • The arc length of a curve can be calculated using a definite integral.. • The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of $y.$. • The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.. • The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals.. Key Equations. • Arc Length of a Function of $x$. $\text{Arc Length}={\displaystyle\int }_{a}^{b}\sqrt{1+{\left[{f}^{\prime }(x)\right]}^{2}}dx$.	• Arc Length of a Function of $y$. $\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy$. • Surface Area of a Function of $x$. $\text{Surface Area}={\displaystyle\int }_{a}^{b}(2\pi f(x)\sqrt{1+{({f}^{\prime }(x))}^{2}})dx$. Glossary. arc length. the arc length of a curve can be thought of as the distance a person would travel along the path of the curve. frustum. a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base. surface area. the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces.
http://www.questionotd.com/2005/08/riddles.html	1,534,363,737,000,000,000	text/html	crawl-data/CC-MAIN-2018-34/segments/1534221210304.2/warc/CC-MAIN-20180815200546-20180815220546-00572.warc.gz	562,773,776	24,123	## Wednesday, August 24, 2005 ### Riddles Since I got such positive feedback on the riddle "Attack!", I thought I'd try a few more: a) What has 4 legs and flies? (Hint: it's not a dead horse!) b) How many times can you subtract 6 from 30? c) What number can you subtract half from to obtain a result that is zero? d) What English word can have 4 of its 5 letters removed and still retain it's original pronunciation? e) You have 10 bags of gold coins, 10 coins per bag, 10 grams per coin, but one bag of coins weigh only 9 grams per coin (because of low quality). How do you find out which bag contains low quality gold coins? You may use a scale only one time. f) How can a woman living in New Jersey, legally marry 3 men, without ever getting a divorce, be widowed, or becoming legally separated? #### 1 comment: 1. a) 2 pairs of pants b) Once. If you subtract 6 from 30, you don't have 30 anymore! c) 8, if you talk the half off of 8, you get 0 on top, and 0 on bottom! d) Queue d) This is the hardest! I actually got this question on a midterm in grad school. step 1: we name all the bag of gold coins as #1, #2, #3......#8, #9, and #10 step 2: we put 1 coin from bag #1, 2 coins from bag #2, 3 coins from bag #3.........8 coins from bag #8, 9 coins from bag #9, and 10 coins from bag 10 onto the scale. Find out the total weight. step 3: the total weight should have been 10 grams X (1+2+3+4+5+6+7+8+9+10=55) = 550 grams if all coins are the same (10grams each). step 4: Subtract the total of step 2 from total of step 3. Conclusion: If step 4 results 1 gram, then bag #1 is the low quality coins, if step 4 results 2 grams, then bag #2 is the one, if step 4 results 3 grams, then bag #3 is the one.......etc. f) It's her job, she's a Justice of the Peace or a Minister. Leave your answer or, if you want to post a question of your own, send me an e-mail. Look in the about section to find my e-mail address. If it's new, I'll post it soon. Please don't leave spam or 'Awesome blog, come visit mine' messages. I'll delete them soon after. Enter your Email and join hundreds of others who get their Question of the Day sent right to their mailbox	623	2,173	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.640625	4	CC-MAIN-2018-34	longest	en	0.928959	## Wednesday, August 24, 2005. ### Riddles. Since I got such positive feedback on the riddle "Attack!", I thought I'd try a few more:. a) What has 4 legs and flies? (Hint: it's not a dead horse!). b) How many times can you subtract 6 from 30?. c) What number can you subtract half from to obtain a result that is zero?. d) What English word can have 4 of its 5 letters removed and still retain it's original pronunciation?. e) You have 10 bags of gold coins, 10 coins per bag, 10 grams per coin, but one bag of coins weigh only 9 grams per coin (because of low quality). How do you find out which bag contains low quality gold coins? You may use a scale only one time.. f) How can a woman living in New Jersey, legally marry 3 men, without ever getting a divorce, be widowed, or becoming legally separated?. #### 1 comment:. 1. a) 2 pairs of pants. b) Once. If you subtract 6 from 30, you don't have 30 anymore!. c) 8, if you talk the half off of 8, you get 0 on top, and 0 on bottom!. d) Queue.	d) This is the hardest! I actually got this question on a midterm in grad school.. step 1: we name all the bag of gold coins as #1, #2, #3......#8, #9, and #10. step 2: we put 1 coin from bag #1, 2 coins from bag #2, 3 coins from bag #3.........8 coins from bag #8, 9 coins from bag #9, and 10 coins from bag 10 onto the scale. Find out the total weight.. step 3: the total weight should have been 10 grams X. (1+2+3+4+5+6+7+8+9+10=55) = 550 grams if all coins are the same (10grams each).. step 4: Subtract the total of step 2 from total of step 3.. Conclusion: If step 4 results 1 gram, then bag #1 is the low quality coins, if step 4 results 2 grams, then bag #2 is the one, if step 4 results 3 grams, then bag #3 is the one.......etc.. f) It's her job, she's a Justice of the Peace or a Minister.. Leave your answer or, if you want to post a question of your own, send me an e-mail. Look in the about section to find my e-mail address. If it's new, I'll post it soon.. Please don't leave spam or 'Awesome blog, come visit mine' messages. I'll delete them soon after.. Enter your Email and join hundreds of others who get their Question of the Day sent right to their mailbox.
https://www.electrical4u.net/calculator/output-power-calculator-formulaoutput-calculation/	1,726,794,840,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700652073.91/warc/CC-MAIN-20240919230146-20240920020146-00852.warc.gz	688,190,667	25,709	# Output Power Calculator, Formula,Output Calculation ## Output Power Calculator: Enter the values of input power, Pi(W) and efficiency, E to determine the value of Output power, Po(W). Enter Input Power W Enter Efficiency: % Result – Output Power: W ### Output Power Formula: Output power signifies the usable or delivered power from a device or system. Measured in watts (W), it reflects the rate at which the device performs work, excluding any energy lost within the system itself. Output power allows us to assess the effectiveness of a device. A higher output power for a given input power indicates better performance. Output power is crucial for calculating efficiency. By comparing it to the input power (energy entering the system), we can determine how much energy is lost during conversion or transmission. Every device or system starts with input power (Pi). This could be electrical energy from the wall outlet, chemical energy from fuel, or solar energy from sunlight. The efficiency (E) of the device or system represents how well it minimizes these energy losses during conversion. A higher efficiency translates to less wasted energy and a higher output power for the same input power. Finally, the output power (Po) represents the usable energy delivered by the device after accounting for these internal losses. It’s the rate at which the device performs its intended function. Output power, Po(W) in watts is calculated by multiplying the input power, Pi(W) in watts with the efficiency, E in percentage. Output power, Po(W) = Pi(W) * E /100 Po(W) = output power in watts, W. Pi(W) = input power in watts, W. E = efficiency. ### Output Power Calculation: 1. A solar panel has an input power rating of 200 watts (Pi) and a rated efficiency of 15%. What is the estimated output power (Po) of the solar panel? Given: Pi(W) = 200 W, E = 15% Output power, Po(W) = Pi(W) * E /100 Po(W) = 200 * 15 / 100 Po(W) = 30 W. 1. An electric motor delivers 80 watts (Po) of usable power and has an efficiency of 80%. What is the minimum input power (Pi) required for the motor? Given: Po(W) = 80 W, E = 80%. Output power, Po(W) = Pi(W) * E /100	501	2,174	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2024-38	latest	en	0.878207	# Output Power Calculator, Formula,Output Calculation. ## Output Power Calculator:. Enter the values of input power, Pi(W) and efficiency, E to determine the value of Output power, Po(W).. Enter Input Power W Enter Efficiency: % Result – Output Power: W. ### Output Power Formula:. Output power signifies the usable or delivered power from a device or system. Measured in watts (W), it reflects the rate at which the device performs work, excluding any energy lost within the system itself.. Output power allows us to assess the effectiveness of a device. A higher output power for a given input power indicates better performance.. Output power is crucial for calculating efficiency. By comparing it to the input power (energy entering the system), we can determine how much energy is lost during conversion or transmission.. Every device or system starts with input power (Pi). This could be electrical energy from the wall outlet, chemical energy from fuel, or solar energy from sunlight.. The efficiency (E) of the device or system represents how well it minimizes these energy losses during conversion. A higher efficiency translates to less wasted energy and a higher output power for the same input power.. Finally, the output power (Po) represents the usable energy delivered by the device after accounting for these internal losses. It’s the rate at which the device performs its intended function.. Output power, Po(W) in watts is calculated by multiplying the input power, Pi(W) in watts with the efficiency, E in percentage.	Output power, Po(W) = Pi(W) * E /100. Po(W) = output power in watts, W.. Pi(W) = input power in watts, W.. E = efficiency.. ### Output Power Calculation:. 1. A solar panel has an input power rating of 200 watts (Pi) and a rated efficiency of 15%. What is the estimated output power (Po) of the solar panel?. Given: Pi(W) = 200 W, E = 15%. Output power, Po(W) = Pi(W) * E /100. Po(W) = 200 * 15 / 100. Po(W) = 30 W.. 1. An electric motor delivers 80 watts (Po) of usable power and has an efficiency of 80%. What is the minimum input power (Pi) required for the motor?. Given: Po(W) = 80 W, E = 80%.. Output power, Po(W) = Pi(W) * E /100.
www.cornerbooks.com.tw	1,723,228,964,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640768597.52/warc/CC-MAIN-20240809173246-20240809203246-00362.warc.gz	554,991,536	36,887	# Intermediate Algebra (4e) #### 優惠價NT\${{ commaFormat(product.member_price) }}NT\${{ commaFormat(product.group_price) }} ###### 商品組合 {{k_row.name}} x {{k_row.qty}} {{k_row.intro}} Intermediate Algebra introduces you to the logic, precision and rigor of mathematics, while helping you build a foundation for future success. With examples and resources that provide step-by-step support, you'll have the tools you need to successfully "do the math". • Preface • R. Review • R.1. Success in Mathematics • R.2. Sets and Classification of Numbers • R.3. Operations on Signed Numbers; Properties of Real Numbers • R.4. Order of Operations • R.5. Algebraic Expressions 1. Linear Equations and Inequalities ### Part I. Linear Equations and Inequalities in One Variable • 1.1. Linear Equations in One Variable • 1.2. An Introduction to Problem Solving • 1.3. Using Formulas to Solve Problems • 1.4. Linear Inequalities in One Variable • Putting the Concepts Together (Sections 1.1–1.4) ### Part II. Linear Equations and Inequalities in Two Variables • 1.5. Rectangular Coordinates and Graphs of Equations • 1.6. Linear Equations in Two Variables • 1.7. Parallel and Perpendicular Lines • 1.8. Linear Inequalities in Two Variables • Chapter 1 Activity: Pass the Paper • Chapter 1 Review • Chapter 1 Test • Cumulative Review Chapters R–1 1. Relations, Functions, and More Inequalities • Putting the Concepts Together (Sections 2.1–2.3) • 2.1. Relations • 2.2. An Introduction to Functions • 2.3. Functions and Their Graphs • 2.4. Linear Functions and Models • 2.5. Compound Inequalities • 2.6. Absolute Value Equations and Inequalities • Chapter 2 Activity: Shifting Discovery • Chapter 2 Review • Chapter 2 Test Systems of Linear Equations and Inequalities • Putting the Concepts Together (Sections 3.1–3.3) • 3.1. Systems of Linear Equations in Two Variables • 3.2. Problem Solving: Systems of Two Linear Equations Containing Two Unknowns • 3.3. Systems of Linear Equations in Three Variables • 3.4. Using Matrices to Solve Systems • 3.5. Determinants and Cramer’s Rule • 3.6. Systems of Linear Inequalities • Chapter 3 Activity: Find the Numbers • Chapter 3 Review • Chapter 3 Test • Cumulative Review Chapters R–3 • Getting Ready for Chapter 4: Laws of Exponents and Scientific Notation Polynomials and Polynomial Functions • Putting the Concepts Together (Sections 4.1–4.3) • 4.1. Adding and Subtracting Polynomials • 4.2. Multiplying Polynomials • 4.3. Dividing Polynomials; Synthetic Division • 4.4. Greatest Common Factor; Factoring by Grouping • 4.5. Factoring Trinomials • 4.6. Factoring Special Products • 4.7. Factoring: A General Strategy • 4.8. Polynomial Equations • Chapter 4 Activity: What Is the Question? • Chapter 4 Review • Chapter 4 Test • Getting Ready for Chapter 5: A Review of Operations on Rational Numbers Rational Expressions and Rational Functions • Putting the Concepts Together (Sections 5.1–5.3) • 5.1. Multiplying and Dividing Rational Expressions • 5.2. Adding and Subtracting Rational Expressions • 5.3. Complex Rational Expressions • 5.4. Rational Equations • 5.5. Rational Inequalities • 5.6. Models Involving Rational Expressions • 5.7. Variation • Chapter 5 Activity: Correct the Quiz • Chapter 5 Review • Chapter 5 Test • Cumulative Review Chapters R–5 • Getting Ready for Chapter 6: Square Roots • Putting the Concepts Together (Sections 6.1–6.5) • 6.1. nth Roots and Rational Exponents • 6.2. Simplifying Expressions Using the Laws of Exponents • 6.7. Radical Equations and Their Applications • 6.8. The Complex Number System • Chapter 6 Activity: Which One Does Not Belong? • Chapter 6 Review • Chapter 6 Test • Putting the Concepts Together (Sections 7.1–7.3) • 7.1. Solving Quadratic Equations by Completing the Square • 7.3. Solving Equations Quadratic in Form • 7.4. Graphing Quadratic Functions Using Transformations • 7.5. Graphing Quadratic Functions Using Properties • 7.6. Polynomial Inequalities • Chapter 7 Activity: Presidential Decision Making • Chapter 7 Review • Chapter 7 Test • Cumulative Review Chapters R–7 Exponential and Logarithmic Functions • Putting the Concepts Together (Sections 8.1–8.3) • 8.1. Composite Functions and Inverse Functions • 8.2. Exponential Functions • 8.3. Logarithmic Functions • 8.4. Properties of Logarithms • 8.5. Exponential and Logarithmic Equations • Chapter 8 Activity: Correct the Quiz • Chapter 8 Review • Chapter 8 Test Conics • Putting the Concepts Together (Sections 9.1–9.5) • 9.1. Distance and Midpoint Formulas • 9.2. Circles • 9.3. Parabolas • 9.4. Ellipses • 9.5. Hyperbolas • 9.6. Systems of Nonlinear Equations • Chapter 9 Activity: How Do You Know That . . . ? • Chapter 9 Review • Chapter 9 Test • Cumulative Review: Chapters R–9 Sequences, Series, and the Binomial Theorem • Putting the Concepts Together (Sections 10.1–10.3) • 10.1. Sequences • 10.2. Arithmetic Sequences • 10.3. Geometric Sequences and Series • 10.4. The Binomial Theorem • Chapter 10 Activity: Discover the Relation • Chapter 10 Review • Chapter 10 Test #### Photo Credits • 出版者 ‏ : ‎ Pearson; 第4版 (2017年 1月 1日) • 語言 ‏ : ‎ English • Hardcover ‏ : ‎ 888 頁 • ISBN-10 ‏ : ‎ 0134555805 • ISBN-13 ‏ : ‎ 978-0134555805 • 商品重量 ‏ : ‎ 1.86 Kilograms • 尺寸 ‏ : ‎ 22.48 x 4.45 x 28.32 cm • 台灣本島訂單金額未滿2,000元，需支付運費100 元, 滿額免運費。 • 台灣外島地區訂單需另外報價運費。 • 國外及大陸地區訂購, 請來電／來信／加入LINE@詢問客服。 • 若商品發生瑕疵請於七日內拍照回傳告知（若外箱有所損壞也請一併拍照），並提供照片向客服人員申請更換新品 , 同時將商品與發票包裝好,我方將派遣貨運公司回收並補寄。 • 本站接受瑕疵換書，暫不接受退款及替換其它書籍。 ※使用超商取貨付款無故未取者，將永久取消其會員資格。	1,808	5,614	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2024-33	latest	en	0.663293	# Intermediate Algebra (4e). #### 優惠價NT\${{ commaFormat(product.member_price) }}NT\${{ commaFormat(product.group_price) }}. ###### 商品組合. {{k_row.name}} x {{k_row.qty}}. {{k_row.intro}}. Intermediate Algebra introduces you to the logic, precision and rigor of mathematics, while helping you build a foundation for future success. With examples and resources that provide step-by-step support, you'll have the tools you need to successfully "do the math".. • Preface. • R. Review. • R.1. Success in Mathematics. • R.2. Sets and Classification of Numbers. • R.3. Operations on Signed Numbers; Properties of Real Numbers. • R.4. Order of Operations. • R.5. Algebraic Expressions. 1. Linear Equations and Inequalities. ### Part I. Linear Equations and Inequalities in One Variable. • 1.1. Linear Equations in One Variable. • 1.2. An Introduction to Problem Solving. • 1.3. Using Formulas to Solve Problems. • 1.4. Linear Inequalities in One Variable. • Putting the Concepts Together (Sections 1.1–1.4). ### Part II. Linear Equations and Inequalities in Two Variables. • 1.5. Rectangular Coordinates and Graphs of Equations. • 1.6. Linear Equations in Two Variables. • 1.7. Parallel and Perpendicular Lines. • 1.8. Linear Inequalities in Two Variables. • Chapter 1 Activity: Pass the Paper. • Chapter 1 Review. • Chapter 1 Test. • Cumulative Review Chapters R–1. 1. Relations, Functions, and More Inequalities. • Putting the Concepts Together (Sections 2.1–2.3). • 2.1. Relations. • 2.2. An Introduction to Functions. • 2.3. Functions and Their Graphs. • 2.4. Linear Functions and Models. • 2.5. Compound Inequalities. • 2.6. Absolute Value Equations and Inequalities. • Chapter 2 Activity: Shifting Discovery. • Chapter 2 Review. • Chapter 2 Test. Systems of Linear Equations and Inequalities. • Putting the Concepts Together (Sections 3.1–3.3). • 3.1. Systems of Linear Equations in Two Variables. • 3.2. Problem Solving: Systems of Two Linear Equations Containing Two Unknowns. • 3.3. Systems of Linear Equations in Three Variables. • 3.4. Using Matrices to Solve Systems. • 3.5. Determinants and Cramer’s Rule. • 3.6. Systems of Linear Inequalities. • Chapter 3 Activity: Find the Numbers. • Chapter 3 Review. • Chapter 3 Test. • Cumulative Review Chapters R–3. • Getting Ready for Chapter 4: Laws of Exponents and Scientific Notation. Polynomials and Polynomial Functions. • Putting the Concepts Together (Sections 4.1–4.3). • 4.1. Adding and Subtracting Polynomials. • 4.2. Multiplying Polynomials. • 4.3. Dividing Polynomials; Synthetic Division. • 4.4. Greatest Common Factor; Factoring by Grouping. • 4.5. Factoring Trinomials. • 4.6. Factoring Special Products. • 4.7. Factoring: A General Strategy. • 4.8. Polynomial Equations. • Chapter 4 Activity: What Is the Question?. • Chapter 4 Review. • Chapter 4 Test. • Getting Ready for Chapter 5: A Review of Operations on Rational Numbers. Rational Expressions and Rational Functions. • Putting the Concepts Together (Sections 5.1–5.3).	• 5.1. Multiplying and Dividing Rational Expressions. • 5.2. Adding and Subtracting Rational Expressions. • 5.3. Complex Rational Expressions. • 5.4. Rational Equations. • 5.5. Rational Inequalities. • 5.6. Models Involving Rational Expressions. • 5.7. Variation. • Chapter 5 Activity: Correct the Quiz. • Chapter 5 Review. • Chapter 5 Test. • Cumulative Review Chapters R–5. • Getting Ready for Chapter 6: Square Roots. • Putting the Concepts Together (Sections 6.1–6.5). • 6.1. nth Roots and Rational Exponents. • 6.2. Simplifying Expressions Using the Laws of Exponents. • 6.7. Radical Equations and Their Applications. • 6.8. The Complex Number System. • Chapter 6 Activity: Which One Does Not Belong?. • Chapter 6 Review. • Chapter 6 Test. • Putting the Concepts Together (Sections 7.1–7.3). • 7.1. Solving Quadratic Equations by Completing the Square. • 7.3. Solving Equations Quadratic in Form. • 7.4. Graphing Quadratic Functions Using Transformations. • 7.5. Graphing Quadratic Functions Using Properties. • 7.6. Polynomial Inequalities. • Chapter 7 Activity: Presidential Decision Making. • Chapter 7 Review. • Chapter 7 Test. • Cumulative Review Chapters R–7. Exponential and Logarithmic Functions. • Putting the Concepts Together (Sections 8.1–8.3). • 8.1. Composite Functions and Inverse Functions. • 8.2. Exponential Functions. • 8.3. Logarithmic Functions. • 8.4. Properties of Logarithms. • 8.5. Exponential and Logarithmic Equations. • Chapter 8 Activity: Correct the Quiz. • Chapter 8 Review. • Chapter 8 Test. Conics. • Putting the Concepts Together (Sections 9.1–9.5). • 9.1. Distance and Midpoint Formulas. • 9.2. Circles. • 9.3. Parabolas. • 9.4. Ellipses. • 9.5. Hyperbolas. • 9.6. Systems of Nonlinear Equations. • Chapter 9 Activity: How Do You Know That . . . ?. • Chapter 9 Review. • Chapter 9 Test. • Cumulative Review: Chapters R–9. Sequences, Series, and the Binomial Theorem. • Putting the Concepts Together (Sections 10.1–10.3). • 10.1. Sequences. • 10.2. Arithmetic Sequences. • 10.3. Geometric Sequences and Series. • 10.4. The Binomial Theorem. • Chapter 10 Activity: Discover the Relation. • Chapter 10 Review. • Chapter 10 Test. #### Photo Credits. • 出版者 ‏ : ‎ Pearson; 第4版 (2017年 1月 1日). • 語言 ‏ : ‎ English. • Hardcover ‏ : ‎ 888 頁. • ISBN-10 ‏ : ‎ 0134555805. • ISBN-13 ‏ : ‎ 978-0134555805. • 商品重量 ‏ : ‎ 1.86 Kilograms. • 尺寸 ‏ : ‎ 22.48 x 4.45 x 28.32 cm. • 台灣本島訂單金額未滿2,000元，需支付運費100 元, 滿額免運費。. • 台灣外島地區訂單需另外報價運費。. • 國外及大陸地區訂購, 請來電／來信／加入LINE@詢問客服。. • 若商品發生瑕疵請於七日內拍照回傳告知（若外箱有所損壞也請一併拍照），並提供照片向客服人員申請更換新品 , 同時將商品與發票包裝好,我方將派遣貨運公司回收並補寄。. • 本站接受瑕疵換書，暫不接受退款及替換其它書籍。. ※使用超商取貨付款無故未取者，將永久取消其會員資格。.
https://nrich.maths.org/public/leg.php?code=5001&cl=1&cldcmpid=5998	1,508,206,472,000,000,000	text/html	crawl-data/CC-MAIN-2017-43/segments/1508187820556.7/warc/CC-MAIN-20171017013608-20171017033608-00038.warc.gz	957,242,490	9,489	Search by Topic Resources tagged with Odd and even numbers similar to Sorting Numbers: Filter by: Content type: Stage: Challenge level: There are 61 results Broad Topics > Numbers and the Number System > Odd and even numbers Sorting Numbers Stage: 1 Challenge Level: Use the interactivity to sort these numbers into sets. Can you give each set a name? Odd Tic Tac Stage: 1 Challenge Level: An odd version of tic tac toe Part the Piles Stage: 2 Challenge Level: Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy? Seven Flipped Stage: 2 Challenge Level: Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time. More Numbers in the Ring Stage: 1 Challenge Level: If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice? Venn Diagrams Stage: 1 and 2 Challenge Level: Use the interactivities to complete these Venn diagrams. Red Even Stage: 2 Challenge Level: You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters? Various Venns Stage: 2 Challenge Level: Use the interactivities to complete these Venn diagrams. Down to Nothing Stage: 2 Challenge Level: A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6. Arrangements Stage: 2 Challenge Level: Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....? Take One Example Stage: 1 and 2 This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure. Number Differences Stage: 2 Challenge Level: Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this? Odd Squares Stage: 2 Challenge Level: Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this? Share Bears Stage: 1 Challenge Level: Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over? I Like ... Stage: 1 Challenge Level: Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea? Light the Lights Stage: 1 Challenge Level: Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights? Carroll Diagrams Stage: 1 Challenge Level: Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers? More Carroll Diagrams Stage: 2 Challenge Level: How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column? Crossings Stage: 2 Challenge Level: In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest? Curious Number Stage: 2 Challenge Level: Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on? Ring a Ring of Numbers Stage: 1 Challenge Level: Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd. Odds and Threes Stage: 2 Challenge Level: A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3. One of Thirty-six Stage: 1 Challenge Level: Can you find the chosen number from the grid using the clues? Becky's Number Plumber Stage: 2 Challenge Level: Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings? Break it Up! Stage: 1 and 2 Challenge Level: In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes. Always, Sometimes or Never? Stage: 1 and 2 Challenge Level: Are these statements relating to odd and even numbers always true, sometimes true or never true? Diagonal Trace Stage: 2 Challenge Level: You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon? Cube Bricks and Daisy Chains Stage: 1 Challenge Level: Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos. Pairs of Numbers Stage: 1 Challenge Level: If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why? Number Detective Stage: 2 Challenge Level: Follow the clues to find the mystery number. Pairs of Legs Stage: 1 Challenge Level: How many legs do each of these creatures have? How many pairs is that? Make 37 Stage: 2 and 3 Challenge Level: Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37. Three Spinners Stage: 2 Challenge Level: These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner? Play to 37 Stage: 2 Challenge Level: In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37. Lots of Biscuits! Stage: 1 Challenge Level: Help share out the biscuits the children have made. What Do You Need? Stage: 2 Challenge Level: Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number? The Set of Numbers Stage: 1 Challenge Level: Can you place the numbers from 1 to 10 in the grid? The Thousands Game Stage: 2 Challenge Level: Each child in Class 3 took four numbers out of the bag. Who had made the highest even number? Multiplication Series: Number Arrays Stage: 1 and 2 This article for teachers describes how number arrays can be a useful reprentation for many number concepts. Numbers as Shapes Stage: 1 Challenge Level: Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares? A Mixed-up Clock Stage: 2 Challenge Level: There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements? What Number? Stage: 1 Short Challenge Level: I am less than 25. My ones digit is twice my tens digit. My digits add up to an even number. Grouping Goodies Stage: 1 Challenge Level: Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had? Sets of Numbers Stage: 2 Challenge Level: How many different sets of numbers with at least four members can you find in the numbers in this box? Number Round Up Stage: 1 Challenge Level: Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre. Two Numbers Under the Microscope Stage: 1 Challenge Level: This investigates one particular property of number by looking closely at an example of adding two odd numbers together. Square Subtraction Stage: 2 Challenge Level: Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it? Take Three Numbers Stage: 2 Challenge Level: What happens when you add three numbers together? Will your answer be odd or even? How do you know? Odd Times Even Stage: 1 Challenge Level: This problem looks at how one example of your choice can show something about the general structure of multiplication. How Odd Stage: 1 Challenge Level: This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?	1,995	8,535	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.4375	4	CC-MAIN-2017-43	longest	en	0.903303	Search by Topic. Resources tagged with Odd and even numbers similar to Sorting Numbers:. Filter by: Content type:. Stage:. Challenge level:. There are 61 results. Broad Topics > Numbers and the Number System > Odd and even numbers. Sorting Numbers. Stage: 1 Challenge Level:. Use the interactivity to sort these numbers into sets. Can you give each set a name?. Odd Tic Tac. Stage: 1 Challenge Level:. An odd version of tic tac toe. Part the Piles. Stage: 2 Challenge Level:. Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?. Seven Flipped. Stage: 2 Challenge Level:. Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.. More Numbers in the Ring. Stage: 1 Challenge Level:. If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?. Venn Diagrams. Stage: 1 and 2 Challenge Level:. Use the interactivities to complete these Venn diagrams.. Red Even. Stage: 2 Challenge Level:. You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?. Various Venns. Stage: 2 Challenge Level:. Use the interactivities to complete these Venn diagrams.. Down to Nothing. Stage: 2 Challenge Level:. A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.. Arrangements. Stage: 2 Challenge Level:. Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?. Take One Example. Stage: 1 and 2. This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.. Number Differences. Stage: 2 Challenge Level:. Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?. Odd Squares. Stage: 2 Challenge Level:. Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?. Share Bears. Stage: 1 Challenge Level:. Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?. I Like .... Stage: 1 Challenge Level:. Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?. Light the Lights. Stage: 1 Challenge Level:. Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?. Carroll Diagrams. Stage: 1 Challenge Level:. Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?. More Carroll Diagrams. Stage: 2 Challenge Level:. How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?. Crossings. Stage: 2 Challenge Level:. In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?. Curious Number. Stage: 2 Challenge Level:. Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?. Ring a Ring of Numbers. Stage: 1 Challenge Level:. Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.. Odds and Threes. Stage: 2 Challenge Level:. A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.. One of Thirty-six. Stage: 1 Challenge Level:. Can you find the chosen number from the grid using the clues?. Becky's Number Plumber. Stage: 2 Challenge Level:. Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?. Break it Up!.	Stage: 1 and 2 Challenge Level:. In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.. Always, Sometimes or Never?. Stage: 1 and 2 Challenge Level:. Are these statements relating to odd and even numbers always true, sometimes true or never true?. Diagonal Trace. Stage: 2 Challenge Level:. You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?. Cube Bricks and Daisy Chains. Stage: 1 Challenge Level:. Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.. Pairs of Numbers. Stage: 1 Challenge Level:. If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?. Number Detective. Stage: 2 Challenge Level:. Follow the clues to find the mystery number.. Pairs of Legs. Stage: 1 Challenge Level:. How many legs do each of these creatures have? How many pairs is that?. Make 37. Stage: 2 and 3 Challenge Level:. Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.. Three Spinners. Stage: 2 Challenge Level:. These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?. Play to 37. Stage: 2 Challenge Level:. In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.. Lots of Biscuits!. Stage: 1 Challenge Level:. Help share out the biscuits the children have made.. What Do You Need?. Stage: 2 Challenge Level:. Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?. The Set of Numbers. Stage: 1 Challenge Level:. Can you place the numbers from 1 to 10 in the grid?. The Thousands Game. Stage: 2 Challenge Level:. Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?. Multiplication Series: Number Arrays. Stage: 1 and 2. This article for teachers describes how number arrays can be a useful reprentation for many number concepts.. Numbers as Shapes. Stage: 1 Challenge Level:. Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?. A Mixed-up Clock. Stage: 2 Challenge Level:. There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?. What Number?. Stage: 1 Short Challenge Level:. I am less than 25. My ones digit is twice my tens digit. My digits add up to an even number.. Grouping Goodies. Stage: 1 Challenge Level:. Pat counts her sweets in different groups and both times she has some left over. How many sweets could she have had?. Sets of Numbers. Stage: 2 Challenge Level:. How many different sets of numbers with at least four members can you find in the numbers in this box?. Number Round Up. Stage: 1 Challenge Level:. Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.. Two Numbers Under the Microscope. Stage: 1 Challenge Level:. This investigates one particular property of number by looking closely at an example of adding two odd numbers together.. Square Subtraction. Stage: 2 Challenge Level:. Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?. Take Three Numbers. Stage: 2 Challenge Level:. What happens when you add three numbers together? Will your answer be odd or even? How do you know?. Odd Times Even. Stage: 1 Challenge Level:. This problem looks at how one example of your choice can show something about the general structure of multiplication.. How Odd. Stage: 1 Challenge Level:. This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?.
https://support.minitab.com/en-us/minitab-express/1/help-and-how-to/basic-statistics/probability-distributions/how-to/cumulative-distribution-function-cdf/perform-the-analysis/select-data-distribution-and-parameters/	1,529,574,664,000,000,000	text/html	crawl-data/CC-MAIN-2018-26/segments/1529267864139.22/warc/CC-MAIN-20180621094633-20180621114633-00384.warc.gz	726,115,385	4,434	# Select the distribution and parameters for Cumulative Distribution Function (CDF) In the Cumulative Distribution Function (CDF) dialog box, select the distribution and the parameters. ## Binomial Complete the following steps to enter the parameters for the binomial distribution. 1. In Number of trials, enter the sample size. 2. In Event probability, enter a number between 0 and 1 for the probability that the outcome you are interested in occurs. An occurrence is called an "event". ## Chi-square Complete the following steps to enter the parameters for the chi-square distribution. 1. In Degrees of freedom, enter the degrees of freedom to define the chi-square distribution. 2. If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0. ## Discrete Complete the following steps to enter the parameters for the discrete distribution. 1. In Values in, enter the column that contains the values to include in the distribution. Usually, values are discrete events or counts that are represented by numeric values. 2. In Probabilities in, enter the column that contains the probabilities for each value. Probabilities must be between 0 and 1, and must sum to 1. ## Exponential Complete the following steps to enter the parameters for the exponential distribution. 1. In Scale, enter the scale parameter. The scale parameter equals the mean when the threshold parameter equals 0. 2. In Threshold, enter the lower bound of the distribution. ## F Complete the following steps to enter the parameters for the F-distribution. 1. In Numerator degrees of freedom and Denominator degrees of freedom, enter the numerator and denominator degrees of freedom to define the F-distribution. 2. If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0. ## Geometric Complete the following steps to enter the parameters for the Geometric distribution. 1. In Event probability, enter a number between 0 and 1 for the probability of an occurrence on each trial. An occurrence is called an "event". 2. From Model, select one of the following to specify the number to model. • Total number of trials: The number of trials includes both events and nonevents. • Only the number of non-events: Do not count the event. For example, this plot shows a geometric distribution that has an event probability of 0.5 and models the total number of trials. ## Integer Complete the following steps to enter the parameters for the integer distribution. 1. In Minimum value, enter the lower end point of the distribution. 2. In Maximum value, enter the upper end point of the distribution. ## Lognormal Complete the following steps to enter the parameters for the lognormal distribution. 1. In Location, enter a value that represents the location of the peak of the related normal distribution. 2. In Scale, enter a value that represents the spread of the related normal distribution. 3. In Threshold, enter the lower bound of the distribution. ## Normal Complete the following steps to enter the parameters for the normal distribution. 1. In Mean, enter the value for the center of the distribution. 2. In Standard deviation, enter the value for the spread of the distribution. ## Poisson In Mean, enter the average rate of occurrence. For more information, go to Poisson distribution. ## t Complete the following steps to enter the parameters for the t-distribution. • In Degrees of freedom, enter the degrees of freedom to define the t-distribution. • If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0. ## Uniform Complete the following steps to enter the parameters for the uniform distribution. 1. In Lower endpoint, enter the minimum value for the distribution. 2. In Upper endpoint, enter the maximum value for the distribution. ## Weibull Complete the following steps to enter the parameters for the Weibull distribution. 1. In Shape parameter, enter the shape parameter to define the Weibull distribution. 2. In Scale parameter, enter the scale parameters to define the Weibull distribution. 3. In Threshold parameter, enter the lower bound of the Weibull distribution. By using this site you agree to the use of cookies for analytics and personalized content. Read our policy	930	4,582	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.671875	4	CC-MAIN-2018-26	latest	en	0.709362	# Select the distribution and parameters for Cumulative Distribution Function (CDF). In the Cumulative Distribution Function (CDF) dialog box, select the distribution and the parameters.. ## Binomial. Complete the following steps to enter the parameters for the binomial distribution.. 1. In Number of trials, enter the sample size.. 2. In Event probability, enter a number between 0 and 1 for the probability that the outcome you are interested in occurs. An occurrence is called an "event".. ## Chi-square. Complete the following steps to enter the parameters for the chi-square distribution.. 1. In Degrees of freedom, enter the degrees of freedom to define the chi-square distribution.. 2. If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0.. ## Discrete. Complete the following steps to enter the parameters for the discrete distribution.. 1. In Values in, enter the column that contains the values to include in the distribution. Usually, values are discrete events or counts that are represented by numeric values.. 2. In Probabilities in, enter the column that contains the probabilities for each value. Probabilities must be between 0 and 1, and must sum to 1.. ## Exponential. Complete the following steps to enter the parameters for the exponential distribution.. 1. In Scale, enter the scale parameter. The scale parameter equals the mean when the threshold parameter equals 0.. 2. In Threshold, enter the lower bound of the distribution.. ## F. Complete the following steps to enter the parameters for the F-distribution.. 1. In Numerator degrees of freedom and Denominator degrees of freedom, enter the numerator and denominator degrees of freedom to define the F-distribution.. 2. If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0.. ## Geometric. Complete the following steps to enter the parameters for the Geometric distribution.. 1. In Event probability, enter a number between 0 and 1 for the probability of an occurrence on each trial. An occurrence is called an "event".. 2. From Model, select one of the following to specify the number to model.. • Total number of trials: The number of trials includes both events and nonevents.	• Only the number of non-events: Do not count the event.. For example, this plot shows a geometric distribution that has an event probability of 0.5 and models the total number of trials.. ## Integer. Complete the following steps to enter the parameters for the integer distribution.. 1. In Minimum value, enter the lower end point of the distribution.. 2. In Maximum value, enter the upper end point of the distribution.. ## Lognormal. Complete the following steps to enter the parameters for the lognormal distribution.. 1. In Location, enter a value that represents the location of the peak of the related normal distribution.. 2. In Scale, enter a value that represents the spread of the related normal distribution.. 3. In Threshold, enter the lower bound of the distribution.. ## Normal. Complete the following steps to enter the parameters for the normal distribution.. 1. In Mean, enter the value for the center of the distribution.. 2. In Standard deviation, enter the value for the spread of the distribution.. ## Poisson. In Mean, enter the average rate of occurrence. For more information, go to Poisson distribution.. ## t. Complete the following steps to enter the parameters for the t-distribution.. • In Degrees of freedom, enter the degrees of freedom to define the t-distribution.. • If you are calculating cumulative probability or inverse cumulative probability, in Noncentrality parameter, enter the noncentrality parameter. Usually, the noncentrality parameter is 0.. ## Uniform. Complete the following steps to enter the parameters for the uniform distribution.. 1. In Lower endpoint, enter the minimum value for the distribution.. 2. In Upper endpoint, enter the maximum value for the distribution.. ## Weibull. Complete the following steps to enter the parameters for the Weibull distribution.. 1. In Shape parameter, enter the shape parameter to define the Weibull distribution.. 2. In Scale parameter, enter the scale parameters to define the Weibull distribution.. 3. In Threshold parameter, enter the lower bound of the Weibull distribution.. By using this site you agree to the use of cookies for analytics and personalized content. Read our policy.
http://www.jiskha.com/display.cgi?id=1351474640	1,498,345,492,000,000,000	text/html	crawl-data/CC-MAIN-2017-26/segments/1498128320362.97/warc/CC-MAIN-20170624221310-20170625001310-00375.warc.gz	556,334,161	3,940	# math posted by . Mrs Olive gave her 4 children a sack of candy to share. Mark ate 1/2 of the candy in the sack and passed the sack to Rachel. Rachel ate 1/3 of the remaining candy and passed the sack to Fred. Fred then ate 1/4 of the candy that remained and passed the sack to Britany. Britany ate the last 12 pieces of candy. How many pieces of candy did Mark eat? • math - original amount = x mark ate x/2, leaving x/2 Rachel ate 1/3(x/2) = x/6, leaving x/3 Fred ate 1/4(x/3) = x/12, leaving x/4 Britany ate 12 = x/4 so, x=48 check Mark ate 24, left 24 Rachel ate 8, left 16 Fred ate 4, left 12 Britany: 12	206	615	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.96875	4	CC-MAIN-2017-26	latest	en	0.94394	# math. posted by .. Mrs Olive gave her 4 children a sack of candy to share. Mark ate 1/2 of the candy in the sack and passed the sack to Rachel. Rachel ate 1/3 of the remaining candy and passed the sack to Fred. Fred then ate 1/4 of the candy that remained and passed the sack to Britany. Britany ate the last 12 pieces of candy. How many pieces of candy did Mark eat?. • math -. original amount = x. mark ate x/2, leaving x/2.	Rachel ate 1/3(x/2) = x/6, leaving x/3. Fred ate 1/4(x/3) = x/12, leaving x/4. Britany ate 12 = x/4. so, x=48. check. Mark ate 24, left 24. Rachel ate 8, left 16. Fred ate 4, left 12. Britany: 12.
https://wentao-shao.gitbook.io/leetcode/data-structure/basic-calculator/772.basic-calculator-iii	1,725,804,987,000,000,000	text/html	crawl-data/CC-MAIN-2024-38/segments/1725700651002.87/warc/CC-MAIN-20240908115103-20240908145103-00408.warc.gz	587,814,304	60,854	# 772.Basic-Calculator-III ## ้ข็ฎๅฐๅ https://leetcode.com/problems/basic-calculator-iii/ ## ้ข็ฎๆ่ฟฐ ``````Implement a basic calculator to evaluate a simple expression string. The expression string may contain open ( and closing parentheses ), the plus + or minus sign -, non-negative integers and empty spaces . The expression string contains only non-negative integers, +, -, , / operators , open ( and closing parentheses ) and empty spaces . The integer division should truncate toward zero. You may assume that the given expression is always valid. All intermediate results will be in the range of [-2147483648, 2147483647]. Some examples: "1 + 1" = 2 " 6-4 / 2 " = 4 "2(5+52)/3+(6/2+8)" = 21 "(2+6 3+5- (314/7+2)5)+3"=-12`````` ## ไปฃ็ Approach 1: Queue + Stack๏ผ้ๅฐไธไธไธชsignๆถ๏ผๅ่ฎก็ฎไนๅsign • Queue: ๅญๅจๆๆ็ๅญ็ฌฆ • Stack ๅญๅจๆๆ็num ``````class Solution { public int calculate(String s) { if (s == null \|\| s.length() == 0) return 0; for (char c : s.toCharArray()) { q.offer(c); } q.offer('+'); return cal(q); } private int cal(Queue<Character> q) { char sign = '+'; Integer num = null; // 1 - 0 - 1 Stack<Integer> stack = new Stack<>(); while (!q.isEmpty()) { char c = q.poll(); if (c == ' ') continue; if (Character.isDigit(c)) { if (num != null) { num = 10 * num + c - '0'; } else { num = c - '0'; } } else if (c == '(') { num = cal(q); } else { // + - * / ) if (sign == '+' && num != null) { stack.push(num); } else if (sign == '-') { if (c == '-' && num == null) { // -1--1 c = '+'; // Negative + negative to positive } else if(num != null){ stack.push(-num); } } else if (sign == '' && num != null) { stack.push(stack.pop() num); } else if (sign == '/' && num != null) { stack.push(stack.pop() / num); } if (c == ')') { break; } num = null; sign = c; } } int sum = 0; while (!stack.isEmpty()) { sum += stack.pop(); } return sum; } }`````` Last updated	677	1,940	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2024-38	latest	en	0.34328	# 772.Basic-Calculator-III. ## ้ข็ฎๅฐๅ. https://leetcode.com/problems/basic-calculator-iii/. ## ้ข็ฎๆ่ฟฐ. ``````Implement a basic calculator to evaluate a simple expression string.. The expression string may contain open ( and closing parentheses ), the plus + or minus sign -, non-negative integers and empty spaces .. The expression string contains only non-negative integers, +, -, , / operators , open ( and closing parentheses ) and empty spaces . The integer division should truncate toward zero.. You may assume that the given expression is always valid. All intermediate results will be in the range of [-2147483648, 2147483647].. Some examples:. "1 + 1" = 2. " 6-4 / 2 " = 4. "2(5+52)/3+(6/2+8)" = 21. "(2+6 3+5- (314/7+2)5)+3"=-12``````. ## ไปฃ็ . Approach 1: Queue + Stack๏ผ้ๅฐไธไธไธชsignๆถ๏ผๅ่ฎก็ฎไนๅsign. • Queue: ๅญๅจๆๆ็ๅญ็ฌฆ. • Stack ๅญๅจๆๆ็num. ``````class Solution {. public int calculate(String s) {. if (s == null \|\| s.length() == 0) return 0;. for (char c : s.toCharArray()) {. q.offer(c);. }. q.offer('+');. return cal(q);. }. private int cal(Queue<Character> q) {. char sign = '+';. Integer num = null; // 1 - 0 - 1. Stack<Integer> stack = new Stack<>();. while (!q.isEmpty()) {. char c = q.poll();. if (c == ' ') continue;. if (Character.isDigit(c)) {. if (num != null) {. num = 10 * num + c - '0';.	} else {. num = c - '0';. }. } else if (c == '(') {. num = cal(q);. } else {. // + - * / ). if (sign == '+' && num != null) {. stack.push(num);. } else if (sign == '-') {. if (c == '-' && num == null) { // -1--1. c = '+'; // Negative + negative to positive. } else if(num != null){. stack.push(-num);. }. } else if (sign == '' && num != null) {. stack.push(stack.pop() num);. } else if (sign == '/' && num != null) {. stack.push(stack.pop() / num);. }. if (c == ')') {. break;. }. num = null;. sign = c;. }. }. int sum = 0;. while (!stack.isEmpty()) {. sum += stack.pop();. }. return sum;. }. }``````. Last updated.
https://www.onlinenotesnepal.com/power-in-ac-circuit	1,631,997,563,000,000,000	text/html	crawl-data/CC-MAIN-2021-39/segments/1631780056572.96/warc/CC-MAIN-20210918184640-20210918214640-00063.warc.gz	947,553,823	14,797	# Power in AC Circuit The power in the AC circuit is explained below: AC circuits are usually three-phase for electrical distribution and electrical transmission purposes. Single-phase circuits are commonly used in our domestic supply system. The total power of a three-phase AC circuit is equal to three times the single-phase power. So if the power in a single phase of a three-phase system is ‘P’, then the total power of the three-phase system would be 3P (provided the three-phase system is perfectly balanced). But if the three-phase system is not exactly balanced, then the total power of the system would be the sum of the power of individual phases. Suppose, in a three-phase system, the power at R phase is PR, at Y phase is PY and at B phase is PB, then total power of the system would be This is a simple scalar sum since power is a scalar quantity. This is the season if we consider only single-phase during calculating and analyzing of three-phase power, it is enough. Let us consider, network A is electrically connected with network B as shown in the figure below: Let us consider the expression of the voltage waveform of a single-phase system is: Where V is the amplitude of the waveform, ω is the angular velocity of propagation of the wave. Now, consider the current of the system is i(t) and this current has a phase difference from the voltage by an angle φ. That means the current wave propagates with φ radiant lag in respect of the voltage. The voltage and current waveform can be represented graphically as shown below: Now, let us plot the term P versus time, It is seen from the graph that, the term P does not have any negative value. So, it will have a nonzero average value. It is sinusoidal with a frequency twice of system frequency. Let us now plot the second term of the power equation, i.e. Q. This is purely sinusoidal and has a zero average value. So from these two graphs, it is clear that P is the component of power in an AC circuit, which actually transported from network A to network B. This power is consumed in network B as electric power. Q, on the other hand, does not really flow from network A to network B. Rather it oscillates between networks A and B. This is also a component of power, actually flowing into and out of the inductor, capacitor-like energy storage elements of the network. Here, P is known as the real or active part of the power, and Q is known as the imaginary or reactive part of the power. Hence, P is called real power or active power, and Q is called imaginary or active power. The unit of active power is Watt, whereas the unit of reactive power is Voltage Ampere Reactive or VAR.	569	2,672	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.125	4	CC-MAIN-2021-39	longest	en	0.950931	# Power in AC Circuit. The power in the AC circuit is explained below:. AC circuits are usually three-phase for electrical distribution and electrical transmission purposes. Single-phase circuits are commonly used in our domestic supply system.. The total power of a three-phase AC circuit is equal to three times the single-phase power.. So if the power in a single phase of a three-phase system is ‘P’, then the total power of the three-phase system would be 3P (provided the three-phase system is perfectly balanced).. But if the three-phase system is not exactly balanced, then the total power of the system would be the sum of the power of individual phases.. Suppose, in a three-phase system, the power at R phase is PR, at Y phase is PY and at B phase is PB, then total power of the system would be. This is a simple scalar sum since power is a scalar quantity. This is the season if we consider only single-phase during calculating and analyzing of three-phase power, it is enough.. Let us consider, network A is electrically connected with network B as shown in the figure below:. Let us consider the expression of the voltage waveform of a single-phase system is:. Where V is the amplitude of the waveform, ω is the angular velocity of propagation of the wave.. Now, consider the current of the system is i(t) and this current has a phase difference from the voltage by an angle φ. That means the current wave propagates with φ radiant lag in respect of the voltage. The voltage and current waveform can be represented graphically as shown below:.	Now, let us plot the term P versus time,. It is seen from the graph that, the term P does not have any negative value. So, it will have a nonzero average value. It is sinusoidal with a frequency twice of system frequency. Let us now plot the second term of the power equation, i.e. Q.. This is purely sinusoidal and has a zero average value. So from these two graphs, it is clear that P is the component of power in an AC circuit, which actually transported from network A to network B. This power is consumed in network B as electric power.. Q, on the other hand, does not really flow from network A to network B. Rather it oscillates between networks A and B. This is also a component of power, actually flowing into and out of the inductor, capacitor-like energy storage elements of the network.. Here, P is known as the real or active part of the power, and Q is known as the imaginary or reactive part of the power.. Hence, P is called real power or active power, and Q is called imaginary or active power. The unit of active power is Watt, whereas the unit of reactive power is Voltage Ampere Reactive or VAR.
http://www.numbersaplenty.com/454235	1,585,820,045,000,000,000	text/html	crawl-data/CC-MAIN-2020-16/segments/1585370506870.41/warc/CC-MAIN-20200402080824-20200402110824-00020.warc.gz	279,534,220	3,412	Search a number 454235 = 590847 BaseRepresentation bin1101110111001011011 3212002002112 41232321123 5104013420 613422535 73601205 oct1567133 9762075 10454235 11290301 1219aa4b 1312b9a2 14bb775 158e8c5 hex6ee5b 454235 has 4 divisors (see below), whose sum is σ = 545088. Its totient is φ = 363384. The previous prime is 454231. The next prime is 454247. The reversal of 454235 is 532454. Adding to 454235 its reverse (532454), we get a palindrome (986689). It is a semiprime because it is the product of two primes. It is a cyclic number. It is not a de Polignac number, because 454235 - 22 = 454231 is a prime. It is a Duffinian number. It is a nialpdrome in base 14. It is a self number, because there is not a number n which added to its sum of digits gives 454235. It is not an unprimeable number, because it can be changed into a prime (454231) by changing a digit. It is a pernicious number, because its binary representation contains a prime number (13) of ones. It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 45419 + ... + 45428. It is an arithmetic number, because the mean of its divisors is an integer number (136272). 2454235 is an apocalyptic number. 454235 is a deficient number, since it is larger than the sum of its proper divisors (90853). 454235 is an equidigital number, since it uses as much as digits as its factorization. 454235 is an odious number, because the sum of its binary digits is odd. The sum of its prime factors is 90852. The product of its digits is 2400, while the sum is 23. The square root of 454235 is about 673.9695838834. The cubic root of 454235 is about 76.8705871396. The spelling of 454235 in words is "four hundred fifty-four thousand, two hundred thirty-five". Divisors: 1 5 90847 454235	535	1,813	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.6875	4	CC-MAIN-2020-16	latest	en	0.894907	Search a number. 454235 = 590847. BaseRepresentation. bin1101110111001011011. 3212002002112. 41232321123. 5104013420. 613422535. 73601205. oct1567133. 9762075. 10454235. 11290301. 1219aa4b. 1312b9a2. 14bb775. 158e8c5. hex6ee5b. 454235 has 4 divisors (see below), whose sum is σ = 545088. Its totient is φ = 363384.. The previous prime is 454231. The next prime is 454247. The reversal of 454235 is 532454.	Adding to 454235 its reverse (532454), we get a palindrome (986689).. It is a semiprime because it is the product of two primes.. It is a cyclic number.. It is not a de Polignac number, because 454235 - 22 = 454231 is a prime.. It is a Duffinian number.. It is a nialpdrome in base 14.. It is a self number, because there is not a number n which added to its sum of digits gives 454235.. It is not an unprimeable number, because it can be changed into a prime (454231) by changing a digit.. It is a pernicious number, because its binary representation contains a prime number (13) of ones.. It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 45419 + ... + 45428.. It is an arithmetic number, because the mean of its divisors is an integer number (136272).. 2454235 is an apocalyptic number.. 454235 is a deficient number, since it is larger than the sum of its proper divisors (90853).. 454235 is an equidigital number, since it uses as much as digits as its factorization.. 454235 is an odious number, because the sum of its binary digits is odd.. The sum of its prime factors is 90852.. The product of its digits is 2400, while the sum is 23.. The square root of 454235 is about 673.9695838834. The cubic root of 454235 is about 76.8705871396.. The spelling of 454235 in words is "four hundred fifty-four thousand, two hundred thirty-five".. Divisors: 1 5 90847 454235.
https://www.roymech.co.uk/Useful_Tables/Maths/fourier/Maths_Fourier_DFT_Transform.html	1,653,799,989,000,000,000	text/html	crawl-data/CC-MAIN-2022-21/segments/1652663039492.94/warc/CC-MAIN-20220529041832-20220529071832-00148.warc.gz	1,109,583,117	7,356	# Discrete Fourier Transforms and Fast Fourier Transforms ### Discrete Fourier Transforsm & Fast Fourier Transforms #### Relationship between Discrete Fourier Transform and Continuous Fourier Tranform Introduction The Discrete Fourier Transform is a periodic and the inverse of the Disrete Fourier Transform is also periodic. Considering an non periodic time domain function and its related non-periodic continuous Fourier Tranform. If the time domain function is sampled a set of discrete values results. If this data is transformed then a periodic frequency domain function results. This data is again in discrete form and the inverse transform of this is a periodic function. The notes and figures below provide an illustration of these operations. The signal corruption resulting from the alias effect and the corruption of waveform resulting from the selection of T0 are very much exagerated. In practice the Discrete fourier Transforms using high values of N provide very accurate data. Typical Aperiodic Fourier transform. The figure below show a typical aperiodic waveform and its Fourier transform. The physical sampling function is mathematical equivalent to mutliplying the function with a Dirac Comb - IIIT( i ) function . Dirac comb. This function and its Fourier Transform is shown below. The sampling interval is T and the sampled function results as follows. The graphical result of this product operation in the time domain and the resulting convolution operation in the frequency domain is shown below. It should be noted the selection of T relates to the alias effect where, in the frequency domain, the adjacent periodic waveforms overlap and signal information is corrupted an lost. Ideally T should be selected such that at a k value of 1/2T the amplitude is effectively zero. The sampling operation does not involve an infinite number of samples. It results from taking N samples over a sample period T0 = N.T. This is mathematically equivalent to truncating the samples using a top hat function. Top hat with a width of T0 and a height 1. This is also called a pulse function. This operation is mathematically represented as shown below The graphical result of this operations is shown below. It is clear that the larger the value T0 the narrower the frequency domain function and the less the corruption of the signal. Ideally, stating the obvious, selecting an infinite T0 and a zero T would clearly result in a Discrete Fourier Transform which would be exactly equal to a continuous Fourier transform In practice sample values comprising the discretised time domain function are tranformed to Discrete Fourier Transforms not Continuous Fourier transforms as shown above. The Discrete Fourier transform is simply the product of the Continuous Fourier Transform and Dirac Comb - IIIa( 1/T0 ) function as illustrated below. The product of the two transforms equates to the convolution of the related time domain functions transforming the original aperiodic function to a periodic function as shown below . The time domain function can be expressed as below Now to derive the frequency domain function from the periodic time domain function. For a periodic time domain function the frequency domain function is a series of pulses. The factor cn is developed as follows The integration is completed over one period and therefore the equation can be simplified to. Because T0 = N.T .The equation can be written The resulting fourier Transform is It can be proved that there are only N distint values computable from this and the equation can be therefore developed into the form. To be continued....	700	3,669	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2022-21	latest	en	0.831417	# Discrete Fourier Transforms and Fast Fourier Transforms. ### Discrete Fourier Transforsm & Fast Fourier Transforms. #### Relationship between Discrete Fourier Transform and Continuous Fourier Tranform. Introduction The Discrete Fourier Transform is a periodic and the inverse of the Disrete Fourier Transform is also periodic. Considering an non periodic time domain function and its related non-periodic continuous Fourier Tranform. If the time domain function is sampled a set of discrete values results. If this data is transformed then a periodic frequency domain function results. This data is again in discrete form and the inverse transform of this is a periodic function. The notes and figures below provide an illustration of these operations. The signal corruption resulting from the alias effect and the corruption of waveform resulting from the selection of T0 are very much exagerated. In practice the Discrete fourier Transforms using high values of N provide very accurate data. Typical Aperiodic Fourier transform. The figure below show a typical aperiodic waveform and its Fourier transform. The physical sampling function is mathematical equivalent to mutliplying the function with a Dirac Comb - IIIT( i ) function . Dirac comb. This function and its Fourier Transform is shown below. The sampling interval is T and the sampled function results as follows. The graphical result of this product operation in the time domain and the resulting convolution operation in the frequency domain is shown below. It should be noted the selection of T relates to the alias effect where, in the frequency domain, the adjacent periodic waveforms overlap and signal information is corrupted an lost. Ideally T should be selected such that at a k value of 1/2T the amplitude is effectively zero.	The sampling operation does not involve an infinite number of samples. It results from taking N samples over a sample period T0 = N.T. This is mathematically equivalent to truncating the samples using a top hat function. Top hat with a width of T0 and a height 1. This is also called a pulse function. This operation is mathematically represented as shown below The graphical result of this operations is shown below. It is clear that the larger the value T0 the narrower the frequency domain function and the less the corruption of the signal. Ideally, stating the obvious, selecting an infinite T0 and a zero T would clearly result in a Discrete Fourier Transform which would be exactly equal to a continuous Fourier transform In practice sample values comprising the discretised time domain function are tranformed to Discrete Fourier Transforms not Continuous Fourier transforms as shown above. The Discrete Fourier transform is simply the product of the Continuous Fourier Transform and Dirac Comb - IIIa( 1/T0 ) function as illustrated below. The product of the two transforms equates to the convolution of the related time domain functions transforming the original aperiodic function to a periodic function as shown below . The time domain function can be expressed as below Now to derive the frequency domain function from the periodic time domain function. For a periodic time domain function the frequency domain function is a series of pulses. The factor cn is developed as follows The integration is completed over one period and therefore the equation can be simplified to. Because T0 = N.T .The equation can be written The resulting fourier Transform is It can be proved that there are only N distint values computable from this and the equation can be therefore developed into the form.. To be continued....
http://www.enotes.com/homework-help/small-object-mass-0-02kg-top-building-160-m-high-445232	1,477,505,234,000,000,000	text/html	crawl-data/CC-MAIN-2016-44/segments/1476988720967.29/warc/CC-MAIN-20161020183840-00326-ip-10-171-6-4.ec2.internal.warc.gz	428,901,230	9,829	# A small object, of mass 0.02kg at the top of a building 160 m high, is dropped from rest. Assuming that the air resistance has magnitude 0·096 N, calculate the height of the object above the... A small object, of mass 0.02kg at the top of a building 160 m high, is dropped from rest. Assuming that the air resistance has magnitude 0·096 N, calculate the height of the object above the ground 4 s after it was dropped Asked on by saj-94 jeew-m \| College Teacher \| (Level 1) Educator Emeritus Posted on Here when the ball falls to the ground two forces will act on the ball. One is the weight of the ball which act downward which drives the ball to ground. In the upward direction air resistance will act on it. Using Newtons second law of motion; `darrF = ma` `0.02xx9.81-0.096 = 0.02xxa` `a = 5.01m/s^2` Using the equations of motion; `darrS = Ut+1/2at^2` `S = 0+1/2xx5.01xx4^2` `S = 40.08` Height above ground `= 160-40.02 = 119.92` So the ball is 199.92m above ground when the time is 4s. Sources: We’ve answered 317,791 questions. We can answer yours, too.	328	1,080	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.71875	4	CC-MAIN-2016-44	latest	en	0.9249	# A small object, of mass 0.02kg at the top of a building 160 m high, is dropped from rest. Assuming that the air resistance has magnitude 0·096 N, calculate the height of the object above the.... A small object, of mass 0.02kg at the top of a building 160 m high, is dropped from rest.. Assuming that the air resistance has magnitude 0·096 N,. calculate the height of the object above the ground 4 s after it was dropped. Asked on by saj-94. jeew-m \| College Teacher \| (Level 1) Educator Emeritus. Posted on. Here when the ball falls to the ground two forces will act on the ball. One is the weight of the ball which act downward which drives the ball to ground. In the upward direction air resistance will act on it.. Using Newtons second law of motion;. `darrF = ma`.	`0.02xx9.81-0.096 = 0.02xxa`. `a = 5.01m/s^2`. Using the equations of motion;. `darrS = Ut+1/2at^2`. `S = 0+1/2xx5.01xx4^2`. `S = 40.08`. Height above ground `= 160-40.02 = 119.92`. So the ball is 199.92m above ground when the time is 4s.. Sources:. We’ve answered 317,791 questions. We can answer yours, too.
https://math.stackexchange.com/questions/3276871/how-is-this-angle-measured-in-the-triangle	1,627,584,074,000,000,000	text/html	crawl-data/CC-MAIN-2021-31/segments/1627046153892.74/warc/CC-MAIN-20210729172022-20210729202022-00380.warc.gz	401,717,125	37,686	# How is this angle measured in the triangle? I am reading George Polya's "How to Solve It". I cannot understand how he is getting to certain solutions. Like the one from the chapter of "Auxiliary Solution". Given the image below: Let the angle at vertex $$A$$ be called $$\alpha$$. Polya says by looking at the diagram and the isoceles triangles $$\triangle{ACE}$$ and $$\triangle{ABD}$$ we can deduce that $$\angle{DAE} = \frac{\alpha}{2} + 90°$$. I just don't understand how that is possible. All I can figure out is that $$\angle{CAE} = \angle{CEA}$$ and $$\angle{BAD} = \angle{BDA}$$, since they are both isoceles. But where will I get $$90°$$ from? Am I missing some important theorem in geometry? Exterior angle of triangle = sum of 2 interior opposite angles $$\angle ACB = 2 \angle EAC \\ \angle ABC = 2 \angle DAB \\$$ $$\Rightarrow \angle EAC + \angle DAB = 0.5(\angle ACB + \angle ABC) = 0.5(180-\alpha)$$ $$\angle DAE = 90-0.5\alpha + \alpha$$ $$\angle DAE = \angle EAC + \angle DAB + \alpha$$ $$= \frac{180°-\angle ACE}{2}+\frac{180°-\angle ABD}{2}+\alpha$$ $$=\frac{\angle ACB+\angle ABC}{2}+\alpha$$ $$=\frac{180°-\alpha}{2}+\alpha$$ $$=90°+\frac{\alpha}{2}$$	381	1,180	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 16, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.5	4	CC-MAIN-2021-31	latest	en	0.80945	# How is this angle measured in the triangle?. I am reading George Polya's "How to Solve It". I cannot understand how he is getting to certain solutions. Like the one from the chapter of "Auxiliary Solution". Given the image below:. Let the angle at vertex $$A$$ be called $$\alpha$$. Polya says by looking at the diagram and the isoceles triangles $$\triangle{ACE}$$ and $$\triangle{ABD}$$ we can deduce that $$\angle{DAE} = \frac{\alpha}{2} + 90°$$.. I just don't understand how that is possible. All I can figure out is that $$\angle{CAE} = \angle{CEA}$$ and $$\angle{BAD} = \angle{BDA}$$, since they are both isoceles.	But where will I get $$90°$$ from?. Am I missing some important theorem in geometry?. Exterior angle of triangle = sum of 2 interior opposite angles $$\angle ACB = 2 \angle EAC \\ \angle ABC = 2 \angle DAB \\$$ $$\Rightarrow \angle EAC + \angle DAB = 0.5(\angle ACB + \angle ABC) = 0.5(180-\alpha)$$ $$\angle DAE = 90-0.5\alpha + \alpha$$. $$\angle DAE = \angle EAC + \angle DAB + \alpha$$ $$= \frac{180°-\angle ACE}{2}+\frac{180°-\angle ABD}{2}+\alpha$$ $$=\frac{\angle ACB+\angle ABC}{2}+\alpha$$ $$=\frac{180°-\alpha}{2}+\alpha$$ $$=90°+\frac{\alpha}{2}$$.
http://math.stackexchange.com/questions/186397/solving-a-non-integer-polynomial-where-the-exponent-is-greater-than-one/203601	1,430,991,874,000,000,000	text/html	crawl-data/CC-MAIN-2015-18/segments/1430461119624.60/warc/CC-MAIN-20150501061839-00097-ip-10-235-10-82.ec2.internal.warc.gz	132,564,774	17,877	# Solving a non-integer polynomial where the exponent is greater than one I'm trying to solve an equation of the form: $ax + bx^{1+c} + d = 0$, where $0 < c < 1$, and the reciprocal of $c$ is not necessarily an integer either. Mathematica protests to me that it is not up to the task of solving this, and I'd like a general solution rather than FindRoot at a particular value. I've poked around and there are a couple work-arounds - usually involving some substitution - for solving a non-integer polynomials but (as far as I can tell - because I'm having trouble applying them to this problem) they all only really seem to work if the non-integer polynomial is less than one and this is definitely more than one. I've come to the point of getting a derivative w.r.t. to one of the parameters of interest with the implicit function theorem, which is fine for my purposes, but obviously getting a general solution would be preferable. Is there a general solution for this? - Already when the reciprocal of $c$ is an integer there is no expression for the solution(s) in terms of elementary functions. – André Nicolas Aug 24 '12 at 17:38 Nitpick: This is not a polynomial unless $c$ is an integer. – Jyrki Lahtonen Aug 28 '12 at 17:38 If $\|b/a\|$ and $\|d/a\|$ are small, you can solve it in series. For convenience, take $a=1$. Then $$x = -d + \sum_{n=0}^\infty \frac{\Gamma(nc+n+1)}{n! \Gamma(nc+2)} (-d)^{nc+1} (-b)^n$$ No, especially in radicals. First of all, if there was a general formula, then it would work for all $c\in\mathbb R$, not just those values in your interval. But if you take $c=\frac 1{n-1}$, you get $bx^\frac n{n-1}+ax+d=0$. Applying the transformation $x=z^{n-1}$ yields $bz^n+az^{n-1}+d=0$. For $n\geq 5$, there is no general solution because of the Abel-Ruffini Theorem.	503	1,802	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.953125	4	CC-MAIN-2015-18	longest	en	0.904059	# Solving a non-integer polynomial where the exponent is greater than one. I'm trying to solve an equation of the form:. $ax + bx^{1+c} + d = 0$, where $0 < c < 1$, and the reciprocal of $c$ is not necessarily an integer either.. Mathematica protests to me that it is not up to the task of solving this, and I'd like a general solution rather than FindRoot at a particular value.. I've poked around and there are a couple work-arounds - usually involving some substitution - for solving a non-integer polynomials but (as far as I can tell - because I'm having trouble applying them to this problem) they all only really seem to work if the non-integer polynomial is less than one and this is definitely more than one.. I've come to the point of getting a derivative w.r.t. to one of the parameters of interest with the implicit function theorem, which is fine for my purposes, but obviously getting a general solution would be preferable.. Is there a general solution for this?. -. Already when the reciprocal of $c$ is an integer there is no expression for the solution(s) in terms of elementary functions.	– André Nicolas Aug 24 '12 at 17:38. Nitpick: This is not a polynomial unless $c$ is an integer. – Jyrki Lahtonen Aug 28 '12 at 17:38. If $\|b/a\|$ and $\|d/a\|$ are small, you can solve it in series. For convenience, take $a=1$. Then $$x = -d + \sum_{n=0}^\infty \frac{\Gamma(nc+n+1)}{n! \Gamma(nc+2)} (-d)^{nc+1} (-b)^n$$. No, especially in radicals. First of all, if there was a general formula, then it would work for all $c\in\mathbb R$, not just those values in your interval. But if you take $c=\frac 1{n-1}$, you get $bx^\frac n{n-1}+ax+d=0$. Applying the transformation $x=z^{n-1}$ yields $bz^n+az^{n-1}+d=0$. For $n\geq 5$, there is no general solution because of the Abel-Ruffini Theorem.
https://www.scribd.com/document/75920279/Time-Dependent-Perturbation-Problems	1,560,993,847,000,000,000	text/html	crawl-data/CC-MAIN-2019-26/segments/1560627999130.50/warc/CC-MAIN-20190620004625-20190620030625-00120.warc.gz	875,892,388	60,759	You are on page 1of 6 # QM II Homework Assignment 8 ## Time Dependent Perturbation Theory: Chris Mueller Dept. of Physics, University of Florida 26 March, 2009 Collaborators: Bobby Bond, Darsa Donelan, Lana Muniz, and Aaron Spector 1. Shankar 18.2.3. Consider a particle in the ground state of a box of length L. Argue on semi-classical grounds that the natural time period associated with it is T nL 2 /. If the box expands symmetrically to double its size in time T what is the probability of catching the particle in the ground state of the new box? The relative semi-classical time scale is the time that it would take a particle to cross the well. The quantum mechanical ground state energy of the innite square well is E 1 = 2 2 2mL 2 . The classical speed of a particle with this energy is v = _ 2E m = mL . The time to travel across the well is therefore T = v L = mL 2 We can use this time scale to judge whether or not a perturbation to the potential is sudden or adiabatic. For changes on much shorter timescales, the classical particle will not have crossed the well and will therefore not know about the change. On the other hand, for changes much longer than this timescale the particle will bounce o of the walls many times during the change and will therefore not gain or lose energy during the perturbation. For this problem, we are given that the time over which the perturbation happens is much less than our timescale; T. This is therefore a sudden perturbation and we will expect the wavefunction of state to be unchanged during the course of the perturbation. The initial (and therefore nal) state of the particle is (x) = _ 2 L cos _ x L _ . The new ground state is 0 (x) = _ 1 L cos _ x 2L _ . 1 Hence, the probability of nding the particle in the ground state is P( 0 ) = _ L/2 L/2 _ 2 L 2 cos _ x 2L _ cos _ x L _ dx 2 = 2 L _ 4 2l 3 _ 2 = _ 8 3 _ 2 0.721. 2. Shankar 18.2.5 An oscillator is in the ground state of H = H 0 + H 1 , where the time-independent per- turbation H 1 is the linear potential V = fx. If at t = 0, H 1 is abruptly turned o, show that the probability that the system is in the nthe eigenstate of H 0 is given by the Poisson distribution P(n) = e n n! where = f 2 2m 3 We need to nd the eigenstates of the perturbed Hamiltonian before we can proceed with the problem. We are fortunate in that this system is exactly solvable. Consider the Hamiltonian, H = p 2 2m + 1 2 m 2 x 2 fx = p 2 2m + 1 2 m 2 _ x f m 2 _ 2 1 2 f 2 m 2 From this form we can see that the perturbed Hamiltonian is simply a harmonic oscillator with a shifted constant energy and a translated center. The perturbed ground state wavefunction can be found from the old unperturbed one by applying the translation operator. \|0 = T _ f m 2 _ \|0 0 = exp _ i f m 2 p _ \|0 0 = exp _ i f m 2 i _ m 2 (a a) _ \|0 0 = exp __ f 2 2m 3 (a a) _ \|0 0 We want to expand the exponential into two terms, but we must be careful because non- commuting operators do not satisfy the usual relation ship. Recall that for two operators A and B, e A+B = e A e B e 1 2 [A, B] . Before we return to the above equation, let us also compute a useful commutator. For operators A and B and a scaler quantity a, [aA, aB] = (aA)(aB) (aB)(aA) = a 2 (AB +BA) = a 2 [B, A]. 2 Applying these two relations, the new state becomes \|0 = exp __ f 2 2m 3 a _ exp _ _ f 2 2m 3 a _ exp _ f 2 4m 2 _ \|0 0 = e a e 1 2 \|0 0 We now want to expand the two exponential terms in terms of their Taylor expansions. Since the operators will be acting on the ground state, all of the terms in the expansion of the second operator will go to zero except for the identity. We can therefore drop it completely. Before we expand the other term, it is helpful to recall that the eigenstates of the harmonic oscillator can be expressed in terms of the ground state and the raising operator as \|n 0 = (a ) n n! \|0 0 ## Lets expand the exponential in the above expression. \|0 = e 1 2 ) n n! (a ) n \|0 0 = e 1 2 ) n n! \|n 0 Finally, since this perturbation is applied suddenly, the system will remain in this state even after the perturbation is applied. The probability of being in one of the new eigenstates of the unperturbed harmonic oscillator is simply its overlap with those states. P(n) = \|n 0 \|0\| 2 = e n n! 3. For a spin 1 2 particle, consider the time dependent Hamiltonian, H(t) = n(t) , where is the vector formed from the Pauli spin matrices. The unit vector n changes with time. Suppose initially n = z, and the spin is in the ground state. Next, n is rotated so that it points in the x direction. What is the wavefunction when n reaches x for the case of (a) a rotation in time T 1 and (b) a rotation in time T 1 ? (Note: you do not need to have the correct phase for the wave function for case (b). The phase will depend on the exact details of how the state is changed form z to x and we will discuss this later.) For both cases, what is the probability that the spin will be in the new ground state ( n = x)? Lets begin this problem by determining the eigenvalues and eigenvectors of the initial and nal Hamiltonians. At time t = 0, n points in the z direction and the Hamiltonian is H(0) = z = _ 0 0 _ . This is clearly already diagonal and so we can identify the eigenvectors and their energies as \| = _ 1 0 _ & \| = _ 0 1 _ . 3 Now for the nal Hamiltonian. At time t = T, n points in the x direction and the Hamiltonian has the form H(T) = x = _ 0 0 _ . We diagonalize this in the usual way, by calculating the characteristic determinant and setting it equal to zero. = 2 2 = ( )( + ) = 0 = Plugging these eigenvalues back in to nd the eigenvectors reveals them to be \| = 1 2 _ 1 1 _ & \| = 1 2 _ 1 1 _ . Part a Since the perturbation in this case is very sudden, we expect the wavefunction to remain unchanged after the perturbation is applied. Hence, the perturbed wavefunction is \| x = _ 1 0 _ . Hence, the probability of it being in the new ground state is P() = _ 1 0 _ 1 2 _ 1 1 _ 2 = 1 2 . Part b In this case, the approximation is adiabatic and we will expect the state of z to track to the ground state of x . Hence, the state after the perturbation will be \| x = 1 2 _ 1 1 _ . Which means that the probability of nding the system in the ground state is one. P() 2 _ 1 1 _ 1 2 _ 1 1 _ 2 = 1 4 4. A time dependent Hamiltonian has the form: H(t) = p 2 2m + 1 2 m 2 (t)x 2 , where the only time dependence comes from (t). This is equivalent to varying the spring constant with time. In this problem, we will make use of equations 7.3.21 and 7.3.22 from the book for the harmonic oscillator wavefunctions, together with the Gaussian integrals: _ dxe ax 2 = _ a _ dxx 2 e ax 2 = 1 2 a 3 (a) From t = 0 to t = T the angular frequency changes from 0 to 2 0 . At t = 0 the system is in the ground state of H(t = 0). If 0 T 1, what is the probability that the system will still be in the ground state at t = T? (b) What is the probability that it will be in the rst excited state at t = T? What is the probability that it will be in the second excited state at t = T? 0 T 1. What is the probability that the system will still be in the ground state at t = T? (d) What is the probability that it will be in the rst excited state at t = T? What is the probability that it will be in the second excited state at t = T? Part a Recall that the wavefunctions of the harmonic oscillator are Gaussians modulated by Hermite polynomials. The ground state is \| 0 ( 0 , x) = _ m 0 _1 4 exp _ m 0 x 2 2 _ . Since the semi-classical time scale of the harmonic oscillator is T = 1 f 1 ## and this perturbation happens on a time scale much less than this, the wavefunction will remain unchanged through the perturbation. Hence, the probability of the particle being in the new ground state is simply the overlap with the wavefunction of the new ground state. P (\| 0 (2 0 , x)) = \| 0 ( 0 , x)\| 0 (2 0 , x)\| 2 = _ _ _ dx 2m 0 exp _ 3m 0 x 2 _ _ _ 2 = 2m 0 2 3m 0 = 2 3 2 3 0.9428 Part b Since this is the same situation as in part a, the system will remain in the original ground state after the perturbation is applied. The probability of being in the rst excited state of the 5 perturbed potential is P (\| 1 (2 0 , x)) = _ m 0 2 5 4 dxx exp _ m 0 x 2 2 _ _ 2 = 0 . So, the particle has a zero probability of ending up in the rst excited state of the perturbed potential. For the second excited state, P (\| 2 (2 0 , x)) = __ m 0 2 _ dx _ 4m 0 x 2 1 _ exp _ 3m 0 x 2 2 __ 2 = m 0 2 _ _ 2 3m 0 _ dx _ 8 3 u 2 1 _ e u 2 du _ 2 = 2 3 _ 8 3 _ 2 = 2 27 0.0524 So we see that the particle is much more likely to end up in the ground state than in the rst excited state. As a check on our calculations we can make sure that these values do not sum up to more than one. Upon checking we nd that P (\| 0 (2 0 , x)) +P (\| 2 (2, x)) 0.9952 which conrms that our calculations are at least close. This also implies that the particle only has a 0.48% chance of ending up in a state other than the ground state or second excited state. Part c Since this perturbation is applied on a much longer time scale than the natural time scale of the system, the particle will end up in the ground state of the perturbed potential. The wavefunction of this perturbed ground state is \| 0 (2 0 , x) = _ 2m 0 _1 4 exp _ m 0 x 2 _ . Hence, the overlap with the perturbed ground state is 1. P (\| 0 (2 0 , x)) = 1 Part d As with part c, the slow perturbation will cause the particle to end up in the new ground state with 100% probability. It will therefore have no overlap with the excited states. P (\| 1 (2 0 , x)) = 0 & P (\| 2 (2 0 , x)) = 0 6	3,083	9,654	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.5625	4	CC-MAIN-2019-26	latest	en	0.927177	You are on page 1of 6. # QM II Homework Assignment 8. ## Time Dependent Perturbation Theory:. Chris Mueller. Dept. of Physics, University of Florida. 26 March, 2009. Collaborators: Bobby Bond, Darsa Donelan, Lana Muniz, and Aaron Spector. 1. Shankar 18.2.3.. Consider a particle in the ground state of a box of length L. Argue on semi-classical. grounds that the natural time period associated with it is T nL. 2. /. If the box expands. symmetrically to double its size in time T what is the probability of catching the. particle in the ground state of the new box?. The relative semi-classical time scale is the time that it would take a particle to cross the well.. The quantum mechanical ground state energy of the innite square well is. E. 1. =. 2. 2. 2mL. 2. .. The classical speed of a particle with this energy is. v =. _. 2E. m. =. mL. .. The time to travel across the well is therefore. T =. v. L. =. mL. 2. We can use this time scale to judge whether or not a perturbation to the potential is sudden or. adiabatic. For changes on much shorter timescales, the classical particle will not have crossed the. well and will therefore not know about the change. On the other hand, for changes much longer. than this timescale the particle will bounce o of the walls many times during the change and will. therefore not gain or lose energy during the perturbation.. For this problem, we are given that the time over which the perturbation happens is much. less than our timescale; T. This is therefore a sudden perturbation and we will expect the. wavefunction of state to be unchanged during the course of the perturbation. The initial (and. therefore nal) state of the particle is. (x) =. _. 2. L. cos. _. x. L. _. .. The new ground state is. 0. (x) =. _. 1. L. cos. _. x. 2L. _. .. 1. Hence, the probability of nding the particle in the ground state is. P(. 0. ) =. _. L/2. L/2. _. 2. L. 2. cos. _. x. 2L. _. cos. _. x. L. _. dx. 2. =. 2. L. _. 4. 2l. 3. _. 2. =. _. 8. 3. _. 2. 0.721.. 2. Shankar 18.2.5. An oscillator is in the ground state of H = H. 0. + H. 1. , where the time-independent per-. turbation H. 1. is the linear potential V = fx. If at t = 0, H. 1. is abruptly turned o,. show that the probability that the system is in the nthe eigenstate of H. 0. is given by the. Poisson distribution. P(n) =. e. n. n!. where =. f. 2. 2m. 3. We need to nd the eigenstates of the perturbed Hamiltonian before we can proceed with the. problem. We are fortunate in that this system is exactly solvable. Consider the Hamiltonian,. H =. p. 2. 2m. +. 1. 2. m. 2. x. 2. fx. =. p. 2. 2m. +. 1. 2. m. 2. _. x. f. m. 2. _. 2. 1. 2. f. 2. m. 2. From this form we can see that the perturbed Hamiltonian is simply a harmonic oscillator with. a shifted constant energy and a translated center. The perturbed ground state wavefunction can. be found from the old unperturbed one by applying the translation operator.. \|0 = T. _. f. m. 2. _. \|0. 0. = exp. _. i. f. m. 2. p. _. \|0. 0. = exp. _. i. f. m. 2. i. _. m. 2. (a. a). _. \|0. 0. = exp. __. f. 2. 2m. 3. (a. a). _. \|0. 0. We want to expand the exponential into two terms, but we must be careful because non-. commuting operators do not satisfy the usual relation ship. Recall that for two operators A and. B,. e. A+B. = e. A. e. B. e. 1. 2. [A, B]. .. Before we return to the above equation, let us also compute a useful commutator. For operators. A and B and a scaler quantity a,. [aA, aB] = (aA)(aB) (aB)(aA) = a. 2. (AB +BA) = a. 2. [B, A].. 2. Applying these two relations, the new state becomes. \|0 = exp. __. f. 2. 2m. 3. a. _. exp. _. _. f. 2. 2m. 3. a. _. exp. _. f. 2. 4m. 2. _. \|0. 0. = e. a. e. 1. 2. \|0. 0. We now want to expand the two exponential terms in terms of their Taylor expansions. Since. the operators will be acting on the ground state, all of the terms in the expansion of the second. operator will go to zero except for the identity. We can therefore drop it completely. Before we. expand the other term, it is helpful to recall that the eigenstates of the harmonic oscillator can be. expressed in terms of the ground state and the raising operator as. \|n. 0. =. (a. ). n. n!. \|0. 0. ## Lets expand the exponential in the above expression.. \|0 = e. 1. 2. ). n. n!. (a. ). n. \|0. 0. = e. 1. 2. ). n. n!. \|n. 0. Finally, since this perturbation is applied suddenly, the system will remain in this state even. after the perturbation is applied. The probability of being in one of the new eigenstates of the. unperturbed harmonic oscillator is simply its overlap with those states.. P(n) = \|n. 0. \|0\|. 2. =. e. n. n!. 3. For a spin. 1. 2. particle, consider the time dependent Hamiltonian,. H(t) = n(t) ,. where is the vector formed from the Pauli spin matrices. The unit vector n changes. with time. Suppose initially n = z, and the spin is in the ground state. Next, n is rotated. so that it points in the x direction. What is the wavefunction when n reaches x for the. case of (a) a rotation in time T. 1. and (b) a rotation in time T. 1. ? (Note: you. do not need to have the correct phase for the wave function for case (b). The phase will. depend on the exact details of how the state is changed form z to x and we will discuss. this later.) For both cases, what is the probability that the spin will be in the new ground. state ( n = x)?. Lets begin this problem by determining the eigenvalues and eigenvectors of the initial and nal. Hamiltonians. At time t = 0, n points in the z direction and the Hamiltonian is. H(0) =. z. =. _. 0. 0. _. .. This is clearly already diagonal and so we can identify the eigenvectors and their energies as. \| =. _. 1. 0. _. & \| =. _. 0.	1. _. .. 3. Now for the nal Hamiltonian. At time t = T, n points in the x direction and the Hamiltonian. has the form. H(T) =. x. =. _. 0. 0. _. .. We diagonalize this in the usual way, by calculating the characteristic determinant and setting it. equal to zero.. =. 2. 2. = ( )( + ) = 0 =. Plugging these eigenvalues back in to nd the eigenvectors reveals them to be. \| =. 1. 2. _. 1. 1. _. & \| =. 1. 2. _. 1. 1. _. .. Part a. Since the perturbation in this case is very sudden, we expect the wavefunction to remain. unchanged after the perturbation is applied. Hence, the perturbed wavefunction is. \|. x. =. _. 1. 0. _. .. Hence, the probability of it being in the new ground state is. P() =. _. 1 0. _. 1. 2. _. 1. 1. _. 2. =. 1. 2. .. Part b. In this case, the approximation is adiabatic and we will expect the state of. z. to track to the. ground state of. x. . Hence, the state after the perturbation will be. \|. x. =. 1. 2. _. 1. 1. _. .. Which means that the probability of nding the system in the ground state is one.. P(). 2. _. 1 1. _. 1. 2. _. 1. 1. _. 2. = 1. 4. 4. A time dependent Hamiltonian has the form:. H(t) =. p. 2. 2m. +. 1. 2. m. 2. (t)x. 2. ,. where the only time dependence comes from (t). This is equivalent to varying the spring. constant with time. In this problem, we will make use of equations 7.3.21 and 7.3.22 from. the book for the harmonic oscillator wavefunctions, together with the Gaussian integrals:. _. dxe. ax. 2. =. _. a. _. dxx. 2. e. ax. 2. =. 1. 2. a. 3. (a) From t = 0 to t = T the angular frequency changes from. 0. to 2. 0. . At t = 0 the. system is in the ground state of H(t = 0). If. 0. T 1, what is the probability that. the system will still be in the ground state at t = T?. (b) What is the probability that it will be in the rst excited state at t = T? What is. the probability that it will be in the second excited state at t = T?. 0. T 1. What is the probability that the system will still be. in the ground state at t = T?. (d) What is the probability that it will be in the rst excited state at t = T? What is. the probability that it will be in the second excited state at t = T?. Part a. Recall that the wavefunctions of the harmonic oscillator are Gaussians modulated by Hermite. polynomials. The ground state is. \|. 0. (. 0. , x) =. _. m. 0. _1. 4. exp. _. m. 0. x. 2. 2. _. .. Since the semi-classical time scale of the harmonic oscillator is T =. 1. f. 1. ## and this perturbation. happens on a time scale much less than this, the wavefunction will remain unchanged through the. perturbation. Hence, the probability of the particle being in the new ground state is simply the. overlap with the wavefunction of the new ground state.. P (\|. 0. (2. 0. , x)) = \|. 0. (. 0. , x)\|. 0. (2. 0. , x)\|. 2. =. _. _. _. dx. 2m. 0. exp. _. 3m. 0. x. 2. _. _. _. 2. =. 2m. 0. 2. 3m. 0. =. 2. 3. 2. 3. 0.9428. Part b. Since this is the same situation as in part a, the system will remain in the original ground. state after the perturbation is applied. The probability of being in the rst excited state of the. 5. perturbed potential is. P (\|. 1. (2. 0. , x)) =. _. m. 0. 2. 5. 4. dxx exp. _. m. 0. x. 2. 2. _. _. 2. = 0 .. So, the particle has a zero probability of ending up in the rst excited state of the perturbed. potential.. For the second excited state,. P (\|. 2. (2. 0. , x)) =. __. m. 0. 2. _. dx. _. 4m. 0. x. 2. 1. _. exp. _. 3m. 0. x. 2. 2. __. 2. =. m. 0. 2. _. _. 2. 3m. 0. _. dx. _. 8. 3. u. 2. 1. _. e. u. 2. du. _. 2. =. 2. 3. _. 8. 3. _. 2. =. 2. 27. 0.0524. So we see that the particle is much more likely to end up in the ground state than in the rst. excited state.. As a check on our calculations we can make sure that these values do not sum up to more than. one. Upon checking we nd that. P (\|. 0. (2. 0. , x)) +P (\|. 2. (2, x)) 0.9952. which conrms that our calculations are at least close. This also implies that the particle only has. a 0.48% chance of ending up in a state other than the ground state or second excited state.. Part c. Since this perturbation is applied on a much longer time scale than the natural time scale of the. system, the particle will end up in the ground state of the perturbed potential. The wavefunction. of this perturbed ground state is. \|. 0. (2. 0. , x) =. _. 2m. 0. _1. 4. exp. _. m. 0. x. 2. _. .. Hence, the overlap with the perturbed ground state is 1.. P (\|. 0. (2. 0. , x)) = 1. Part d. As with part c, the slow perturbation will cause the particle to end up in the new ground state. with 100% probability. It will therefore have no overlap with the excited states.. P (\|. 1. (2. 0. , x)) = 0 & P (\|. 2. (2. 0. , x)) = 0. 6.
https://www.spss-tutorials.com/spss-sum-cautionary-note/comment-page-1/	1,596,445,120,000,000,000	text/html	crawl-data/CC-MAIN-2020-34/segments/1596439735792.85/warc/CC-MAIN-20200803083123-20200803113123-00519.warc.gz	840,826,394	11,805	SPSS TUTORIALS # SPSS Sum – Cautionary Note ## Summary In SPSS, `SUM(v1,v2)` is not always equivalent to `v1 + v2`. This tutorial explains the difference and shows how to make the right choice here. Different Ways of Taking Sums have Different Outcomes when Missing Values are Present ## Explanation • In SPSS, `v1 + v2 + v3` will result in a system missing value if at least one missing value is present in v1, v2 or v3. • The first alternative, `SUM(v1, v2, v3)` implicitly replaces missing values with zeroes. • The second alternative, `MEAN(v1, v2, v3) * 3` implicitly replaces missing values with the mean of the non missing values. • The third alternative, `MEAN.2(v1, v2, v3) * 3` is almost similar to the second. However, by suffixing `MEAN` by `.2`, you ensure that a mean is only calculated if at least two non missing values are present in v1, v2 and v3. • These points are demonstrated by the syntax below. ## SPSS Syntax Demonstration data list free/v1 v2 v3. begin data 1 3 5 1 3 '' 1 '' '' end data. compute sum_by_sum = sum(v1,v2,v3). compute sum_by_plus = v1 + v2 + v3. compute sum_by_mean = mean(v1 to v3) * 3. compute sum_by_mean.2 = mean.2(v1 to v3) * 3. exe. ## So Which one Is Best? • This question is rather hard to answer. It may depend on the meaning of the missing values (question skipped? technical problem?). Also, what are the individual questions and the sum supposed to reflect? • Second, the amount of missing values and sample size may be taken into account. Does it permit excluding some observations with missing values? Will this affect representativity and if so, is that a real problem? • For one thing, sums calculated by `SUM` may be biased towards zero. For instance, if v1 through v3 measure components of satisfaction, repondents will be seen as "less satisfied" insofar they have more missing values. That conclusion may be misleading. • Using the `+` operator does not induce such bias but may result in many missing values in the sum. This problem becomes larger as more missing values are present in the input variables and a sum is taken over more variables. • Multiplying the mean by the number of variables, may be a better alternative. However, it will always come up with with a sum if there's at least one non missing value. Especially with many input variables, a single value may be jugded insufficient for inferring a summation measure. • But perhaps none of these options is expected to yield sufficiently accurate results. In this case, one could partly circumvent the problem with a (multiple) imputation of missing values. # Tell us what you think! *Required field. Your comment will show up after approval from a moderator. # THIS TUTORIAL HAS 2 COMMENTS: • ### By Mengesha Abrha on January 6th, 2017 This tutor is very fantastic and still we want to explore more. thanks in advance • ### By Vu on November 14th, 2018 Super page. The way you explained things is so simple but easy to understand. Thank so much!	745	2,990	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.671875	4	CC-MAIN-2020-34	latest	en	0.84446	SPSS TUTORIALS. # SPSS Sum – Cautionary Note. ## Summary. In SPSS, `SUM(v1,v2)` is not always equivalent to `v1 + v2`. This tutorial explains the difference and shows how to make the right choice here.. Different Ways of Taking Sums have Different Outcomes when Missing Values are Present. ## Explanation. • In SPSS, `v1 + v2 + v3` will result in a system missing value if at least one missing value is present in v1, v2 or v3.. • The first alternative, `SUM(v1, v2, v3)` implicitly replaces missing values with zeroes.. • The second alternative, `MEAN(v1, v2, v3) * 3` implicitly replaces missing values with the mean of the non missing values.. • The third alternative, `MEAN.2(v1, v2, v3) * 3` is almost similar to the second. However, by suffixing `MEAN` by `.2`, you ensure that a mean is only calculated if at least two non missing values are present in v1, v2 and v3.. • These points are demonstrated by the syntax below.. ## SPSS Syntax Demonstration. data list free/v1 v2 v3.. begin data. 1 3 5. 1 3 ''. 1 '' ''. end data.. compute sum_by_sum = sum(v1,v2,v3).. compute sum_by_plus = v1 + v2 + v3.. compute sum_by_mean = mean(v1 to v3) * 3.. compute sum_by_mean.2 = mean.2(v1 to v3) * 3.. exe.. ## So Which one Is Best?. • This question is rather hard to answer.	It may depend on the meaning of the missing values (question skipped? technical problem?). Also, what are the individual questions and the sum supposed to reflect?. • Second, the amount of missing values and sample size may be taken into account. Does it permit excluding some observations with missing values? Will this affect representativity and if so, is that a real problem?. • For one thing, sums calculated by `SUM` may be biased towards zero. For instance, if v1 through v3 measure components of satisfaction, repondents will be seen as "less satisfied" insofar they have more missing values. That conclusion may be misleading.. • Using the `+` operator does not induce such bias but may result in many missing values in the sum. This problem becomes larger as more missing values are present in the input variables and a sum is taken over more variables.. • Multiplying the mean by the number of variables, may be a better alternative. However, it will always come up with with a sum if there's at least one non missing value. Especially with many input variables, a single value may be jugded insufficient for inferring a summation measure.. • But perhaps none of these options is expected to yield sufficiently accurate results. In this case, one could partly circumvent the problem with a (multiple) imputation of missing values.. # Tell us what you think!. *Required field. Your comment will show up after approval from a moderator.. # THIS TUTORIAL HAS 2 COMMENTS:. • ### By Mengesha Abrha on January 6th, 2017. This tutor is very fantastic and still we want to explore more. thanks in advance. • ### By Vu on November 14th, 2018. Super page. The way you explained things is so simple but easy to understand. Thank so much!.
https://math.libretexts.org/Bookshelves/Pre-Algebra/Book%3A_Prealgebra_(OpenStax)	1,591,087,084,000,000,000	text/html	crawl-data/CC-MAIN-2020-24/segments/1590347423915.42/warc/CC-MAIN-20200602064854-20200602094854-00022.warc.gz	420,342,359	24,621	Skip to main content $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ # Book: Prealgebra (OpenStax) $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.	589	1,797	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.890625	4	CC-MAIN-2020-24	latest	en	0.352494	Skip to main content. $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$. # Book: Prealgebra (OpenStax). $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\\| #1 \\|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$.	Prealgebra is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.
http://math.stackexchange.com/questions/tagged/residue-calculus+trigonometry	1,411,304,127,000,000,000	text/html	crawl-data/CC-MAIN-2014-41/segments/1410657135549.24/warc/CC-MAIN-20140914011215-00298-ip-10-234-18-248.ec2.internal.warc.gz	178,170,859	12,015	# Tagged Questions 46 views ### Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$ I'd like to calculate the following integral on the interval $[0,2\pi]$: $$I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi.$$ 86 views ### Show these approximations of $\cos$, $\sin$ and $\tan$ are exact. A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: c_n(x) = \frac{\prod_{i=1}^n ... 151 views ### Use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$? How do I use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ... ### Singularities of $\ \frac{z-1}{z^2 \sin z} \$ Find all singularities of $\ \frac{z-1}{z^2 \sin z} \$ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, ... The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...	414	1,245	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.609375	4	CC-MAIN-2014-41	latest	en	0.845352	# Tagged Questions. 46 views. ### Computing $\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t$. I'd like to calculate the following integral on the interval $[0,2\pi]$: $$I=\int_0^{2\pi}(1+2\cos t)^n\cos nt\ \mathrm{d}t = 2\pi.$$. 86 views. ### Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.. A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: c_n(x) = \frac{\prod_{i=1}^n .... 151 views. ### Use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$?.	How do I use Residue Theorem to evaluate $\ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \$ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise .... ### Singularities of $\ \frac{z-1}{z^2 \sin z} \$. Find all singularities of $\ \frac{z-1}{z^2 \sin z} \$ Determine if they are isolated or nonisolated. This is not hard, it is z = 0 and z = k*pi. But how do I: For isolated singularities, .... The sum of the squares of the reciprocals of the positive fixed points of the tangent function is $1/10$. I've seen this proved by means of residues, but I don't remember the details. I've also ...
https://www.jiskha.com/similar?question=Show+that+the+equation+3x%5E2+-+x%5E3+%2B+3+%3D+0+can+be+rearranged+to+give%3A+x+%3D+3+%2B+3+----+x%5E2+x%5E3+%3D+Divide+by+%3F+%3D+x+%3D+3+%2B+3+----+x%5E2+Using+Xn%2B1+%3D+3&page=41	1,563,246,547,000,000,000	text/html	crawl-data/CC-MAIN-2019-30/segments/1563195524475.48/warc/CC-MAIN-20190716015213-20190716041213-00394.warc.gz	751,720,211	19,037	# Show that the equation 3x^2 - x^3 + 3 = 0 can be rearranged to give: x = 3 + 3 ---- x^2 x^3 = Divide by ? = x = 3 + 3 ---- x^2 Using Xn+1 = 3 62,835 questions, page 41 1. ## math help find an equation of the line passing through the given points. Write the equation in function notation (4,5) and (2,9) f(x)= asked by Anonymous on October 11, 2010 2. ## HELP what is an equation in slope intercept for. for the line that passes through the points 1,-3 and 3,1 y=3x+1 y=x-3 y=2x+5 y=2x-5 For the equation -4y=8x what is the constant of variation? -4 -2 1 2 Are they right asked by Jason on February 10, 2017 3. ## math find an equation of the line with the given slope that passes through the given point. write the equation in the form Ax+By=C M=3/2, (7,-3) asked by ann on October 29, 2013 4. ## Pre-Calculus Find the equation of the hyperbola whose vertices are at (-1,-5) and (-1,1) with a focus at (-1,-7). So far I have the center at (-1,-2) and part of the equation is (y+2)^2 - (x+1)^2 but do not know how to figure a^2, b^2, or c^2. Help please.... asked by Lucy on October 12, 2008 5. ## math find an equation of the line containing the given pair of points. Use function notation to write the equation. (5/7,10/14) and (-1/7,13/14) f(x) asked by kate on October 8, 2010 6. ## calculus what is the equation for taking the derivitive of -x^2+x just equation setup i don't need an answer from power rool asked by jerry on December 30, 2010 7. ## College Algebra Given the equation x^2+y^2-8x-6y=0 (a) Write the equation in standard form. (b) State the center, radius, and intercepts. asked by sunny on January 4, 2012 8. ## Calculus I need to solve this equation for P. I've got it partially solved, but I'm having trouble on the last part. Here's the equation. 4 = -200P^-3 I can get it down to: -50 = P^-3 I think? But, I don't know how to get the -3 away from the P. 9. ## calculus supose that (f(x+h)-f(x))/h=3x^2-6h+5 and f(1)=8. ind the equation of the tangent line to the graph of y=f(x) at x=1. if we write this equation in the form y=mx+b then m=? and b=? asked by Thalia on September 30, 2010 10. ## Math Explain how to solve? Which equation below shows the equation y = – 1/4x + 4 written in standard form using integers? A) 2x + 3y = 21 B) 3x – 2y = 21 C) –2x + 3y = 21 D) –2x – 3y = 21 asked by Jane on January 5, 2012 11. ## Algebra Write an equation that expresses the relationship. Then solve the equation for c. q varies jointly as c and the square of h. asked by Jacqueline on October 23, 2014 12. ## math find an equation of the line passing through the given points. Write the equation in function notation (4,5) and (2,9) f(x)= asked by Anonymous on October 11, 2010 13. ## algebra Find algebraically the equation of the axis of symmetry and the coordinates of the vertex of the parabola whose equation is y= -2x-8x+3 asked by KR on February 1, 2010 14. ## College Algebra Solve each equation. If an equation has no solution, so indicate. (no complex #) a - 1 = _ 3a + 2 a+2 a2 + 4a + 4 reads: a over a+2 minus 1 equals -3a+2 over a2 (a squared) + 4a + 4 asked by Colby on September 16, 2012 15. ## Calculus Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1) asked by Anonymous on February 18, 2011 16. ## math Can someone please help? Solve the differential equation dy/dx = 6xy with the condition y(0) = 40 Find the solution to the equation y= ______ asked by mark on May 19, 2013 17. ## Algebra For the following system, if you isolated x in the first equation to use the Substitution Method, what expression would you substitute into the second equation? −x − 2y = −4 3x + y = 12 −2y − 4 2y − 4 2y + 4 −2y + 4 --- I think it'd be this asked by anonymouse on December 13, 2016 18. ## algebra Given the equation x^2+y^2-8x-6y=0 (a.) Write the equation in standard form . (b.) State the center, radius, and intercepts. asked by sarah on January 18, 2012 19. ## Chemistry Why would potassium no longer appear in the equation of a net ionic equation with KIO3 being the oxidizing agent? asked by Teel on April 20, 2010 20. ## math How do I convert the polar equation r - 3cos(theta) = 5sin to cartesian equation form? asked by karen on March 22, 2010 21. ## Math Express the given equation in rectangular coordinate equation and sketch the curve. r^2=1+sin(theta) 22. ## algebra ll the first equation of the system is mutiplied by 4, by what number would you mutiply the second equation to eliminate the y variable by adding? first: 2x+5y=16 second:8x - 4y =10 i got 2 0r 5 im not sure which one asked by Anonymous, on August 15, 2008 23. ## Calculus Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1) y = asked by Ant on June 13, 2013 24. ## Algebra I have a parabola that I have to write as an equation I found the vertex as (-3,-2) and the following points (-4,-1)(-2,-1),(-5,2),(-1,2). I can't seem to come up with the correct equation. I'm using the formula y=a(h-k)^2+k but I'm not doing something asked by Jennifer on May 15, 2011 25. ## ALGEBRA 2 The equation y = mx + b is known as the slope-intercept form of a line. Which equation shows y = mx + b correctly solved for x? asked by KELR on January 12, 2012 26. ## Calculus Find the equation of the tangent plane and symmetric equation for the normal line to the given surface at P: xy^2 + zy^2 + 4y -xz^2 = 18 P(-2,0,3) asked by Elyse on June 2, 2013 27. ## Algebra Write a trigonometric equation for which the solutions lie in quadrant I and II. Then solve the equation for 0 asked by Blank on March 12, 2019 28. ## Geometry Point J is between H and K on line HK. Use the given information to write an equation in terms of x. Solve the equation. Then find HJ and JK HJ=5x-3 JK=x-9 KH=5x asked by anonymous on August 27, 2015 29. ## science a =vf-vi/t is the equation for calculating the acceleration of an object. Write out the relationship shown in the equation, using words. asked by Milli on February 22, 2012 30. ## math Find an equation of the tangent line to the graph of y = g(x) at x = 2 if g(2) = −6 and g'(2) = 5. (Enter your answer as an equation in terms of y and x.) asked by Yuxiang Nie on February 21, 2019 31. ## Geometry Find the equation of a line that is perpendicular to y=2x+7, and passes through the point (-4, 7) I got the answer y=-1/2x+5, is that the correct equation?? 32. ## Algebra Is this equation true or false? Why? 20-4ã2=16ã2 With like radicands should I add the 4 to the other side of the equation? I'm really lost. Can someone please explain this to me? asked by C on September 9, 2010 33. ## Polar Equations Identify each polar equation by changing the equation to rectangular coordinates. Let t = theta (1) t = -pi/4 (2) r(sin(t)) = -2 asked by sharkman on February 23, 2008 34. ## chemistry the net ionic equation for the following molecular equation KNO2(aq) + HCl(aq) KCl(aq) + HNO2(aq) asked by Jasmine on October 10, 2012 35. ## Calculus Find the equation of the tangent plane and symmetric equation for the normal line to the given surface at P: xy^2 + zy^2 + 4y -xz^2 = 18 P(-2,0,3) asked by Kim on June 2, 2013 36. ## Precalc i need help with this equation anything help will be greatly appreciated find the standard equation of a parabola with a focus (-2,0) directrix x=3 asked by Natali on November 13, 2009 Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1) asked by Anonymous on February 18, 2011 38. ## Calculus Find inverse equation of y= 3e^(x-1)^5 Answer I got is (ln(x/3))^(1/5) +5 =y But I think it's wrong because the domain and range is really different from the original equation asked by Po on January 22, 2017 39. ## Maths A differential equation question Find the general solution to the equation (dy/dx)^2 + sinxcos^2x(dy/dx) - sin^4x = 0 Thanks asked by Peirs on March 16, 2018 40. ## college algebra Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation -5r^(4)+6r^(3)+9r-15=0 asked by samantha on October 27, 2010 41. ## Algebra II Write an equation for a parabola that opens upward from its vertex at (-4,3). What is the equation of its line of symmetry? asked by Sam on November 27, 2012 42. ## Math Write an equation to represent the relationship " a number decreased by 7 is the same thing as 28" and then solve the equation asked by Ivory on September 9, 2015 43. ## Alg. Find the coordinates of the vertex and the equation of the axis of symmetry of the parabola given by the equation. y = − 1/7 x^2 + x − 3 and graph it asked by K- ASAP on May 7, 2014 44. ## Math Can someone please help me? 33 1/3% of 7,500 is what number? (Write the equation and solve it ) this is my equation: 33 1/3% = 7500 100 Base= x asked by Anonymous on August 21, 2009 45. ## Algebra Suppose that Y varies directly with X, and Y=16 when X=10. (a) Write a direct equation that relates X and Y Equation: (b) Find Y when X=15 Y= asked by Anonymous on March 11, 2016 46. ## Chemistry Complete and balance the following molecular equation. Cu(OH)2(s)+HClO4(aq)→ And write the net ionic equation for it. asked by Thomas on February 26, 2019 47. ## maths convert polar equation into rectangular equation? theta= 4pi/3. Confused kindly help asked by Asfand on November 25, 2016 48. ## Chemistry Complete and balance the following molecular equation. Al(OH)3(s)+HNO3(aq)→ And write the net ionic equation for it. asked by Thomas on February 26, 2019 49. ## Math 11 The equation y= x^2-8 is translated 7 units to the left. Which equation describes the graph after it has undergone this translation? a) y=x^2-1 b)y=x^2-15 c) y=(x+7)^2-8 d) y=(x-7)^2-8 e) y=-7x^2-8 asked by julie on September 17, 2007 50. ## geometry using the slope y-intercert equation, how do you write an equation that passes through point p and is parallel to the line. P(2,4), y = -3x. Is it y = -3x + 10? Thanks asked by jill on September 28, 2016 51. ## Mechanics 2) A geometry textbook gives the equation of a parabola as y =x2, where x and y are measured in centimetres. How can this equation be dimensionally correct? asked by Sinawo on August 18, 2016 52. ## chemistry Balance the following equation by partial equation method:- P4 + HNO3= H3PO4 + NO2 + H2O asked by simran on May 27, 2014 53. ## chemistry The Rydberg equation (1/lambda=R/ni^2–R/nf^2) can be treated as a line equation. What is the value of nf as a function of the slope (m) and y-intercept(b)? asked by Katherine on January 10, 2010 54. ## pre cal The given equation is a partial answer to a calculus problem. Solve the equation for the symbol y'. 6y2y' − y − xy' = x y' = 55. ## math Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation -5r^(4)+6r^(3)+9r-15=0 asked by Allie on October 26, 2010 56. ## math help & correction Problem #1 Is this correct or wrong? Find the slope of the line passing through the points(-1, -1)and(-1, 2). Write the equation of the line. For this one I KEEP GETTING Y= - (3)/(2)x-2.5 Problem #2 Find the y-intecept and slope of 7x+9y=72 My answer: asked by jacky on March 31, 2007 57. ## MATH Carolyn and Kim are selling lemonade this summer.  It costs \$0.10 to make each cup of lemonade.  They are going to sell each cup of lemonade for \$0.25. a. What is the total cost to make 55 cups of lemonade? Show or explain your work.\$5.50 b. If they asked by NINA on February 20, 2012 58. ## Science * P is a a black powder, which, when fused with a mixture of potassium hydroxide and potassium nitrate and then treated with water yields a green solution. The green solution turn purple when acidified. -identify P, the green solution and purple solution. asked by Piom on July 30, 2011 59. ## Intermediate Algebra If the discriminant of a quadratic equation has the given value, determine the number and type of solutions of the equation. A.48 B.0 asked by Ralph on April 21, 2012 60. ## Math Solve the equation (5y+1)^2 + 5(5y+1) + 6 = 0. Do not expand first. I solved the equation up to 25y^2 + 35y + 12 = 0 and not sure how to proceed from here on. asked by Valdes on September 4, 2015 61. ## algebra Find the equation of the line that passes through the points ( 8, - 15 )and(-2,5). write the equation in y=mx = b form. asked by Julie on February 19, 2011 62. ## algebra write each equation in slope-intercept form. then graph the line described by the equation. 2y=4x+6 asked by genny on January 23, 2014 63. ## algebra write each equation in slope-intercept form. then graph the line described by the equation. 3y+2x=12 asked by genny on January 23, 2014 64. ## Math 7d-4 (d+8) = -8 ??? When I did this equation I got 8=d I know I did something wrong. I just don't know any other way to do it. I did basic equation-solving rules and got his answer. Help? asked by Andrey on January 6, 2015 65. ## Math Multiply the equation by a power of 10 to write an equivalnet equation with integers coefficients. 6.2x + 4.5 = 3.8x+ 7.9 asked by Ronique on January 7, 2008 66. ## Precalculus Solve the equation (x+3)(x-3) = -9. What are the solutions to the equation if you replace x first with sinx and then with cosx, where 0≤x≤2pi? asked by Melissa on January 26, 2012 67. ## math Describe the difference between an open equation and a closed equation. And on my last quest did steve mean 4,0 and 3, -2? asked by Jman on August 29, 2013 68. ## algebra Replace each equation with constants to complete the square and form a true equation x^2+2x+--=(x+---)^2 asked by mema on May 26, 2011 69. ## Math I need help with an equation problem. Write the equation of the line that passes through each pair of points. (-2, -7) and (3,8) asked by Johnnie on November 28, 2011 70. ## MATH Julie is solving the equation x2 + 6x + 9 = 0 and notices that the discriminant b2 - 4ac has a value of 0. This tells her that the equation has... asked by LINDSEY on May 16, 2013 71. ## Precalculus Find the equation of the ellipse with the center at (2,-3), one focus at (3,-3), and one vertex at (5,-3). Also graph the equation. asked by Anonymous on May 13, 2014 72. ## Mathematics The equation of a circle is given by D: x^2 + y^2 - 2x - 4y - 20 = 0 Use implicit differentiation to find the equation of the normal D at the point S(5, -1) asked by James on February 25, 2016 73. ## Math Help What is the value of x in the equation 3(2x + 4) = −6? A. −3 B. 1 C. 12 D. 19 Please explain this I'm having difficulties with this equation, I don't want to fail and loose my A in the class. Thank you. asked by Alex on March 30, 2017 74. ## Algebra help Match the equation 4x^6-2x^3+1=0. With a substitution from the column that could be used to reduced the equation to quadratic form: a) u=x^-1/3 b) u=x^1/3 c) u=x^-2 d) u=x^2 e) u=x^-2/3 f) u=x^3 g) u=x^2/3 h) u= x^4 asked by Miguel on March 24, 2011 75. ## Algebra help Match the equation 4x^6-2x^3+1=0. With a substitution from the column that could be used to reduced the equation to quadratic form: a) u=x^-1/3 b) u=x^1/3 c) u=x^-2 d) u=x^2 e) u=x^-2/3 f) u=x^3 g) u=x^2/3 h) u= x^4 asked by Miguel on March 24, 2011 76. ## math find an equation of the line passing through the pair of points. write the equation in Ax+By=C (-9,7), (-10,-4) asked by ann on October 29, 2013 77. ## Math Which equation is the equation of a line that passes through (–10, 3) and is perpendicular to y = 5x – 7? y = 5x + 53 y = –x – 7 y = –x + 1 y = x + 5 Help - I don't understand...being told it is C but why? asked by Deborah on November 8, 2012 78. ## maths a)If one root of the equation x^2 + a(3a-5)x = 2(x+4a) is negative of the other. Find the values of a . b) If a>0, use the result of (a) to solve the equation. 79. ## algebra If is a line whose equation is y= 2x - 1, find the equation of the image of under each of the following translations: in book explanation a. (x, y) S (x, y - 2) b. (x, y) S (x +3, y) c. (x, y) S (x - 3, y + 2) asked by Rebecca on March 3, 2010 80. ## College Algebra For parts (a)–(c) use the following equation: y = 2/(x + 6). (a) Find the ordered pair that represents a solution to the equation when the value of x is 0. asked by Autumn on September 7, 2008 81. ## Math Find an equation of the line. Write the equation in the standard form. Through (7,6); parallel to 4x-y=9 asked by Charlie on August 22, 2018 82. ## math A circle has centre(5,12) and its tangent to the line with equation 2x-y+3=0. Wtite equation of the cirle asked by donald on December 28, 2016 83. ## Precalculus Identify the graph of the equation. What is the angle of rotation for the equation? y^2+8x=0 How do i figure this out? I think it has to be a hyperbola. asked by Beatrice on August 19, 2011 equation 1.5(8x)=300 It tells me to simplify the left side of the equation and solve. Can you help? asked by tito on October 14, 2008 85. ## Chemistry When working with the combined gas law you would use the following equation: (P1)(V1)/T1 = (P2)(V2)/T2 I am not quite sure how you would rearrange this equation to solve for T2. Please Help ! :) asked by Anonymous on January 7, 2009 86. ## math when I try to solve the equation 9x-5y=4, my answer is y=9/5+4/5. is this correct because I am having trouble trying to do a table of values for this equation. asked by malia on January 1, 2008 87. ## math A circle has centre(5,12) and its tangent to the line with equation 2x-y+3=0. Wtite equation of the cirle asked by donald on December 30, 2016 88. ## Maths Consider the equation in x: (2k-3)x^2 + (k+1)x - 1 = 0. For what value of k is this a linear equation. I don't understand what the question is asking and I'm so confused. Any help is appreciated. asked by Lenard on May 14, 2019 89. ## Math Sketch a graph of the equation y=-x^2 + x + 10 for x=0...x=4 i think that the equation y=-x2+x+10 is a parabola and for the x values that were given am i suppose to plug in? asked by Anonymous on September 9, 2010 90. ## maths physics chemistry State the principle on which each of the following is based. (a) the equation of continuity, Av = constant (b) the Bernoulli equation asked by Joseph on December 29, 2015 91. ## math When a situation can be modeled by a linear equation, what information do you need in order to find an equation? asked by Carly on October 29, 2016 92. ## Converting Cartesian Equations to Polar Equations Convert the Cartesian equation x^2 - y^2 = 16 to a polar equation. r² = 16/cos2è r = -4 r² = 8 help please im stuck between 2 answers. 93. ## math The x-intercepts of a particular equation are x = p, q. Suggest a possible equation. Quite confused on this question... Any idea? Thanks in advance. asked by Anonymous on June 29, 2018 94. ## math When a situation can be modeled by a linear equation, what information do you need in order to find an equation? asked by vick on July 20, 2017 95. ## math I need help solving this equation please. Use the qudratic formula to find any x-intercepton the graph of the equation. Y=3x^2+9-1 asked by Sam on April 24, 2013 96. ## Chemistry Find the product for the equation and then write a balanced equation. Cl2 + FeBr3 -> ? asked by Zoe on January 15, 2012 97. ## math solve each equation.write each equation in simplest form 7x/8=21 a.24 b.18.375 c.1.857 d.19 asked by chen on December 2, 2015 98. ## Math How do I solve this simultaneous equation 3x-2y=5 (1) 2y-5x=9 (2) I managed to find the value of x and that is -7 but I am having problems to substitute it's value in equation (1) asked by Vince on May 16, 2010 99. ## Math Julie is solving the equation x2 + 6x + 9 = 0 and notices that the discriminant b2 - 4ac has a value of 0. This tells her that the equation has asked by Jack on October 23, 2012 100. ## analytic geometry FINd an equation in x and y that has the same graph as the polar equation r=Î± sin theta, with Î± is not equal to 0. asked by Anonymous on September 17, 2015	6,442	20,226	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.671875	4	CC-MAIN-2019-30	latest	en	0.938636	# Show that the equation 3x^2 - x^3 + 3 = 0 can be rearranged to give: x = 3 + 3 ---- x^2 x^3 = Divide by ? = x = 3 + 3 ---- x^2 Using Xn+1 = 3. 62,835 questions, page 41. 1. ## math help. find an equation of the line passing through the given points. Write the equation in function notation (4,5) and (2,9) f(x)=. asked by Anonymous on October 11, 2010. 2. ## HELP. what is an equation in slope intercept for. for the line that passes through the points 1,-3 and 3,1 y=3x+1 y=x-3 y=2x+5 y=2x-5 For the equation -4y=8x what is the constant of variation? -4 -2 1 2 Are they right. asked by Jason on February 10, 2017. 3. ## math. find an equation of the line with the given slope that passes through the given point. write the equation in the form Ax+By=C M=3/2, (7,-3). asked by ann on October 29, 2013. 4. ## Pre-Calculus. Find the equation of the hyperbola whose vertices are at (-1,-5) and (-1,1) with a focus at (-1,-7). So far I have the center at (-1,-2) and part of the equation is (y+2)^2 - (x+1)^2 but do not know how to figure a^2, b^2, or c^2. Help please..... asked by Lucy on October 12, 2008. 5. ## math. find an equation of the line containing the given pair of points. Use function notation to write the equation. (5/7,10/14) and (-1/7,13/14) f(x). asked by kate on October 8, 2010. 6. ## calculus. what is the equation for taking the derivitive of -x^2+x just equation setup i don't need an answer from power rool. asked by jerry on December 30, 2010. 7. ## College Algebra. Given the equation x^2+y^2-8x-6y=0 (a) Write the equation in standard form. (b) State the center, radius, and intercepts.. asked by sunny on January 4, 2012. 8. ## Calculus. I need to solve this equation for P. I've got it partially solved, but I'm having trouble on the last part. Here's the equation. 4 = -200P^-3 I can get it down to: -50 = P^-3 I think? But, I don't know how to get the -3 away from the P.. 9. ## calculus. supose that (f(x+h)-f(x))/h=3x^2-6h+5 and f(1)=8. ind the equation of the tangent line to the graph of y=f(x) at x=1. if we write this equation in the form y=mx+b then m=? and b=?. asked by Thalia on September 30, 2010. 10. ## Math. Explain how to solve? Which equation below shows the equation y = – 1/4x + 4 written in standard form using integers? A) 2x + 3y = 21 B) 3x – 2y = 21 C) –2x + 3y = 21 D) –2x – 3y = 21. asked by Jane on January 5, 2012. 11. ## Algebra. Write an equation that expresses the relationship. Then solve the equation for c. q varies jointly as c and the square of h.. asked by Jacqueline on October 23, 2014. 12. ## math. find an equation of the line passing through the given points. Write the equation in function notation (4,5) and (2,9) f(x)=. asked by Anonymous on October 11, 2010. 13. ## algebra. Find algebraically the equation of the axis of symmetry and the coordinates of the vertex of the parabola whose equation is y= -2x-8x+3. asked by KR on February 1, 2010. 14. ## College Algebra. Solve each equation. If an equation has no solution, so indicate. (no complex #) a - 1 = _ 3a + 2 a+2 a2 + 4a + 4 reads: a over a+2 minus 1 equals -3a+2 over a2 (a squared) + 4a + 4. asked by Colby on September 16, 2012. 15. ## Calculus. Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1). asked by Anonymous on February 18, 2011. 16. ## math. Can someone please help? Solve the differential equation dy/dx = 6xy with the condition y(0) = 40 Find the solution to the equation y= ______. asked by mark on May 19, 2013. 17. ## Algebra. For the following system, if you isolated x in the first equation to use the Substitution Method, what expression would you substitute into the second equation? −x − 2y = −4 3x + y = 12 −2y − 4 2y − 4 2y + 4 −2y + 4 --- I think it'd be this. asked by anonymouse on December 13, 2016. 18. ## algebra. Given the equation x^2+y^2-8x-6y=0 (a.) Write the equation in standard form . (b.) State the center, radius, and intercepts.. asked by sarah on January 18, 2012. 19. ## Chemistry. Why would potassium no longer appear in the equation of a net ionic equation with KIO3 being the oxidizing agent?. asked by Teel on April 20, 2010. 20. ## math. How do I convert the polar equation r - 3cos(theta) = 5sin to cartesian equation form?. asked by karen on March 22, 2010. 21. ## Math. Express the given equation in rectangular coordinate equation and sketch the curve. r^2=1+sin(theta). 22. ## algebra ll. the first equation of the system is mutiplied by 4, by what number would you mutiply the second equation to eliminate the y variable by adding? first: 2x+5y=16 second:8x - 4y =10 i got 2 0r 5 im not sure which one. asked by Anonymous, on August 15, 2008. 23. ## Calculus. Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1) y =. asked by Ant on June 13, 2013. 24. ## Algebra. I have a parabola that I have to write as an equation I found the vertex as (-3,-2) and the following points (-4,-1)(-2,-1),(-5,2),(-1,2). I can't seem to come up with the correct equation. I'm using the formula y=a(h-k)^2+k but I'm not doing something. asked by Jennifer on May 15, 2011. 25. ## ALGEBRA 2. The equation y = mx + b is known as the slope-intercept form of a line. Which equation shows y = mx + b correctly solved for x?. asked by KELR on January 12, 2012. 26. ## Calculus. Find the equation of the tangent plane and symmetric equation for the normal line to the given surface at P: xy^2 + zy^2 + 4y -xz^2 = 18 P(-2,0,3). asked by Elyse on June 2, 2013. 27. ## Algebra. Write a trigonometric equation for which the solutions lie in quadrant I and II. Then solve the equation for 0. asked by Blank on March 12, 2019. 28. ## Geometry. Point J is between H and K on line HK. Use the given information to write an equation in terms of x. Solve the equation. Then find HJ and JK HJ=5x-3 JK=x-9 KH=5x. asked by anonymous on August 27, 2015. 29. ## science. a =vf-vi/t is the equation for calculating the acceleration of an object. Write out the relationship shown in the equation, using words.. asked by Milli on February 22, 2012. 30. ## math. Find an equation of the tangent line to the graph of y = g(x) at x = 2 if g(2) = −6 and g'(2) = 5. (Enter your answer as an equation in terms of y and x.). asked by Yuxiang Nie on February 21, 2019. 31. ## Geometry. Find the equation of a line that is perpendicular to y=2x+7, and passes through the point (-4, 7) I got the answer y=-1/2x+5, is that the correct equation??. 32. ## Algebra. Is this equation true or false? Why? 20-4ã2=16ã2 With like radicands should I add the 4 to the other side of the equation? I'm really lost. Can someone please explain this to me?. asked by C on September 9, 2010. 33. ## Polar Equations. Identify each polar equation by changing the equation to rectangular coordinates. Let t = theta (1) t = -pi/4 (2) r(sin(t)) = -2. asked by sharkman on February 23, 2008. 34. ## chemistry. the net ionic equation for the following molecular equation KNO2(aq) + HCl(aq) KCl(aq) + HNO2(aq). asked by Jasmine on October 10, 2012. 35. ## Calculus. Find the equation of the tangent plane and symmetric equation for the normal line to the given surface at P: xy^2 + zy^2 + 4y -xz^2 = 18 P(-2,0,3). asked by Kim on June 2, 2013. 36. ## Precalc. i need help with this equation anything help will be greatly appreciated find the standard equation of a parabola with a focus (-2,0) directrix x=3. asked by Natali on November 13, 2009. Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1). asked by Anonymous on February 18, 2011. 38. ## Calculus. Find inverse equation of y= 3e^(x-1)^5 Answer I got is (ln(x/3))^(1/5) +5 =y But I think it's wrong because the domain and range is really different from the original equation. asked by Po on January 22, 2017. 39. ## Maths. A differential equation question Find the general solution to the equation (dy/dx)^2 + sinxcos^2x(dy/dx) - sin^4x = 0 Thanks. asked by Peirs on March 16, 2018. 40. ## college algebra. Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation -5r^(4)+6r^(3)+9r-15=0. asked by samantha on October 27, 2010. 41. ## Algebra II. Write an equation for a parabola that opens upward from its vertex at (-4,3). What is the equation of its line of symmetry?. asked by Sam on November 27, 2012. 42. ## Math. Write an equation to represent the relationship " a number decreased by 7 is the same thing as 28" and then solve the equation. asked by Ivory on September 9, 2015. 43. ## Alg.. Find the coordinates of the vertex and the equation of the axis of symmetry of the parabola given by the equation. y = − 1/7 x^2 + x − 3 and graph it. asked by K- ASAP on May 7, 2014. 44. ## Math. Can someone please help me? 33 1/3% of 7,500 is what number? (Write the equation and solve it ) this is my equation: 33 1/3% = 7500 100 Base= x. asked by Anonymous on August 21, 2009. 45. ## Algebra. Suppose that Y varies directly with X, and Y=16 when X=10. (a) Write a direct equation that relates X and Y Equation: (b) Find Y when X=15 Y=. asked by Anonymous on March 11, 2016. 46. ## Chemistry. Complete and balance the following molecular equation. Cu(OH)2(s)+HClO4(aq)→ And write the net ionic equation for it.. asked by Thomas on February 26, 2019. 47. ## maths. convert polar equation into rectangular equation? theta= 4pi/3. Confused kindly help. asked by Asfand on November 25, 2016. 48. ## Chemistry. Complete and balance the following molecular equation. Al(OH)3(s)+HNO3(aq)→ And write the net ionic equation for it.. asked by Thomas on February 26, 2019. 49. ## Math 11. The equation y= x^2-8 is translated 7 units to the left. Which equation describes the graph after it has undergone this translation? a) y=x^2-1 b)y=x^2-15 c) y=(x+7)^2-8 d) y=(x-7)^2-8 e) y=-7x^2-8. asked by julie on September 17, 2007. 50. ## geometry. using the slope y-intercert equation, how do you write an equation that passes through point p and is parallel to the line. P(2,4), y = -3x. Is it y = -3x + 10? Thanks. asked by jill on September 28, 2016. 51. ## Mechanics. 2) A geometry textbook gives the equation of a parabola as y =x2, where x and y are measured in centimetres. How can this equation be dimensionally correct?. asked by Sinawo on August 18, 2016. 52. ## chemistry. Balance the following equation by partial equation method:- P4 + HNO3= H3PO4 + NO2 + H2O. asked by simran on May 27, 2014.	53. ## chemistry. The Rydberg equation (1/lambda=R/ni^2–R/nf^2) can be treated as a line equation. What is the value of nf as a function of the slope (m) and y-intercept(b)?. asked by Katherine on January 10, 2010. 54. ## pre cal. The given equation is a partial answer to a calculus problem. Solve the equation for the symbol y'. 6y2y' − y − xy' = x y' =. 55. ## math. Use Descarte's rule of signs to discuss the possibilities for the roots of the equation. Do not solve the equation -5r^(4)+6r^(3)+9r-15=0. asked by Allie on October 26, 2010. 56. ## math help & correction. Problem #1 Is this correct or wrong? Find the slope of the line passing through the points(-1, -1)and(-1, 2). Write the equation of the line. For this one I KEEP GETTING Y= - (3)/(2)x-2.5 Problem #2 Find the y-intecept and slope of 7x+9y=72 My answer:. asked by jacky on March 31, 2007. 57. ## MATH. Carolyn and Kim are selling lemonade this summer.  It costs \$0.10 to make each cup of lemonade.  They are going to sell each cup of lemonade for \$0.25. a. What is the total cost to make 55 cups of lemonade? Show or explain your work.\$5.50 b. If they. asked by NINA on February 20, 2012. 58. ## Science. * P is a a black powder, which, when fused with a mixture of potassium hydroxide and potassium nitrate and then treated with water yields a green solution. The green solution turn purple when acidified. -identify P, the green solution and purple solution.. asked by Piom on July 30, 2011. 59. ## Intermediate Algebra. If the discriminant of a quadratic equation has the given value, determine the number and type of solutions of the equation. A.48 B.0. asked by Ralph on April 21, 2012. 60. ## Math. Solve the equation (5y+1)^2 + 5(5y+1) + 6 = 0. Do not expand first. I solved the equation up to 25y^2 + 35y + 12 = 0 and not sure how to proceed from here on.. asked by Valdes on September 4, 2015. 61. ## algebra. Find the equation of the line that passes through the points ( 8, - 15 )and(-2,5). write the equation in y=mx = b form.. asked by Julie on February 19, 2011. 62. ## algebra. write each equation in slope-intercept form. then graph the line described by the equation. 2y=4x+6. asked by genny on January 23, 2014. 63. ## algebra. write each equation in slope-intercept form. then graph the line described by the equation. 3y+2x=12. asked by genny on January 23, 2014. 64. ## Math. 7d-4 (d+8) = -8 ??? When I did this equation I got 8=d I know I did something wrong. I just don't know any other way to do it. I did basic equation-solving rules and got his answer. Help?. asked by Andrey on January 6, 2015. 65. ## Math. Multiply the equation by a power of 10 to write an equivalnet equation with integers coefficients. 6.2x + 4.5 = 3.8x+ 7.9. asked by Ronique on January 7, 2008. 66. ## Precalculus. Solve the equation (x+3)(x-3) = -9. What are the solutions to the equation if you replace x first with sinx and then with cosx, where 0≤x≤2pi?. asked by Melissa on January 26, 2012. 67. ## math. Describe the difference between an open equation and a closed equation. And on my last quest did steve mean 4,0 and 3, -2?. asked by Jman on August 29, 2013. 68. ## algebra. Replace each equation with constants to complete the square and form a true equation x^2+2x+--=(x+---)^2. asked by mema on May 26, 2011. 69. ## Math. I need help with an equation problem. Write the equation of the line that passes through each pair of points. (-2, -7) and (3,8). asked by Johnnie on November 28, 2011. 70. ## MATH. Julie is solving the equation x2 + 6x + 9 = 0 and notices that the discriminant b2 - 4ac has a value of 0. This tells her that the equation has.... asked by LINDSEY on May 16, 2013. 71. ## Precalculus. Find the equation of the ellipse with the center at (2,-3), one focus at (3,-3), and one vertex at (5,-3). Also graph the equation.. asked by Anonymous on May 13, 2014. 72. ## Mathematics. The equation of a circle is given by D: x^2 + y^2 - 2x - 4y - 20 = 0 Use implicit differentiation to find the equation of the normal D at the point S(5, -1). asked by James on February 25, 2016. 73. ## Math Help. What is the value of x in the equation 3(2x + 4) = −6? A. −3 B. 1 C. 12 D. 19 Please explain this I'm having difficulties with this equation, I don't want to fail and loose my A in the class. Thank you.. asked by Alex on March 30, 2017. 74. ## Algebra help. Match the equation 4x^6-2x^3+1=0. With a substitution from the column that could be used to reduced the equation to quadratic form: a) u=x^-1/3 b) u=x^1/3 c) u=x^-2 d) u=x^2 e) u=x^-2/3 f) u=x^3 g) u=x^2/3 h) u= x^4. asked by Miguel on March 24, 2011. 75. ## Algebra help. Match the equation 4x^6-2x^3+1=0. With a substitution from the column that could be used to reduced the equation to quadratic form: a) u=x^-1/3 b) u=x^1/3 c) u=x^-2 d) u=x^2 e) u=x^-2/3 f) u=x^3 g) u=x^2/3 h) u= x^4. asked by Miguel on March 24, 2011. 76. ## math. find an equation of the line passing through the pair of points. write the equation in Ax+By=C (-9,7), (-10,-4). asked by ann on October 29, 2013. 77. ## Math. Which equation is the equation of a line that passes through (–10, 3) and is perpendicular to y = 5x – 7? y = 5x + 53 y = –x – 7 y = –x + 1 y = x + 5 Help - I don't understand...being told it is C but why?. asked by Deborah on November 8, 2012. 78. ## maths. a)If one root of the equation x^2 + a(3a-5)x = 2(x+4a) is negative of the other. Find the values of a . b) If a>0, use the result of (a) to solve the equation.. 79. ## algebra. If is a line whose equation is y= 2x - 1, find the equation of the image of under each of the following translations: in book explanation a. (x, y) S (x, y - 2) b. (x, y) S (x +3, y) c. (x, y) S (x - 3, y + 2). asked by Rebecca on March 3, 2010. 80. ## College Algebra. For parts (a)–(c) use the following equation: y = 2/(x + 6). (a) Find the ordered pair that represents a solution to the equation when the value of x is 0.. asked by Autumn on September 7, 2008. 81. ## Math. Find an equation of the line. Write the equation in the standard form. Through (7,6); parallel to 4x-y=9. asked by Charlie on August 22, 2018. 82. ## math. A circle has centre(5,12) and its tangent to the line with equation 2x-y+3=0. Wtite equation of the cirle. asked by donald on December 28, 2016. 83. ## Precalculus. Identify the graph of the equation. What is the angle of rotation for the equation? y^2+8x=0 How do i figure this out? I think it has to be a hyperbola.. asked by Beatrice on August 19, 2011. equation 1.5(8x)=300 It tells me to simplify the left side of the equation and solve. Can you help?. asked by tito on October 14, 2008. 85. ## Chemistry. When working with the combined gas law you would use the following equation: (P1)(V1)/T1 = (P2)(V2)/T2 I am not quite sure how you would rearrange this equation to solve for T2. Please Help ! :). asked by Anonymous on January 7, 2009. 86. ## math. when I try to solve the equation 9x-5y=4, my answer is y=9/5+4/5. is this correct because I am having trouble trying to do a table of values for this equation.. asked by malia on January 1, 2008. 87. ## math. A circle has centre(5,12) and its tangent to the line with equation 2x-y+3=0. Wtite equation of the cirle. asked by donald on December 30, 2016. 88. ## Maths. Consider the equation in x: (2k-3)x^2 + (k+1)x - 1 = 0. For what value of k is this a linear equation. I don't understand what the question is asking and I'm so confused. Any help is appreciated.. asked by Lenard on May 14, 2019. 89. ## Math. Sketch a graph of the equation y=-x^2 + x + 10 for x=0...x=4 i think that the equation y=-x2+x+10 is a parabola and for the x values that were given am i suppose to plug in?. asked by Anonymous on September 9, 2010. 90. ## maths physics chemistry. State the principle on which each of the following is based. (a) the equation of continuity, Av = constant (b) the Bernoulli equation. asked by Joseph on December 29, 2015. 91. ## math. When a situation can be modeled by a linear equation, what information do you need in order to find an equation?. asked by Carly on October 29, 2016. 92. ## Converting Cartesian Equations to Polar Equations. Convert the Cartesian equation x^2 - y^2 = 16 to a polar equation. r² = 16/cos2è r = -4 r² = 8 help please im stuck between 2 answers.. 93. ## math. The x-intercepts of a particular equation are x = p, q. Suggest a possible equation. Quite confused on this question... Any idea? Thanks in advance.. asked by Anonymous on June 29, 2018. 94. ## math. When a situation can be modeled by a linear equation, what information do you need in order to find an equation?. asked by vick on July 20, 2017. 95. ## math. I need help solving this equation please. Use the qudratic formula to find any x-intercepton the graph of the equation. Y=3x^2+9-1. asked by Sam on April 24, 2013. 96. ## Chemistry. Find the product for the equation and then write a balanced equation. Cl2 + FeBr3 -> ?. asked by Zoe on January 15, 2012. 97. ## math. solve each equation.write each equation in simplest form 7x/8=21 a.24 b.18.375 c.1.857 d.19. asked by chen on December 2, 2015. 98. ## Math. How do I solve this simultaneous equation 3x-2y=5 (1) 2y-5x=9 (2) I managed to find the value of x and that is -7 but I am having problems to substitute it's value in equation (1). asked by Vince on May 16, 2010. 99. ## Math. Julie is solving the equation x2 + 6x + 9 = 0 and notices that the discriminant b2 - 4ac has a value of 0. This tells her that the equation has. asked by Jack on October 23, 2012. 100. ## analytic geometry. FINd an equation in x and y that has the same graph as the polar equation r=Î± sin theta, with Î± is not equal to 0.. asked by Anonymous on September 17, 2015.
http://www.toontricks.com/2019/05/tutorial-fibonacci-using-1-variable.html	1,600,714,220,000,000,000	text/html	crawl-data/CC-MAIN-2020-40/segments/1600400202007.15/warc/CC-MAIN-20200921175057-20200921205057-00381.warc.gz	237,979,795	41,540	# Tutorial :Fibonacci using 1 variable ### Question: I was asked the following question in an interview: Is there any way in which Fibonacci series can be generated using only 1 variable ? I didn't know what to answer. What should I have said? ### Solution:1 Yes, you can used the closed-form expression: where You can calculate the expression using a `double` and round the result to the nearest integer. Because of the finite precision of floating point arithmetic this formula will give a wrong answer for large enough n, but I think it will work in the case when the result fits into a Java 32-bit integer. ### Solution:2 Up to a point, yes (though in C, you could convert it to Java - it would look much uglier). ``#include <stdio.h> #include <stdlib.h> int main (void) { unsigned long i = 1; printf ("0\n"); while (((i & 0xffff0000) >> 16) + (i & 0xffff) <= 0xffff) { printf ("%d\n", i & 0xffff); i = ((i & 0xffff) << 16) \| ((i >> 16) + (i & 0xffff)); } return 0; } `` which produces: ``0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 `` :-) The real question, of course, is: Why would you want to? If you're curious as to how it works, it's really quite simple. The one variable is actually divided into two parts and those two parts maintain the individual values for the Fibonacci sequence. It's still technically one variable, we've just imposed some extra structure on top of it to achieve our ends. ### Solution:3 Sure, using recursion: ``public class Test { public static int fib(int n) { return n < 2 ? n : fib(n-1) + fib(n-2); } public static void main(String[] args) { for(int i = 0; i <= 10; i++) { System.out.print(fib(i)+", "); } System.out.println("..."); } } // 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... `` ### Solution:4 Yes, but you still need to remember 2 values. You could take a 64-bit variable and use it as 2 32-bit vars. ### Solution:5 The answer is "yes", but maybe you could be more specific. The first example I could think of, using double recursion (that leads to an exponential complexity, not recommended): ``int fib(int a) { if (a < 2) { return 1 } else { return fib(a-1) + fib(a-2); } } `` Assuming a >= 0 (you could add a check for that). (Edit - used the wrong convention of F(0) undefined, F(1) = 1) ### Solution:6 After the initial `1 1`, it is in theory possible to generate one value from the previous one (until machine precision comes around to bite you) via: ``f = Math.round(f * PHI) `` where `PHI` is the constant defined in another comment: `static final double PHI = (1 + Math.sqrt(5))/2;` ### Solution:7 You can always do something like this: `` String theOneVar = "100;0;1"; while (true) { if (theOneVar.split(";")[0].equals("0")) break; System.out.println(theOneVar.split(";")[1]); theOneVar = String.format("%s;%s;%s", Integer.parseInt(theOneVar.split(";")[0]) - 1, theOneVar.split(";")[2], new BigInteger(theOneVar.split(";")[1]).add( new BigInteger(theOneVar.split(";")[2]) ) ); } `` This prints (as seen on ideone.com): ``0 1 1 2 3 5 8 13 : : 83621143489848422977 135301852344706746049 218922995834555169026 `` This uses only one explicit variable, and it's essentially a linear non-recursive algorithm. It needs to be said that this is an abuse of `String`, though. ### Solution:8 So this is evil, but: ``static String fibRecord = "x;"; static int nextFib() { try { return fibRecord.indexOf(';'); } finally { fibRecord = fibRecord.replaceAll("(x);(x)", "\$1\$2;\$1"); } } public static void main(String[] ignored) { for (int i=0; i < 30; i++) { System.out.println(nextFib()); } } `` My machine here starts to fall over around the 38th Fibonacci number. ### Solution:9 Here's an example in C#. Shows the first 100 terms. The ratio between terms in the Fibonacci approaches the golden ratio (1.618033...), so a single variable approach simply requires a multiplication by a constant for each term. Yay math! ``double fib = 1; for (int i = 0; i < 100; i++) { Console.WriteLine("" + fib); fib = Math.Round(fib = 1.6180339887d); } `` ### Solution:10 ``class fibo{ public static void main (String args[]) { long i = 1; while (((i & 0xffff0000) >> 16) + (i & 0xffff) <= 0xffff) { System.out.println(i & 0xffff); i = ((i & 0xffff) << 16) \| ((i >> 16) + (i & 0xffff)); } } } `` Here is the java code of Fibonacci series using one variable. ### Solution:11 THE PROGRAM IS FOR PRINTING UP TO 10 NUMBER BUT YOU CAN CHANGE IT. ``import java. i o.; class q { public static void main()throws IO Exception { int n=0; for(int i=1; i<=10 ; i++) { System.out.print(n +" "); n=(int)Math.round(n*1.618) } } } 1.618 = PHI `` the program has some mistakes in import and in main statement but the body is full correct ### Solution:12 ``public class test { public static void main(String[] args) { int arr[]=new int[13]; arr[0]=0; arr[1]=1; for(int i=2;i<=12;i++){ arr[i]=arr[i-1]+arr[i-2]; } for(int i=0;i<=arr.length-1;i++){ System.out.print(arr[i]+" "); } } } `` Note:If u also have question or solution just comment us below or mail us on toontricks1994@gmail.com Previous Next Post »	1,655	5,588	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2020-40	longest	en	0.838789	# Tutorial :Fibonacci using 1 variable. ### Question:. I was asked the following question in an interview:. Is there any way in which Fibonacci series can be generated using only 1 variable ?. I didn't know what to answer. What should I have said?. ### Solution:1. Yes, you can used the closed-form expression:. where. You can calculate the expression using a `double` and round the result to the nearest integer. Because of the finite precision of floating point arithmetic this formula will give a wrong answer for large enough n, but I think it will work in the case when the result fits into a Java 32-bit integer.. ### Solution:2. Up to a point, yes (though in C, you could convert it to Java - it would look much uglier).. ``#include <stdio.h> #include <stdlib.h> int main (void) { unsigned long i = 1; printf ("0\n"); while (((i & 0xffff0000) >> 16) + (i & 0xffff) <= 0xffff) { printf ("%d\n", i & 0xffff); i = ((i & 0xffff) << 16) \| ((i >> 16) + (i & 0xffff)); } return 0; } ``. which produces:. ``0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 ``. :-). The real question, of course, is: Why would you want to?. If you're curious as to how it works, it's really quite simple. The one variable is actually divided into two parts and those two parts maintain the individual values for the Fibonacci sequence. It's still technically one variable, we've just imposed some extra structure on top of it to achieve our ends.. ### Solution:3. Sure, using recursion:. ``public class Test { public static int fib(int n) { return n < 2 ? n : fib(n-1) + fib(n-2); } public static void main(String[] args) { for(int i = 0; i <= 10; i++) { System.out.print(fib(i)+", "); } System.out.println("..."); } } // 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... ``. ### Solution:4. Yes, but you still need to remember 2 values. You could take a 64-bit variable and use it as 2 32-bit vars.. ### Solution:5. The answer is "yes", but maybe you could be more specific.. The first example I could think of, using double recursion (that leads to an exponential complexity, not recommended):. ``int fib(int a) { if (a < 2) { return 1 } else { return fib(a-1) + fib(a-2); } } ``. Assuming a >= 0 (you could add a check for that).. (Edit - used the wrong convention of F(0) undefined, F(1) = 1). ### Solution:6.	After the initial `1 1`, it is in theory possible to generate one value from the previous one (until machine precision comes around to bite you) via:. ``f = Math.round(f * PHI) ``. where `PHI` is the constant defined in another comment:. `static final double PHI = (1 + Math.sqrt(5))/2;`. ### Solution:7. You can always do something like this:. `` String theOneVar = "100;0;1"; while (true) { if (theOneVar.split(";")[0].equals("0")) break; System.out.println(theOneVar.split(";")[1]); theOneVar = String.format("%s;%s;%s", Integer.parseInt(theOneVar.split(";")[0]) - 1, theOneVar.split(";")[2], new BigInteger(theOneVar.split(";")[1]).add( new BigInteger(theOneVar.split(";")[2]) ) ); } ``. This prints (as seen on ideone.com):. ``0 1 1 2 3 5 8 13 : : 83621143489848422977 135301852344706746049 218922995834555169026 ``. This uses only one explicit variable, and it's essentially a linear non-recursive algorithm. It needs to be said that this is an abuse of `String`, though.. ### Solution:8. So this is evil, but:. ``static String fibRecord = "x;"; static int nextFib() { try { return fibRecord.indexOf(';'); } finally { fibRecord = fibRecord.replaceAll("(x);(x)", "\$1\$2;\$1"); } } public static void main(String[] ignored) { for (int i=0; i < 30; i++) { System.out.println(nextFib()); } } ``. My machine here starts to fall over around the 38th Fibonacci number.. ### Solution:9. Here's an example in C#. Shows the first 100 terms. The ratio between terms in the Fibonacci approaches the golden ratio (1.618033...), so a single variable approach simply requires a multiplication by a constant for each term.. Yay math!. ``double fib = 1; for (int i = 0; i < 100; i++) { Console.WriteLine("" + fib); fib = Math.Round(fib = 1.6180339887d); } ``. ### Solution:10. ``class fibo{ public static void main (String args[]) { long i = 1; while (((i & 0xffff0000) >> 16) + (i & 0xffff) <= 0xffff) { System.out.println(i & 0xffff); i = ((i & 0xffff) << 16) \| ((i >> 16) + (i & 0xffff)); } } } ``. Here is the java code of Fibonacci series using one variable.. ### Solution:11. THE PROGRAM IS FOR PRINTING UP TO 10 NUMBER BUT YOU CAN CHANGE IT.. ``import java. i o.; class q { public static void main()throws IO Exception { int n=0; for(int i=1; i<=10 ; i++) { System.out.print(n +" "); n=(int)Math.round(n*1.618) } } } 1.618 = PHI ``. the program has some mistakes in import and in main statement but the body is full correct. ### Solution:12. ``public class test { public static void main(String[] args) { int arr[]=new int[13]; arr[0]=0; arr[1]=1; for(int i=2;i<=12;i++){ arr[i]=arr[i-1]+arr[i-2]; } for(int i=0;i<=arr.length-1;i++){ System.out.print(arr[i]+" "); } } } ``. Note:If u also have question or solution just comment us below or mail us on toontricks1994@gmail.com. Previous. Next Post ».
https://answers.yahoo.com/question/index?qid=20140905000051KK00047	1,582,119,287,000,000,000	text/html	crawl-data/CC-MAIN-2020-10/segments/1581875144150.61/warc/CC-MAIN-20200219122958-20200219152958-00230.warc.gz	280,022,701	26,008	THYE asked in 科學及數學其他 - 科學 · 6 years ago # Physics Mechanics 1. A wedge of mass M lies on a smooth horizontal plane while a block of mass m=M/2 is in contact with the smooth inclined face of the wedge . The inclination of this face to the horizontal is 30°. The system is released from rest with the block at a vertical height H above the horizontal plane. When the block reaches the horizontal plane, find (i) the speed of the wedge, and (ii) the distance the wedge has travelled. Update: My approach is like this: Normal reaction acting on the wedge by block = Force acting on the block = mg cos30 By basic geometry, horizontal component of the force= sin30 mg cos30 Acceleration= 1/2(mgcos30)/(M)=1/2(mgcos30)/(2m) Update 2: Considering the motion of the block: s=0.5 at^2+ut H/sin30=1/2(0.5g)(t^2) H=1/8(gt^2) t= 2sqrt(2H/g) By v=u+at, v=at=1/2(gcos30)(2sqrt(2H/g) =sqrt(6gH)/4 =1.918sqrtH m/s Update 3: By s=0.5at^2+ut distance= 0.5(1/4)gcos30 (8H/g) = gcos30H/g =cos30H =0.866H m Anyone can tell me whether I am correct or not? Thanks a lot! Rating • 天同 Lv 7 6 years ago Consider the block of mass m, acceleration of block down the wedge = g.sin(30) Use equation of motion: s = ut + (1/2)at^2 with s = H/sin(30), u = 0 m/s, a = g.sin(30) m/s^2, t=? (where t is the time the block reaches the horizontal plane) hence, H/sin(30) = (1/2).(g.sin(30))t^2 (take g = 10 m/s^2) t = 0.8944[sqrt(H)] where "sqrt" stands for "square-root" Consider the wedge, force acting onto the inclined face = mg.cos(30) Hence, horizontal force pushing the wedge to move = [mg.cos(30)].sin(30) Horizontal acceleration of wedge = mg.cos(30).sin(30)/M = g.cos(30).sin(30)/2 = 0.2165 m/s^2 (take g = 10 m/s^2) (i) Speed of wedge when the block reaches the plane = 0.2165 x 0.8944(sqrt(H)) = 0.1936.sqrt(H) (ii) Distance travelled by the wedge = (1/2).(0.2165).[0.8944sqrt(H)]^2 (use s = ut + (1/2)at^2 ) = 0.0866H 2014-09-05 19:49:19 補充： sorry, I made a wrong numerical calculation in the wedge acceleration. It should be, Horizontal acceleration of wedge = mg.cos(30).sin(30)/M = g.cos(30).sin(30)/2 = 2.165 m/s^2 (take g = 10 m/s^2) 2014-09-05 19:49:43 補充： (cont'd)... Hence, the speed in (i) and distance in (ii) should be 10 times the given figures, i.e. 1.936sqrt(H) and 0.866H respectively.	820	2,324	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.9375	4	CC-MAIN-2020-10	latest	en	0.754359	THYE asked in 科學及數學其他 - 科學 · 6 years ago. # Physics Mechanics. 1. A wedge of mass M lies on a smooth horizontal plane while a block of mass m=M/2 is in contact with the smooth inclined face of the wedge . The inclination of this face to the horizontal is 30°. The system is released from rest with the block at a vertical height H above the horizontal plane. When the block reaches the horizontal plane, find (i) the speed of the wedge, and (ii) the distance the wedge has travelled.. Update:. My approach is like this:. Normal reaction acting on the wedge by block. = Force acting on the block. = mg cos30. By basic geometry, horizontal component of the force= sin30 mg cos30. Acceleration= 1/2(mgcos30)/(M)=1/2(mgcos30)/(2m). Update 2:. Considering the motion of the block:. s=0.5 at^2+ut. H/sin30=1/2(0.5g)(t^2). H=1/8(gt^2). t= 2sqrt(2H/g). By v=u+at,. v=at=1/2(gcos30)(2sqrt(2H/g). =sqrt(6gH)/4. =1.918sqrtH m/s. Update 3:. By s=0.5at^2+ut. distance= 0.5(1/4)gcos30 (8H/g). = gcos30H/g. =cos30H. =0.866H m. Anyone can tell me whether I am correct or not?.	Thanks a lot!. Rating. • 天同. Lv 7. 6 years ago. Consider the block of mass m, acceleration of block down the wedge = g.sin(30). Use equation of motion: s = ut + (1/2)at^2. with s = H/sin(30), u = 0 m/s, a = g.sin(30) m/s^2, t=? (where t is the time the block reaches the horizontal plane). hence, H/sin(30) = (1/2).(g.sin(30))t^2 (take g = 10 m/s^2). t = 0.8944[sqrt(H)] where "sqrt" stands for "square-root". Consider the wedge, force acting onto the inclined face = mg.cos(30). Hence, horizontal force pushing the wedge to move. = [mg.cos(30)].sin(30). Horizontal acceleration of wedge = mg.cos(30).sin(30)/M = g.cos(30).sin(30)/2. = 0.2165 m/s^2 (take g = 10 m/s^2). (i) Speed of wedge when the block reaches the plane. = 0.2165 x 0.8944(sqrt(H)) = 0.1936.sqrt(H). (ii) Distance travelled by the wedge. = (1/2).(0.2165).[0.8944sqrt(H)]^2 (use s = ut + (1/2)at^2 ). = 0.0866H. 2014-09-05 19:49:19 補充：. sorry, I made a wrong numerical calculation in the wedge acceleration. It should be,. Horizontal acceleration of wedge = mg.cos(30).sin(30)/M = g.cos(30).sin(30)/2. = 2.165 m/s^2 (take g = 10 m/s^2). 2014-09-05 19:49:43 補充：. (cont'd)... Hence, the speed in (i) and distance in (ii) should be 10 times the given figures, i.e. 1.936sqrt(H) and 0.866H respectively.
https://www.helpingwithmath.com/resources/games/drag_add_to10/AddingToTen.html	1,544,926,683,000,000,000	text/html	crawl-data/CC-MAIN-2018-51/segments/1544376827175.38/warc/CC-MAIN-20181216003916-20181216025916-00336.warc.gz	912,491,906	10,325	# Number Bonds To 10 - Drag and Drop Drag and drop the number from the right to the number on the left so they both add up to 10. In other words: Click and hold the button down while moving the number. Release the button when over the other number. Number Bonds To 10 \| Number Bonds To 20 \| Number Bonds To 30 \| Number Bonds To 40 \| Number Bonds To 50 \| Number Bonds To 100 An Addition Game - By HelpingWithMath.com ### Related Resources The resources listed below are aligned to the same standard, (1OA06) re: Common Core Standards For Mathematics as the Addition and subtraction game shown above. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).	320	1,111	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2018-51	longest	en	0.821656	# Number Bonds To 10 - Drag and Drop. Drag and drop the number from the right to the number on the left so they both add up to 10.. In other words: Click and hold the button down while moving the number. Release the button when over the other number.. Number Bonds To 10 \| Number Bonds To 20 \| Number Bonds To 30 \|. Number Bonds To 40 \| Number Bonds To 50 \| Number Bonds To 100.	An Addition Game - By HelpingWithMath.com. ### Related Resources. The resources listed below are aligned to the same standard, (1OA06) re: Common Core Standards For Mathematics as the Addition and subtraction game shown above.. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).
https://tianrunhe.wordpress.com/2012/03/25/nth-fibonacci-number/	1,529,347,874,000,000,000	text/html	crawl-data/CC-MAIN-2018-26/segments/1529267860776.63/warc/CC-MAIN-20180618183714-20180618203714-00511.warc.gz	723,412,048	16,586	## Nth Fibonacci number Write a method to generate the $n$th Fibonacci number. My initial thoughts: • Base case: F(0) = 0, F(1) = 1. • Recursion: F(n) = F(n-1) + F(n-2) My initial codes: ``` public static int Fibonacci(int n) { if (n < 0) return -1; if (n == 0) return 0; if (n == 1) return 1; return Fibonacci(n - 1) + Fibonacci(n - 2); } Solution: Recursive solution: public static int fibo(int n) { if (n == 0) { return 0; // f(0) = 0 } else if (n == 1) { return 1; // f(1) = 1 } else if (n > 1) { return fibo(n - 1) + fibo(n - 2); // f(n) = f(n—1) + f(n-2) } else { return -1; // Error condition } } Handles more error conditions than mine. Time complexity: $O(2^{n})$; Space complexity $O(n)$ considering the function call stack. Iterative Solution: public static int IterativeFibo(int n) { if (n < 0) return -1; // Error condition. if (n == 0) return 0; int a = 1, b = 1; for (int i = 3; i <= n; i++) { int c = a + b; a = b; b = c; } return b; } Time complexity: $O(n)$; Space complexity $O(1)$. __ATA.cmd.push(function() { __ATA.initVideoSlot('atatags-370373-5b27ff224ea1b', { sectionId: '370373', }); }); __ATA.cmd.push(function() { __ATA.initSlot('atatags-26942-5b27ff224ea49', { collapseEmpty: 'before', sectionId: '26942', width: 300, height: 250 }); }); __ATA.cmd.push(function() { __ATA.initSlot('atatags-114160-5b27ff224ea4b', { collapseEmpty: 'before', sectionId: '114160', width: 300, height: 250 }); });	516	1,435	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2018-26	latest	en	0.232456	## Nth Fibonacci number. Write a method to generate the $n$th Fibonacci number.. My initial thoughts:. • Base case: F(0) = 0, F(1) = 1.. • Recursion: F(n) = F(n-1) + F(n-2). My initial codes:. ``` public static int Fibonacci(int n) {. if (n < 0). return -1;. if (n == 0). return 0;. if (n == 1). return 1;. return Fibonacci(n - 1) + Fibonacci(n - 2);. }. Solution:. Recursive solution:. public static int fibo(int n) {. if (n == 0) {. return 0; // f(0) = 0. } else if (n == 1) {. return 1; // f(1) = 1. } else if (n > 1) {. return fibo(n - 1) + fibo(n - 2); // f(n) = f(n—1) + f(n-2). } else {. return -1; // Error condition. }. }. Handles more error conditions than mine.. Time complexity: $O(2^{n})$; Space complexity $O(n)$ considering the function call stack.. Iterative Solution:. public static int IterativeFibo(int n) {. if (n < 0). return -1; // Error condition.	if (n == 0). return 0;. int a = 1, b = 1;. for (int i = 3; i <= n; i++) {. int c = a + b;. a = b;. b = c;. }. return b;. }. Time complexity: $O(n)$; Space complexity $O(1)$.. __ATA.cmd.push(function() {. __ATA.initVideoSlot('atatags-370373-5b27ff224ea1b', {. sectionId: '370373',. });. });. __ATA.cmd.push(function() {. __ATA.initSlot('atatags-26942-5b27ff224ea49', {. collapseEmpty: 'before',. sectionId: '26942',. width: 300,. height: 250. });. });. __ATA.cmd.push(function() {. __ATA.initSlot('atatags-114160-5b27ff224ea4b', {. collapseEmpty: 'before',. sectionId: '114160',. width: 300,. height: 250. });. });.
https://mcqslearn.com/electronics/advance-electromagnetic-theory/quiz/quiz-questions-and-answers.php?page=28	1,709,576,592,000,000,000	text/html	crawl-data/CC-MAIN-2024-10/segments/1707947476464.74/warc/CC-MAIN-20240304165127-20240304195127-00602.warc.gz	384,627,661	19,978	Engineering Courses Electromagnetic Theory Certification Exam Tests Electromagnetic Theory Practice Test 28 # Gauss's Law MCQ Questions and Answers PDF - 28 Books: Apps: The e-Book Gauss's Law MCQ Questions, gauss's law quiz answers PDF download chapter 4-28 to learn online electromagnetic theory degree programs. Solve Time Varying and Harmonic Electromagnetic Fields Test PDF, Gauss's Law Multiple Choice Questions (MCQ Quiz) for online college degrees. The Gauss's Law MCQ Quiz App Download: Free certification app for dielectric constitutive relationship, metamaterials: electric and magnetic responses, maxwell's equations, snell's law, gauss's law test prep for online engineering associate's degree programs. The MCQ Quiz ∇·B=ρmv, is termed as: ampere's law, faraday's law of induction, gauss's law (electrical) and gauss's law for magnetism with "Gauss's Law" App APK Download (Free) to study online educational courses. Study time varying and harmonic electromagnetic fields questions and answers, Apple Book to download free sample for college entrance test. ## Gauss's Law MCQ with Answers : Quiz 28 MCQ 136: ∇·B=ρmv, is termed as B) Ampere's Law C) Gauss's Law (electrical) D) Gauss's Law for magnetism MCQ 137: It is Pendry et al., who first proposed to use A) insulators B) natural materials C) artificial materials D) some ferrites MCQ 138: Maxwell's equations can be written in A) integral form B) differential form C) logical form D) either in integral or differential form MCQ 139: Under the free electron approximation, resonant frequency is always taken as A) zero B) 1 C) 10 D) infinite MCQ 140: In D=∊E, 'E' is A) electric flux density B) electric field intensity C) magnetic field intensity D) magnetic dipole moment ### Gauss's Law Learning App & Free Study Apps Download Advance Electromagnetic Theory MCQs App to learn Gauss's Law MCQ, Electronic Devices Quiz App, and Digital Electronics MCQ App (Android & iOS). The free "Gauss's Law MCQs" App includes complete analytics of history with interactive assessments. Download Play Store & App Store learning Apps & enjoy 100% functionality with subscriptions! ALL-in-ONE Learning App (Android & iOS) Advance Electromagnetic Theory App (Android & iOS) Electronic Devices App (Android & iOS) Digital Electronics App (Android & iOS)	567	2,328	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.546875	4	CC-MAIN-2024-10	latest	en	0.795938	Engineering Courses. Electromagnetic Theory Certification Exam Tests. Electromagnetic Theory Practice Test 28. # Gauss's Law MCQ Questions and Answers PDF - 28. Books:. Apps:. The e-Book Gauss's Law MCQ Questions, gauss's law quiz answers PDF download chapter 4-28 to learn online electromagnetic theory degree programs. Solve Time Varying and Harmonic Electromagnetic Fields Test PDF, Gauss's Law Multiple Choice Questions (MCQ Quiz) for online college degrees. The Gauss's Law MCQ Quiz App Download: Free certification app for dielectric constitutive relationship, metamaterials: electric and magnetic responses, maxwell's equations, snell's law, gauss's law test prep for online engineering associate's degree programs.. The MCQ Quiz ∇·B=ρmv, is termed as: ampere's law, faraday's law of induction, gauss's law (electrical) and gauss's law for magnetism with "Gauss's Law" App APK Download (Free) to study online educational courses. Study time varying and harmonic electromagnetic fields questions and answers, Apple Book to download free sample for college entrance test.. ## Gauss's Law MCQ with Answers : Quiz 28. MCQ 136: ∇·B=ρmv, is termed as. B) Ampere's Law. C) Gauss's Law (electrical). D) Gauss's Law for magnetism. MCQ 137: It is Pendry et al., who first proposed to use. A) insulators. B) natural materials. C) artificial materials. D) some ferrites. MCQ 138: Maxwell's equations can be written in.	A) integral form. B) differential form. C) logical form. D) either in integral or differential form. MCQ 139: Under the free electron approximation, resonant frequency is always taken as. A) zero. B) 1. C) 10. D) infinite. MCQ 140: In D=∊E, 'E' is. A) electric flux density. B) electric field intensity. C) magnetic field intensity. D) magnetic dipole moment. ### Gauss's Law Learning App & Free Study Apps. Download Advance Electromagnetic Theory MCQs App to learn Gauss's Law MCQ, Electronic Devices Quiz App, and Digital Electronics MCQ App (Android & iOS). The free "Gauss's Law MCQs" App includes complete analytics of history with interactive assessments. Download Play Store & App Store learning Apps & enjoy 100% functionality with subscriptions!. ALL-in-ONE Learning App (Android & iOS). Advance Electromagnetic Theory App (Android & iOS). Electronic Devices App (Android & iOS). Digital Electronics App (Android & iOS).
https://interviewmania.com/discussion/207-aptitude-profit-and-loss	1,545,199,561,000,000,000	text/html	crawl-data/CC-MAIN-2018-51/segments/1544376831334.97/warc/CC-MAIN-20181219045716-20181219071716-00155.warc.gz	644,633,967	6,167	## Discussion 1. A retailer increase the selling price by 25% due to which his profit percentage increases from 20% to 25%. What is the percentage increase in cost price ? 1. 20% 2. 30% 3. 25% 4. 50% 1. ##### Correct Option: A In beginning Cost Price (CP) = Rs. 100 Profit % = 20 Selling Price (SP) = 120 When profit increases from 20% to 25% Cost Price (CP) = y Profit % = 25 Selling Price (SP) after 25% increase = 120 + 25% = Rs. 150 From question y + 25 % profit = 150 1.25 y = 150 so y = 120 % change in cost price = [ (120 - 100) / 100 ] x 100 = 20%	198	561	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.734375	4	CC-MAIN-2018-51	longest	en	0.849084	## Discussion. 1. A retailer increase the selling price by 25% due to which his profit percentage increases from 20% to 25%. What is the percentage increase in cost price ?. 1. 20%. 2. 30%. 3. 25%. 4. 50%. 1. ##### Correct Option: A. In beginning. Cost Price (CP) = Rs.	100. Profit % = 20. Selling Price (SP) = 120. When profit increases from 20% to 25%. Cost Price (CP) = y. Profit % = 25. Selling Price (SP) after 25% increase = 120 + 25% = Rs. 150. From question. y + 25 % profit = 150. 1.25 y = 150. so y = 120. % change in cost price = [ (120 - 100) / 100 ] x 100 = 20%.
http://slideplayer.com/slide/9289385/	1,708,872,438,000,000,000	text/html	crawl-data/CC-MAIN-2024-10/segments/1707947474617.27/warc/CC-MAIN-20240225135334-20240225165334-00806.warc.gz	37,199,879	19,940	# Solving Rational Equations ## Presentation on theme: "Solving Rational Equations"— Presentation transcript: Solving Rational Equations Section 6.6 Solving Rational Equations Definition A rational equation is an equation that contains one or more rational expressions. Examples: Goal: Clear Fractions Solving Rational Equations 1) Factor each denominator completely to find LCD 2) Multiply both sides by the LCD to clear the fractions 3) Solve the resulting equation 4) Check all possible solutions in the original equation Example Solve the equation 1 Multiply both sides by the LCD 7 3 1 LCD is 21x 21x 21x 21x 1 1 1 Solve the resulting equation Example Solve 1 Multiply both sides by the LCD 1 1 1 LCD is 5b 5b 5b Solve the resulting equation Check the solution in the original equation Example Solve the equation Multiply both sides by the LCD 1 1 (q+5) LCD is (q+5) (q+5) (q+5) 1 1 Solve the resulting equation No Solution More Examples Solve the following equations Solving for a specified variable Isolate the specified variable in a formula The answer is another formula (not a numerical value) Example Solve for R Multiply both sides by the LCD 1 LCD is (R+r) NO! Subtracting rI on both sides Dividing both sides by I Example Solve for b 1 Multiply both sides by the LCD 1 1 1 abf LCD is Solve the resulting equation Move the term containing b to the left Factor out common factor b Divide both sides by f – a More Examples Solve the following equations	365	1,475	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.90625	4	CC-MAIN-2024-10	latest	en	0.87223	# Solving Rational Equations. ## Presentation on theme: "Solving Rational Equations"— Presentation transcript:. Solving Rational Equations. Section 6.6 Solving Rational Equations. Definition A rational equation is an equation that contains. one or more rational expressions. Examples: Goal: Clear Fractions. Solving Rational Equations. 1) Factor each denominator completely to find LCD 2) Multiply both sides by the LCD to clear the fractions 3) Solve the resulting equation 4) Check all possible solutions in the original equation. Example Solve the equation 1 Multiply both sides by the LCD 7 3 1. LCD is 21x 21x 21x 21x 1 1 1 Solve the resulting equation. Example Solve 1 Multiply both sides by the LCD 1 1 1 LCD is 5b 5b 5b.	Solve the resulting equation Check the solution in the original equation. Example Solve the equation Multiply both sides by the LCD 1 1 (q+5). LCD is (q+5) (q+5) (q+5) 1 1 Solve the resulting equation No Solution. More Examples Solve the following equations. Solving for a specified variable. Isolate the specified variable in a formula The answer is another formula (not a numerical value). Example Solve for R Multiply both sides by the LCD 1 LCD is (R+r). NO! Subtracting rI on both sides Dividing both sides by I. Example Solve for b 1 Multiply both sides by the LCD 1 1 1 abf LCD is. Solve the resulting equation Move the term containing b to the left Factor out common factor b Divide both sides by f – a. More Examples Solve the following equations.
http://clay6.com/qa/38539/find-the-general-solution-for-each-of-the-following-equation-sin-x-sin-3x-s	1,544,543,000,000,000,000	text/html	crawl-data/CC-MAIN-2018-51/segments/1544376823657.20/warc/CC-MAIN-20181211151237-20181211172737-00582.warc.gz	59,589,184	8,387	Comment Share Q) # Find the general solution for each of the following equation $\sin x+\sin 3x+\sin 5x=0$ $\begin{array}{1 1}(A)\;n\pi\pm \large\frac{\pi}{6}&(B)\;n\pi\pm \large\frac{\pi}{3}\\(C)\;2n\pi\pm \large\frac{\pi}{2}&(D)\;n\pi\pm \large\frac{\pi}{4}\end{array}$ • $\sin A+\sin B=2\sin \large\frac{A+B}{2}$$\cos \large\frac{A-B}{2} \sin x+\sin 3x+\sin 5x=0 (\sin 5x+\sin x)+\sin 3x=0 2\sin \large\frac{5x+x}{2}$$\cos \large\frac{5x-x}{2}$$+\sin 3x=0 2\sin 3x\cos 2x+\sin 3x=0 \Rightarrow \sin 3x(2\cos 2x+1)=0 If \sin 3x=0,3x=n\pi or x=\large\frac{n\pi}{3} If 2\cos 2x+1=0,\cos 2x=-\large\frac{1}{2}$$=\cos (\pi-\large\frac{\pi}{3})$ $\Rightarrow \cos \large\frac{2\pi}{3}$ $2x=2n\pi+\large\frac{2\pi}{3}$ or $x=n\pi\pm \large\frac{\pi}{3}$	379	752	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.09375	4	CC-MAIN-2018-51	latest	en	0.207051	Comment. Share. Q). # Find the general solution for each of the following equation $\sin x+\sin 3x+\sin 5x=0$. $\begin{array}{1 1}(A)\;n\pi\pm \large\frac{\pi}{6}&(B)\;n\pi\pm \large\frac{\pi}{3}\\(C)\;2n\pi\pm \large\frac{\pi}{2}&(D)\;n\pi\pm \large\frac{\pi}{4}\end{array}$.	• $\sin A+\sin B=2\sin \large\frac{A+B}{2}$$\cos \large\frac{A-B}{2} \sin x+\sin 3x+\sin 5x=0 (\sin 5x+\sin x)+\sin 3x=0 2\sin \large\frac{5x+x}{2}$$\cos \large\frac{5x-x}{2}$$+\sin 3x=0 2\sin 3x\cos 2x+\sin 3x=0 \Rightarrow \sin 3x(2\cos 2x+1)=0 If \sin 3x=0,3x=n\pi or x=\large\frac{n\pi}{3} If 2\cos 2x+1=0,\cos 2x=-\large\frac{1}{2}$$=\cos (\pi-\large\frac{\pi}{3})$. $\Rightarrow \cos \large\frac{2\pi}{3}$. $2x=2n\pi+\large\frac{2\pi}{3}$ or $x=n\pi\pm \large\frac{\pi}{3}$.
http://www.kidport.com/RefLib/Math/Equations/GuessingSymbol.htm	1,532,270,674,000,000,000	text/html	crawl-data/CC-MAIN-2018-30/segments/1531676593302.74/warc/CC-MAIN-20180722135607-20180722155607-00261.warc.gz	492,237,461	4,095	# Guessing a Symbol Sometimes you can easily find the value of a symbol through Guess and Check. In Guess and Check, you guess at what you believe is the answer, then plug it into the equation to see if you are correct. Custom Search ## Other Math Activities: Fun with Equations Arithmetic Let's do the following problem with standard arithematic. A puppy weighed 15 pounds when Jeff bought it. It gained 3 pounds. How much does the puppy weigh now? Let the new weight of the puppy be x. We determine the answer by adding 3 to 15, as in the following math sentence: 15 + 3 = x Solving the math equation we get: 15 + 3 = 18 Jeff's puppy now weighs 18 lbs. Algebra (guessing) Now let's try a word problem using Guess and Check. Jane's puppy gained 5 lbs, and now weighs 22 lbs. What did the puppy weigh when she bought it? Let the weight of the puppy be x, when he bought it. So, the math sentence is: x + 5 = 22 Let's guess the answer is 18. Substituting 18 for x makes the math sentence: 18(?) + 5 = 22, which is not correct since 18 + 5 = 23 So, let's guess 17. 17 + 5 = 22, which is true. So, x = 17 The puppy weighed 17 pounds when Jane bought the puppy. This was a simple problem, and clearly you could have solved it using subtraction (22 - 5 = 17). However, sometimes if you don't know what to do, Guess and Check is a potential way to find the answer.	368	1,382	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.53125	5	CC-MAIN-2018-30	latest	en	0.967658	# Guessing a Symbol. Sometimes you can easily find the value of a symbol through Guess and Check. In Guess and Check, you guess at what you believe is the answer, then plug it into the equation to see if you are correct.. Custom Search. ## Other Math Activities:. Fun with Equations. Arithmetic. Let's do the following problem with standard arithematic.. A puppy weighed 15 pounds when Jeff bought it. It gained 3 pounds. How much does the puppy weigh now? Let the new weight of the puppy be x.. We determine the answer by adding 3 to 15, as in the following math sentence:. 15 + 3 = x. Solving the math equation we get:. 15 + 3 = 18. Jeff's puppy now weighs 18 lbs.	Algebra (guessing). Now let's try a word problem using Guess and Check.. Jane's puppy gained 5 lbs, and now weighs 22 lbs. What did the puppy weigh when she bought it?. Let the weight of the puppy be x, when he bought it. So, the math sentence is:. x + 5 = 22. Let's guess the answer is 18. Substituting 18 for x makes the math sentence:. 18(?) + 5 = 22, which is not correct since 18 + 5 = 23. So, let's guess 17.. 17 + 5 = 22, which is true.. So, x = 17. The puppy weighed 17 pounds when Jane bought the puppy.. This was a simple problem, and clearly you could have solved it using subtraction (22 - 5 = 17). However, sometimes if you don't know what to do, Guess and Check is a potential way to find the answer.
https://www.gradesaver.com/textbooks/math/precalculus/precalculus-mathematics-for-calculus-7th-edition/chapter-1-section-1-5-equations-1-5-exercises-page-56/43	1,537,631,421,000,000,000	text/html	crawl-data/CC-MAIN-2018-39/segments/1537267158450.41/warc/CC-MAIN-20180922142831-20180922163231-00158.warc.gz	760,250,413	14,269	## Precalculus: Mathematics for Calculus, 7th Edition $t=\dfrac{-v_{0}\pm\sqrt{v^{2}_{0}+2gh}}{g}$ $h=\dfrac{1}{2}gt^{2}+v_{0}t$; for $t$ Take $h$ to the right side of the equation to set it equal to $0$: $\dfrac{1}{2}gt^{2}+v_{0}t-h=0$ Multiply the whole equation by $2$, to avoid working with fractions: $gt^{2}+2v_{0}t-2h=0$ This is a quadratic equation. Let's solve it using the quadratic formula, which is: $t=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ For this particular equation, $a=g$, $b=2v_{0}$ and $c=-2h$ Substitute: $t=\dfrac{-2v_{0}\pm\sqrt{(2v_{0})^{2}-4g(-2h)}}{2g}$ Simplify: $t=\dfrac{-2v_{0}\pm\sqrt{4v^{2}_{0}+8gh}}{2g}$ $t=\dfrac{-2v_{0}\pm\sqrt{4(v^{2}_{0}+2gh)}}{2g}$ $t=\dfrac{-2v_{0}\pm2\sqrt{v^{2}_{0}+2gh}}{2g}$ $t=\dfrac{-v_{0}\pm\sqrt{v^{2}_{0}+2gh}}{g}$	361	777	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.5	4	CC-MAIN-2018-39	longest	en	0.570177	## Precalculus: Mathematics for Calculus, 7th Edition. $t=\dfrac{-v_{0}\pm\sqrt{v^{2}_{0}+2gh}}{g}$. $h=\dfrac{1}{2}gt^{2}+v_{0}t$; for $t$ Take $h$ to the right side of the equation to set it equal to $0$: $\dfrac{1}{2}gt^{2}+v_{0}t-h=0$ Multiply the whole equation by $2$, to avoid working with fractions: $gt^{2}+2v_{0}t-2h=0$ This is a quadratic equation.	Let's solve it using the quadratic formula, which is: $t=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ For this particular equation, $a=g$, $b=2v_{0}$ and $c=-2h$ Substitute: $t=\dfrac{-2v_{0}\pm\sqrt{(2v_{0})^{2}-4g(-2h)}}{2g}$ Simplify: $t=\dfrac{-2v_{0}\pm\sqrt{4v^{2}_{0}+8gh}}{2g}$ $t=\dfrac{-2v_{0}\pm\sqrt{4(v^{2}_{0}+2gh)}}{2g}$ $t=\dfrac{-2v_{0}\pm2\sqrt{v^{2}_{0}+2gh}}{2g}$ $t=\dfrac{-v_{0}\pm\sqrt{v^{2}_{0}+2gh}}{g}$.
https://www.physicsforums.com/threads/question-about-a-spring.622360/	1,510,950,530,000,000,000	text/html	crawl-data/CC-MAIN-2017-47/segments/1510934803906.12/warc/CC-MAIN-20171117185611-20171117205611-00154.warc.gz	832,384,782	16,090	1. Jul 20, 2012 ### cragar 1. The problem statement, all variables and given/known data The left end of a spring is attached to a wall. When bob pulls on the right end with 200N force, he stretches the spring by 20cm. The same spring is then used for tug of war between bob and Carlos. Each pulls on the spring with a 200N force. How far does Carlos's end of the spring move? explain 3. The attempt at a solution If I think of this as the spring hanging in a gravitational field. I put a weight on it that was 200 N it will pull it down 20cm. but if my spring is hanging on a nail, that is also providing a force of 200N in the opposite direction. So i think that the spring will be stretched still by 20cm. Although I am not sure if it will be stretched by 10cm at Carloss's end or zero. 2. Jul 20, 2012 ### TSny Yes, the spring will still stretch a total of 20 cm. I think the two guys are assumed to move equal distances. 3. Jul 20, 2012 ### pgardn Maybe you want to do this in the horizontal to make it easier. (No gravity) Also you might want to ask yourself how hard the wall pulls on the spring before Carlos comes into play. Oops I see someone has already posted. I will now kindly drop out. 4. Jul 20, 2012 ### TSny Hi, pgardn. Don't drop out. The more the merrier! 5. Jul 20, 2012 ### cragar so then each person will move 10cm 6. Jul 20, 2012 ### PeterO Probably - but you can certainly be sure the ends of the spring will be 20cm further apart. The common, incorrect response to this question is that each move 20cm - as I said, and you seem to realise, that answer is incorrect. 7. Jul 20, 2012 ### TSny I second that. 8. Jul 20, 2012 ### pgardn I was just going to replace the wall with Carlos. And there you have it. All is in order. 9. Jul 20, 2012 ### azizlwl Using work-energy analysis could also be used.	517	1,852	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.625	4	CC-MAIN-2017-47	longest	en	0.945632	1. Jul 20, 2012. ### cragar. 1. The problem statement, all variables and given/known data. The left end of a spring is attached to a wall. When bob pulls on the right end with 200N force,. he stretches the spring by 20cm. The same spring is then used for tug of war between bob and Carlos. Each pulls on the spring with a 200N force. How far does Carlos's end of the spring move? explain. 3. The attempt at a solution. If I think of this as the spring hanging in a gravitational field. I put a weight on it that was 200 N it will pull it down 20cm. but if my spring is hanging on a nail, that is also providing a force of 200N in the opposite direction. So i think that the spring will be stretched still by 20cm. Although I am not sure if it will be stretched by 10cm at Carloss's end or zero.. 2. Jul 20, 2012. ### TSny. Yes, the spring will still stretch a total of 20 cm. I think the two guys are assumed to move equal distances.. 3. Jul 20, 2012. ### pgardn. Maybe you want to do this in the horizontal to make it easier. (No gravity). Also you might want to ask yourself how hard the wall pulls on the spring before Carlos comes into play.. Oops I see someone has already posted. I will now kindly drop out.	4. Jul 20, 2012. ### TSny. Hi, pgardn. Don't drop out. The more the merrier!. 5. Jul 20, 2012. ### cragar. so then each person will move 10cm. 6. Jul 20, 2012. ### PeterO. Probably - but you can certainly be sure the ends of the spring will be 20cm further apart.. The common, incorrect response to this question is that each move 20cm - as I said, and you seem to realise, that answer is incorrect.. 7. Jul 20, 2012. ### TSny. I second that.. 8. Jul 20, 2012. ### pgardn. I was just going to replace the wall with Carlos. And there you have it.. All is in order.. 9. Jul 20, 2012. ### azizlwl. Using work-energy analysis could also be used.
https://nrich.maths.org/public/leg.php?code=6&cl=2&cldcmpid=1171	1,526,989,174,000,000,000	text/html	crawl-data/CC-MAIN-2018-22/segments/1526794864725.4/warc/CC-MAIN-20180522112148-20180522132148-00318.warc.gz	595,151,272	9,293	# Search by Topic #### Resources tagged with Place value similar to The Clockmaker's Birthday Cake: Filter by: Content type: Stage: Challenge level: ### There are 72 results Broad Topics > Numbers and the Number System > Place value ### Napier's Bones ##### Stage: 2 Challenge Level: The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications? ### Oddly ##### Stage: 2 Challenge Level: Find the sum of all three-digit numbers each of whose digits is odd. ### Six Is the Sum ##### Stage: 2 Challenge Level: What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros? ### Being Resilient - Primary Number ##### Stage: 1 and 2 Challenge Level: Number problems at primary level that may require resilience. ### Being Resourceful - Primary Number ##### Stage: 1 and 2 Challenge Level: Number problems at primary level that require careful consideration. ### Being Collaborative - Primary Number ##### Stage: 1 and 2 Challenge Level: Number problems at primary level to work on with others. ### ABC ##### Stage: 2 Challenge Level: In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication? ### All the Digits ##### Stage: 2 Challenge Level: This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures? ### Spell by Numbers ##### Stage: 2 Challenge Level: Can you substitute numbers for the letters in these sums? ### Trebling ##### Stage: 2 Challenge Level: Can you replace the letters with numbers? Is there only one solution in each case? ### What Do You Need? ##### Stage: 2 Challenge Level: Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number? ### The Thousands Game ##### Stage: 2 Challenge Level: Each child in Class 3 took four numbers out of the bag. Who had made the highest even number? ### Multiply Multiples 2 ##### Stage: 2 Challenge Level: Can you work out some different ways to balance this equation? ### Diagonal Sums ##### Stage: 2 Challenge Level: In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice? ### Multiply Multiples 1 ##### Stage: 2 Challenge Level: Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it? ### Coded Hundred Square ##### Stage: 2 Challenge Level: This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up? ### Nice or Nasty for Two ##### Stage: 2 Challenge Level: Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent. ### Round the Dice Decimals 1 ##### Stage: 2 Challenge Level: Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number? ### Multiply Multiples 3 ##### Stage: 2 Challenge Level: Have a go at balancing this equation. Can you find different ways of doing it? ### Round the Dice Decimals 2 ##### Stage: 2 Challenge Level: What happens when you round these numbers to the nearest whole number? ### Round the Three Dice ##### Stage: 2 Challenge Level: What happens when you round these three-digit numbers to the nearest 100? ### Song Book ##### Stage: 2 Challenge Level: A school song book contains 700 songs. The numbers of the songs are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed? ### Which Is Quicker? ##### Stage: 2 Challenge Level: Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why? ### One Million to Seven ##### Stage: 2 Challenge Level: Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like? ### Number Detective ##### Stage: 2 Challenge Level: Follow the clues to find the mystery number. ##### Stage: 1 and 2 Challenge Level: Who said that adding couldn't be fun? ### Becky's Number Plumber ##### Stage: 2 Challenge Level: Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings? ### Three Times Seven ##### Stage: 3 Challenge Level: A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why? ### Subtraction Surprise ##### Stage: 2 and 3 Challenge Level: Try out some calculations. Are you surprised by the results? ### Football Sum ##### Stage: 3 Challenge Level: Find the values of the nine letters in the sum: FOOT + BALL = GAME ### Arrange the Digits ##### Stage: 3 Challenge Level: Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500? ### Digit Sum ##### Stage: 3 Challenge Level: What is the sum of all the digits in all the integers from one to one million? ### Repeaters ##### Stage: 3 Challenge Level: Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13. ### Alien Counting ##### Stage: 2 Challenge Level: Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7. ### Being Curious - Primary Number ##### Stage: 1 and 2 Challenge Level: Number problems for inquiring primary learners. ### Which Scripts? ##### Stage: 2 Challenge Level: There are six numbers written in five different scripts. Can you sort out which is which? ### Cayley ##### Stage: 3 Challenge Level: The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"? ##### Stage: 3 Challenge Level: Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true. ### Dicey Operations for Two ##### Stage: 2 Challenge Level: Dicey Operations for an adult and child. Can you get close to 1000 than your partner? ### Skeleton ##### Stage: 3 Challenge Level: Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum. ### Even Up ##### Stage: 3 Challenge Level: Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number? ### X Marks the Spot ##### Stage: 3 Challenge Level: When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" . ### Calculator Bingo ##### Stage: 2 Challenge Level: A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins. ### Tis Unique ##### Stage: 3 Challenge Level: This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility. ##### Stage: 2 and 3 Challenge Level: Watch our videos of multiplication methods that you may not have met before. Can you make sense of them? ### Reach 100 ##### Stage: 2 and 3 Challenge Level: Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100. ### Two and Two ##### Stage: 3 Challenge Level: How many solutions can you find to this sum? Each of the different letters stands for a different number. ### Eleven ##### Stage: 3 Challenge Level: Replace each letter with a digit to make this addition correct. ### Lesser Digits ##### Stage: 3 Challenge Level: How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9? ### Six Times Five ##### Stage: 3 Challenge Level: How many six digit numbers are there which DO NOT contain a 5?	1,994	8,543	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.0625	4	CC-MAIN-2018-22	latest	en	0.828509	# Search by Topic. #### Resources tagged with Place value similar to The Clockmaker's Birthday Cake:. Filter by: Content type:. Stage:. Challenge level:. ### There are 72 results. Broad Topics > Numbers and the Number System > Place value. ### Napier's Bones. ##### Stage: 2 Challenge Level:. The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?. ### Oddly. ##### Stage: 2 Challenge Level:. Find the sum of all three-digit numbers each of whose digits is odd.. ### Six Is the Sum. ##### Stage: 2 Challenge Level:. What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?. ### Being Resilient - Primary Number. ##### Stage: 1 and 2 Challenge Level:. Number problems at primary level that may require resilience.. ### Being Resourceful - Primary Number. ##### Stage: 1 and 2 Challenge Level:. Number problems at primary level that require careful consideration.. ### Being Collaborative - Primary Number. ##### Stage: 1 and 2 Challenge Level:. Number problems at primary level to work on with others.. ### ABC. ##### Stage: 2 Challenge Level:. In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?. ### All the Digits. ##### Stage: 2 Challenge Level:. This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?. ### Spell by Numbers. ##### Stage: 2 Challenge Level:. Can you substitute numbers for the letters in these sums?. ### Trebling. ##### Stage: 2 Challenge Level:. Can you replace the letters with numbers? Is there only one solution in each case?. ### What Do You Need?. ##### Stage: 2 Challenge Level:. Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?. ### The Thousands Game. ##### Stage: 2 Challenge Level:. Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?. ### Multiply Multiples 2. ##### Stage: 2 Challenge Level:. Can you work out some different ways to balance this equation?. ### Diagonal Sums. ##### Stage: 2 Challenge Level:. In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?. ### Multiply Multiples 1. ##### Stage: 2 Challenge Level:. Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?. ### Coded Hundred Square. ##### Stage: 2 Challenge Level:. This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?. ### Nice or Nasty for Two. ##### Stage: 2 Challenge Level:. Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.. ### Round the Dice Decimals 1. ##### Stage: 2 Challenge Level:. Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?. ### Multiply Multiples 3. ##### Stage: 2 Challenge Level:. Have a go at balancing this equation. Can you find different ways of doing it?. ### Round the Dice Decimals 2. ##### Stage: 2 Challenge Level:. What happens when you round these numbers to the nearest whole number?. ### Round the Three Dice. ##### Stage: 2 Challenge Level:. What happens when you round these three-digit numbers to the nearest 100?. ### Song Book. ##### Stage: 2 Challenge Level:. A school song book contains 700 songs. The numbers of the songs are displayed by combining special small single-digit boards. What is the minimum number of small boards that is needed?. ### Which Is Quicker?. ##### Stage: 2 Challenge Level:. Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?. ### One Million to Seven. ##### Stage: 2 Challenge Level:. Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?.	### Number Detective. ##### Stage: 2 Challenge Level:. Follow the clues to find the mystery number.. ##### Stage: 1 and 2 Challenge Level:. Who said that adding couldn't be fun?. ### Becky's Number Plumber. ##### Stage: 2 Challenge Level:. Becky created a number plumber which multiplies by 5 and subtracts 4. What do you notice about the numbers that it produces? Can you explain your findings?. ### Three Times Seven. ##### Stage: 3 Challenge Level:. A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?. ### Subtraction Surprise. ##### Stage: 2 and 3 Challenge Level:. Try out some calculations. Are you surprised by the results?. ### Football Sum. ##### Stage: 3 Challenge Level:. Find the values of the nine letters in the sum: FOOT + BALL = GAME. ### Arrange the Digits. ##### Stage: 3 Challenge Level:. Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?. ### Digit Sum. ##### Stage: 3 Challenge Level:. What is the sum of all the digits in all the integers from one to one million?. ### Repeaters. ##### Stage: 3 Challenge Level:. Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.. ### Alien Counting. ##### Stage: 2 Challenge Level:. Investigate the different ways these aliens count in this challenge. You could start by thinking about how each of them would write our number 7.. ### Being Curious - Primary Number. ##### Stage: 1 and 2 Challenge Level:. Number problems for inquiring primary learners.. ### Which Scripts?. ##### Stage: 2 Challenge Level:. There are six numbers written in five different scripts. Can you sort out which is which?. ### Cayley. ##### Stage: 3 Challenge Level:. The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?. ##### Stage: 3 Challenge Level:. Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.. ### Dicey Operations for Two. ##### Stage: 2 Challenge Level:. Dicey Operations for an adult and child. Can you get close to 1000 than your partner?. ### Skeleton. ##### Stage: 3 Challenge Level:. Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.. ### Even Up. ##### Stage: 3 Challenge Level:. Consider all of the five digit numbers which we can form using only the digits 2, 4, 6 and 8. If these numbers are arranged in ascending order, what is the 512th number?. ### X Marks the Spot. ##### Stage: 3 Challenge Level:. When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .. ### Calculator Bingo. ##### Stage: 2 Challenge Level:. A game to be played against the computer, or in groups. Pick a 7-digit number. A random digit is generated. What must you subract to remove the digit from your number? the first to zero wins.. ### Tis Unique. ##### Stage: 3 Challenge Level:. This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.. ##### Stage: 2 and 3 Challenge Level:. Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?. ### Reach 100. ##### Stage: 2 and 3 Challenge Level:. Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.. ### Two and Two. ##### Stage: 3 Challenge Level:. How many solutions can you find to this sum? Each of the different letters stands for a different number.. ### Eleven. ##### Stage: 3 Challenge Level:. Replace each letter with a digit to make this addition correct.. ### Lesser Digits. ##### Stage: 3 Challenge Level:. How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?. ### Six Times Five. ##### Stage: 3 Challenge Level:. How many six digit numbers are there which DO NOT contain a 5?.
https://www.quantumstudy.com/an-electron-is-accelerated-through-a-potential-difference-of-v-volt-it-has-a-wavelength-%CE%BB-associated-with-it-through-what-potential-difference/	1,695,729,248,000,000,000	text/html	crawl-data/CC-MAIN-2023-40/segments/1695233510208.72/warc/CC-MAIN-20230926111439-20230926141439-00599.warc.gz	1,054,571,965	49,343	# An electron is accelerated through a potential difference of V volt. It has a wavelength λ associated with it….. Q. An electron is accelerated through a potential difference of V volt. It has a wavelength λ associated with it. Through what potential difference an electron must be accelerated so that its de-Broglie wavelength is the same as that of a proton? Take mass of proton to be 1837 times larger than the mass of electron. (a) V volt (b) 1837 V volt (c) V/1837 volt (d) √(1837)volt Ans: (d) Sol: λ = h/√(2meV) λ’ = h/√(2m’eV’) Since , λ = λ’ h/√(2meV) = h/√(2m’eV’) m V =m’V’ m V = m×1837×V’ V’ = V/1837 volt	201	631	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.765625	4	CC-MAIN-2023-40	latest	en	0.789042	# An electron is accelerated through a potential difference of V volt. It has a wavelength λ associated with it…... Q. An electron is accelerated through a potential difference of V volt. It has a wavelength λ associated with it. Through what potential difference an electron must be accelerated so that its de-Broglie wavelength is the same as that of a proton? Take mass of proton to be 1837 times larger than the mass of electron.. (a) V volt. (b) 1837 V volt. (c) V/1837 volt. (d) √(1837)volt.	Ans: (d). Sol: λ = h/√(2meV). λ’ = h/√(2m’eV’). Since , λ = λ’. h/√(2meV) = h/√(2m’eV’). m V =m’V’. m V = m×1837×V’. V’ = V/1837 volt.
https://sciencing.com/roots-quadratic-8116675.html	1,722,894,980,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640455981.23/warc/CC-MAIN-20240805211050-20240806001050-00247.warc.gz	409,359,406	87,221	# How to Find the Roots of a Quadratic ••• Jupiterimages/Polka Dot/Getty Images Print A quadratic equation, or a quadratic in short, is an equation in the form of ax^2 + bx + c = 0, where a is not equal to zero. The “roots” of the quadratic are the numbers that satisfy the quadratic equation. There are always two roots for any quadratic equation, although sometimes they may coincide. You solve quadratic equations by completing the squares, factoring and by using the quadratic formula. However, since completing the squares and factoring are not universally applicable, it is best to learn and use the quadratic formula to find the roots of any quadratic equation. The roots of any quadratic equation is given by: x = [-b +/- sqrt(-b^2 - 4ac)]/2a. Write down the quadratic in the form of ax^2 + bx + c = 0. If the equation is in the form y = ax^2 + bx +c, simply replace the y with 0. This is done because the roots of the equation are the values where the y axis is equal to 0. For example, suppose the quadratic is 2x^2 - 20x + 5 = 0, where a = 2, b = -20, and c = 5. Calculate the first root by using the formula x = [-b + sqrt(-b^2 - 4ac)]/2a. Substitute the values of a, b, and c. In our example, x = [20 + sqrt (20_20 - 4_2_5)]/2_5, which equals 9.7. Note that in order to find the first root, the first item inside the big brackets has changed its signs (because of double negative) and added to the second item. Determine the second root by using the formula: x = [-b + sqrt(-b^2 - 4ac)]/2a. Note that the first item inside the big brackets is subtracted from the second to find the second root. In our example, x = [20 - sqrt(20_20 - 4_2_5)]/2_5, which equals 0.26. Access the quadratic equation solver at Mathworld and enter the values of a, b, and c. Use this option if you do not want to use a calculator. • Pen • Paper • Calculator #### Warnings • Negative numbers squared becomes positive. Make sure that you use correct signs.	531	1,956	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.6875	5	CC-MAIN-2024-33	latest	en	0.884749	# How to Find the Roots of a Quadratic. ••• Jupiterimages/Polka Dot/Getty Images. Print. A quadratic equation, or a quadratic in short, is an equation in the form of ax^2 + bx + c = 0, where a is not equal to zero. The “roots” of the quadratic are the numbers that satisfy the quadratic equation. There are always two roots for any quadratic equation, although sometimes they may coincide.. You solve quadratic equations by completing the squares, factoring and by using the quadratic formula. However, since completing the squares and factoring are not universally applicable, it is best to learn and use the quadratic formula to find the roots of any quadratic equation.. The roots of any quadratic equation is given by: x = [-b +/- sqrt(-b^2 - 4ac)]/2a.. Write down the quadratic in the form of ax^2 + bx + c = 0. If the equation is in the form y = ax^2 + bx +c, simply replace the y with 0. This is done because the roots of the equation are the values where the y axis is equal to 0. For example, suppose the quadratic is 2x^2 - 20x + 5 = 0, where a = 2, b = -20, and c = 5.. Calculate the first root by using the formula x = [-b + sqrt(-b^2 - 4ac)]/2a. Substitute the values of a, b, and c.	In our example, x = [20 + sqrt (20_20 - 4_2_5)]/2_5, which equals 9.7. Note that in order to find the first root, the first item inside the big brackets has changed its signs (because of double negative) and added to the second item.. Determine the second root by using the formula: x = [-b + sqrt(-b^2 - 4ac)]/2a. Note that the first item inside the big brackets is subtracted from the second to find the second root. In our example, x = [20 - sqrt(20_20 - 4_2_5)]/2_5, which equals 0.26.. Access the quadratic equation solver at Mathworld and enter the values of a, b, and c. Use this option if you do not want to use a calculator.. • Pen. • Paper. • Calculator. #### Warnings. • Negative numbers squared becomes positive. Make sure that you use correct signs.
https://www.esaral.com/q/if-a-is-a-square-matrix-using-mathematical-induction-prove-that-39547	1,723,170,652,000,000,000	text/html	crawl-data/CC-MAIN-2024-33/segments/1722640751424.48/warc/CC-MAIN-20240809013306-20240809043306-00777.warc.gz	610,990,420	11,511	# If A is a square matrix, using mathematical induction prove that Question: If $A$ is a square matrix, using mathematical induction prove that $\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. Solution: Let the given statement $P(n)$, be given as $P(n):\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$. We observe that $P(1):\left(A^{T}\right)^{1}=A^{T}=\left(A^{1}\right)^{\top}$ Thus, $P(n)$ is true for $n=1$. Assume that $P(n)$ is true for $n=k \in N$. i.e., $P(k):\left(A^{T}\right)^{k}=\left(A^{k}\right)^{T}$ To prove that $P(k+1)$ is true, we have $\left(A^{T}\right)^{k+1}=\left(A^{T}\right)^{k} \cdot\left(A^{T}\right)^{1}$ $=\left(A^{k}\right)^{T} \cdot\left(A^{1}\right)^{T}$ $=\left(A^{k+1}\right)^{T}$ Thus, $P(k+1)$ is true, whenever $P(k)$ is true. Hence, by the Principle of mathematical induction, $P(n)$ is true for all $n \in N$.	345	899	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.21875	4	CC-MAIN-2024-33	latest	en	0.492859	# If A is a square matrix, using mathematical induction prove that. Question:. If $A$ is a square matrix, using mathematical induction prove that $\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$.. Solution:. Let the given statement $P(n)$, be given as. $P(n):\left(A^{T}\right)^{n}=\left(A^{n}\right)^{T}$ for all $n \in N$.. We observe that. $P(1):\left(A^{T}\right)^{1}=A^{T}=\left(A^{1}\right)^{\top}$. Thus, $P(n)$ is true for $n=1$.	Assume that $P(n)$ is true for $n=k \in N$.. i.e., $P(k):\left(A^{T}\right)^{k}=\left(A^{k}\right)^{T}$. To prove that $P(k+1)$ is true, we have. $\left(A^{T}\right)^{k+1}=\left(A^{T}\right)^{k} \cdot\left(A^{T}\right)^{1}$. $=\left(A^{k}\right)^{T} \cdot\left(A^{1}\right)^{T}$. $=\left(A^{k+1}\right)^{T}$. Thus, $P(k+1)$ is true, whenever $P(k)$ is true.. Hence, by the Principle of mathematical induction, $P(n)$ is true for all $n \in N$.
http://www.accountingweb.com/article/excel-tip-determining-remaining-length-loan-using-nper/221766	1,433,260,784,000,000,000	text/html	crawl-data/CC-MAIN-2015-22/segments/1433195035877.3/warc/CC-MAIN-20150601214355-00109-ip-10-180-206-219.ec2.internal.warc.gz	195,611,328	13,704	Excel Tip: Determining the Remaining Length of a Loan Using NPER \| AccountingWEB ## Excel Tip: Determining the Remaining Length of a Loan Using NPER By David Ringstrom, CPA Previously I’ve explained how you can use worksheet functions in Excel to determine the payment for a loan, as well as how to calculate total interest in a single worksheet cell. This time around I’m going to use the NPER function in Excel to show you how you can determine just how long it will take you to payoff that credit card bill on which you’re making monthly payments. You can get a refresher on the PMT and CUMIPMT functions, but any typical loan has four key values: • Interest rate • Term of loan • Amount borrowed • Monthly payment As discussed in my previous article, if you have the interest rate, term, and loan amount, you can then use the PMT function to solve for the payment. Sometimes, such as with a credit card balance, you’ll know the interest rate, payment, and loan amount, but not the term. In this case, Excel’s NPER function (short for number of periods) enables you to calculate the fourth value. NPER has 3 required and 2 optional arguments: rate - The interest rate for the loan expressed as a monthly rate. pmt - The monthly payment, which should always be shown as a negative amount. pv - The current loan balance fv - This optional argument allows you to specify a future value if a balloon amount is due at the end of the loan. Omitting this argument implicitly states that the loan is to be paid down to 0. type - This optional argument allows you to specify if payments are made at the beginning of each period, or you can omit the argument to indicate that payments are made at the end of each period. You may also specify 0 in this position to explicitly indicate that payments are made at the end of each period. As shown in Figure 1, it will take 36 months to pay back \$20,000 with a monthly payment of \$586.04 at an interest rate of 3.5 percent. Always be sure that the payment is shown as a negative number, otherwise NPER may show a slightly longer payment term. I omitted the 2 optional arguments, so in this case, the PMT function assumes the loan is paid to 0 and payments are made at the end of each period. Figure 1: Use Excel’s NPER function to calculate the payment term for a loan. For longer term loans, NPER may return a large number of months, such as 94, which can be difficult to convert to months in your head. Let’s extend our calculations to make the output more user-friendly. First, as shown in Figure 2, we’ll add the ROUNDUP function to our NPER formula. Loan periods will typically involve some fraction of the final month, which for our purposes we want to treat as a whole month. The ROUNDUP function rounds a number up, as opposed to the commonly used ROUND function that may round numbers up or down. ROUNDUP has two arguments: number – In this case the result of NPER will be our number num_digits – in this case we’ll use zero, as we wish to round up to the next whole month. If you wanted to round a number to say the nearest thousand, you’d use -3 instead. Figure 2: Use the ROUNDUP function with the NPER formula to convert to months. Next, we need to convert the result that ROUNDUP/NPER return into a number of years and months. To do so, we can use the TRUNC function. This function converts a number to an integer by removing the decimal or fractional portion. You could also use ROUNDDOWN and specify zero as the number of digits to accomplish the same effect. To calculate the number of whole years in the loan, we can use this formula: =TRUNC(B4/12) Or this would work as well: =ROUNDDOWN(B4/12,0) In either case we’re taking the number of periods returned by NPER, dividing it by 12, and then truncating the decimal places. Use this formula to calculate the number of months remaining after the whole years: =ROUNDUP(B4-TRUNC(B4/12)12,0) We can then string all of this together into a tidy format, as shown in Figure 3: ="months, or "&TRUNC(B4/12)& " Years, "&ROUNDUP(B4-TRUNC(B4/12)12,0)&" Months" In this we’re using the ampersand to concatenate, or join together, text and calculations into an understandable phrase. I much prefer using the ampersand to join text together, but if you’re a fan of Excel’s CONCATENATE function, the formula would take this form: =CONCATENATE("months, or ",TRUNC(B4/12), " Years, ",ROUNDUP(B4-TRUNC(B4/12)*12,0)," Months") Figure 3: Use the ampersand sign to concatenate text and calculations into a logical phrase. Read more articles by David Ringstrom. David H. Ringstrom, CPA heads up Accounting Advisors, Inc., an Atlanta-based software and database consulting firm providing training and consulting services nationwide. Contact David at david@acctadv.com or follow him on Twitter. David speaks at conferences about Microsoft Excel, and presents webcasts for several CPE providers, including AccountingWEB partner CPE Link. Wait, there's more! There's always more at AccountingWEB. We're an active community of financial professionals and journalists who strive to bring you valuable content every day. If you'd like, let us know your interests and we'll send you a few articles every week either in taxation, practice excellence, or just our most popular stories from that week. It's free to sign up and to be a part of our community. ## Editor's Choice WHAT KIND OF FIRM ARE YOU? As part of our continued effort to provide valuable resources and insight to our subscribers, we're conducting this brief survey to learn more about your personal experiences in the accounting profession. We will be giving away five \$50 Amazon gift cards, and a \$250 Amazon gift card to one lucky participant. This is strictly for internal use and data will not be sold or shared with any third parties.	1,335	5,824	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.515625	4	CC-MAIN-2015-22	latest	en	0.910895	Excel Tip: Determining the Remaining Length of a Loan Using NPER \| AccountingWEB. ## Excel Tip: Determining the Remaining Length of a Loan Using NPER. By David Ringstrom, CPA. Previously I’ve explained how you can use worksheet functions in Excel to determine the payment for a loan, as well as how to calculate total interest in a single worksheet cell. This time around I’m going to use the NPER function in Excel to show you how you can determine just how long it will take you to payoff that credit card bill on which you’re making monthly payments.. You can get a refresher on the PMT and CUMIPMT functions, but any typical loan has four key values:. • Interest rate. • Term of loan. • Amount borrowed. • Monthly payment. As discussed in my previous article, if you have the interest rate, term, and loan amount, you can then use the PMT function to solve for the payment. Sometimes, such as with a credit card balance, you’ll know the interest rate, payment, and loan amount, but not the term. In this case, Excel’s NPER function (short for number of periods) enables you to calculate the fourth value.. NPER has 3 required and 2 optional arguments:. rate - The interest rate for the loan expressed as a monthly rate.. pmt - The monthly payment, which should always be shown as a negative amount.. pv - The current loan balance. fv - This optional argument allows you to specify a future value if a balloon amount is due at the end of the loan. Omitting this argument implicitly states that the loan is to be paid down to 0.. type - This optional argument allows you to specify if payments are made at the beginning of each period, or you can omit the argument to indicate that payments are made at the end of each period. You may also specify 0 in this position to explicitly indicate that payments are made at the end of each period.. As shown in Figure 1, it will take 36 months to pay back \$20,000 with a monthly payment of \$586.04 at an interest rate of 3.5 percent. Always be sure that the payment is shown as a negative number, otherwise NPER may show a slightly longer payment term. I omitted the 2 optional arguments, so in this case, the PMT function assumes the loan is paid to 0 and payments are made at the end of each period.. Figure 1:. Use Excel’s NPER function to calculate the payment term for a loan.. For longer term loans, NPER may return a large number of months, such as 94, which can be difficult to convert to months in your head. Let’s extend our calculations to make the output more user-friendly.. First, as shown in Figure 2, we’ll add the ROUNDUP function to our NPER formula. Loan periods will typically involve some fraction of the final month, which for our purposes we want to treat as a whole month. The ROUNDUP function rounds a number up, as opposed to the commonly used ROUND function that may round numbers up or down.. ROUNDUP has two arguments:. number – In this case the result of NPER will be our number. num_digits – in this case we’ll use zero, as we wish to round up to the next whole month. If you wanted to round a number to say the nearest thousand, you’d use -3 instead.. Figure 2:. Use the ROUNDUP function with the NPER formula to convert to months.	Next, we need to convert the result that ROUNDUP/NPER return into a number of years and months. To do so, we can use the TRUNC function. This function converts a number to an integer by removing the decimal or fractional portion. You could also use ROUNDDOWN and specify zero as the number of digits to accomplish the same effect.. To calculate the number of whole years in the loan, we can use this formula:. =TRUNC(B4/12). Or this would work as well:. =ROUNDDOWN(B4/12,0). In either case we’re taking the number of periods returned by NPER, dividing it by 12, and then truncating the decimal places.. Use this formula to calculate the number of months remaining after the whole years:. =ROUNDUP(B4-TRUNC(B4/12)12,0). We can then string all of this together into a tidy format, as shown in Figure 3:. ="months, or "&TRUNC(B4/12)& " Years, "&ROUNDUP(B4-TRUNC(B4/12)12,0)&" Months". In this we’re using the ampersand to concatenate, or join together, text and calculations into an understandable phrase. I much prefer using the ampersand to join text together, but if you’re a fan of Excel’s CONCATENATE function, the formula would take this form:. =CONCATENATE("months, or ",TRUNC(B4/12), " Years, ",ROUNDUP(B4-TRUNC(B4/12)*12,0)," Months"). Figure 3:. Use the ampersand sign to concatenate text and calculations into a logical phrase.. Read more articles by David Ringstrom.. David H. Ringstrom, CPA heads up Accounting Advisors, Inc., an Atlanta-based software and database consulting firm providing training and consulting services nationwide. Contact David at david@acctadv.com or follow him on Twitter. David speaks at conferences about Microsoft Excel, and presents webcasts for several CPE providers, including AccountingWEB partner CPE Link.. Wait, there's more!. There's always more at AccountingWEB. We're an active community of financial professionals and journalists who strive to bring you valuable content every day. If you'd like, let us know your interests and we'll send you a few articles every week either in taxation, practice excellence, or just our most popular stories from that week. It's free to sign up and to be a part of our community.. ## Editor's Choice. WHAT KIND OF FIRM ARE YOU?. As part of our continued effort to provide valuable resources and insight to our subscribers, we're conducting this brief survey to learn more about your personal experiences in the accounting profession. We will be giving away five \$50 Amazon gift cards, and a \$250 Amazon gift card to one lucky participant.. 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https://edurev.in/v/240516/Problems-on-Ages-Tips-and-Tricks-for-Government-Exams	1,709,326,447,000,000,000	text/html	crawl-data/CC-MAIN-2024-10/segments/1707947475701.61/warc/CC-MAIN-20240301193300-20240301223300-00497.warc.gz	217,070,897	58,090	Tips & Tricks: Problems on Ages # Problems on Ages Tips and Tricks for Government Exams ## Tips & Tricks for Government Exams 65 videos\|65 docs ## FAQs on Problems on Ages Tips and Tricks for Government Exams 1. What are some common problems people face when solving age-related problems? Ans. Some common problems people face when solving age-related problems include confusion in setting up equations, difficulty in interpreting and translating word problems into mathematical equations, and challenges in determining the correct variables to use in the equations. 2. How can I approach age-related problems to solve them effectively? Ans. To solve age-related problems effectively, start by carefully reading and understanding the problem statement. Identify the unknown variables and assign them appropriate variables. Set up equations based on the given information and use algebraic techniques to solve for the unknowns. Finally, verify the solution by checking if it satisfies the given conditions. 3. What are some tips for setting up equations in age-related problems? Ans. When setting up equations in age-related problems, it is helpful to represent the ages using variables, such as "x" and "y." Consider the relationships between the ages given in the problem statement and express them using mathematical expressions or equations. Additionally, pay attention to keywords like "older," "younger," "twice," or "half," as they provide clues for determining the relationships between ages. 4. How can I improve my problem-solving skills for age-related problems? Ans. To improve problem-solving skills for age-related problems, practice solving a variety of problems from different sources. Break down complex problems into smaller, manageable steps. Seek out resources such as textbooks, online tutorials, or educational videos that provide examples and explanations for solving age-related problems. Additionally, work on developing logical reasoning and critical thinking skills, as they are valuable for solving mathematical problems. 5. Are there any specific strategies or techniques to solve age-related problems quickly? Ans. Yes, there are some strategies that can help you solve age-related problems quickly. One such strategy is to assign variables to the ages and use a system of equations to represent the given information. Another technique is to use ratios or proportions to solve the problems. It is also important to identify the key information and relationships in the problem statement to avoid unnecessary calculations. Practice and familiarity with different types of age-related problems can also help in solving them quickly. ## Tips & Tricks for Government Exams 65 videos\|65 docs ### Up next Explore Courses for Bank Exams exam Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests. 10M+ students study on EduRev Track your progress, build streaks, highlight & save important lessons and more! Related Searches , , , , , , , , , , , , , , , , , , , , , ;	591	3,076	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.25	4	CC-MAIN-2024-10	latest	en	0.942266	Tips & Tricks: Problems on Ages. # Problems on Ages Tips and Tricks for Government Exams. ## Tips & Tricks for Government Exams. 65 videos\|65 docs. ## FAQs on Problems on Ages Tips and Tricks for Government Exams. 1. What are some common problems people face when solving age-related problems?. Ans. Some common problems people face when solving age-related problems include confusion in setting up equations, difficulty in interpreting and translating word problems into mathematical equations, and challenges in determining the correct variables to use in the equations.. 2. How can I approach age-related problems to solve them effectively?. Ans. To solve age-related problems effectively, start by carefully reading and understanding the problem statement. Identify the unknown variables and assign them appropriate variables. Set up equations based on the given information and use algebraic techniques to solve for the unknowns. Finally, verify the solution by checking if it satisfies the given conditions.. 3. What are some tips for setting up equations in age-related problems?. Ans. When setting up equations in age-related problems, it is helpful to represent the ages using variables, such as "x" and "y." Consider the relationships between the ages given in the problem statement and express them using mathematical expressions or equations. Additionally, pay attention to keywords like "older," "younger," "twice," or "half," as they provide clues for determining the relationships between ages.. 4. How can I improve my problem-solving skills for age-related problems?. Ans. To improve problem-solving skills for age-related problems, practice solving a variety of problems from different sources. Break down complex problems into smaller, manageable steps. Seek out resources such as textbooks, online tutorials, or educational videos that provide examples and explanations for solving age-related problems. Additionally, work on developing logical reasoning and critical thinking skills, as they are valuable for solving mathematical problems.. 5. Are there any specific strategies or techniques to solve age-related problems quickly?. Ans. Yes, there are some strategies that can help you solve age-related problems quickly. One such strategy is to assign variables to the ages and use a system of equations to represent the given information. Another technique is to use ratios or proportions to solve the problems.	It is also important to identify the key information and relationships in the problem statement to avoid unnecessary calculations. Practice and familiarity with different types of age-related problems can also help in solving them quickly.. ## Tips & Tricks for Government Exams. 65 videos\|65 docs. ### Up next. Explore Courses for Bank Exams exam. Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.. 10M+ students study on EduRev. Track your progress, build streaks, highlight & save important lessons and more!. Related Searches. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ;.
https://math.answers.com/Q/Add_the_following_fractions_2_over_3_plus_1_over_6_plus_1_over_6	1,713,822,699,000,000,000	text/html	crawl-data/CC-MAIN-2024-18/segments/1712296818374.84/warc/CC-MAIN-20240422211055-20240423001055-00690.warc.gz	328,376,831	47,705	0 # Add the following fractions 2 over 3 plus 1 over 6 plus 1 over 6? Updated: 9/17/2023 Wiki User 14y ago 1 Wiki User 14y ago Earn +20 pts Q: Add the following fractions 2 over 3 plus 1 over 6 plus 1 over 6? Submit Still have questions? Related questions 0.9375 ### Add the following fractions 1 over 2 plus 1 over 4 plus 1 over 8 plus 1 over 16? its going to be 4over 27 ---- ### What is 1 over 3 plus 1 over 6 plus 1 12? Convert all the fractions to a common denominator. Then add. ### What is 1 and 5 over 8 plus 1 and 5 over 8? To do additions with mixed fractions, you should add the integer (whole) part, and the fractions, separately. ### 1 over 9 plus 4 over 9? If you are using the word "over" to describe that these numbers are fractions, then 1/9 plus 4/9 is 5/9. Since the denominators are the same, you simply add the numerators of both fractions, and you have your answer, 5 over 9. ### What is 2 over 3 plus 2 over 3? 4 over 3. In adding fractions, you just add the numerators together if the denominators are the same. ### How do you add fractions step by step? you make fractions equivalent denominators, you add the numerators and put it over the denominator ### How do you Add 6 one -half plus 6 one-half plus 6 three-fourth plus 6 three -fourth? Carry out the following steps:Convert each of the mixed fractions into improper fractions.Rename these fractions so that their denominators are the same (= d).Add the numerators together (= n)The answer is n/d, which will be an improper fraction and you may wish to convert that to a mixed fraction. 40 ### Is 1whole equal to 1third plus 1halve plus 1sixth? Yes because the fractions add up to 1 ### How do you add and subtract by fractions? If the fractions have the same denominator, add and subtract the numerators as if the denominators weren't there and put the result over that denominator. Reduce if possible. If the fractions have different denominators, find the LCM of the denominators and convert the fractions to equivalent fractions with like denominators. Then add and subtract the numerators as if the denominators weren't there and put the result over that denominator. Reduce if possible. ### What is the answer to 5 over 6 plus 4 over 6? For adding fractions, you need to make both denominators the same, then add the numerators. In this case 5/6 and 4/6 have the sum 9/6, which can be simplified to 1 1/2 (fractions are difficult in these answer windows).	658	2,471	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.40625	4	CC-MAIN-2024-18	latest	en	0.846658	0. # Add the following fractions 2 over 3 plus 1 over 6 plus 1 over 6?. Updated: 9/17/2023. Wiki User. 14y ago. 1. Wiki User. 14y ago. Earn +20 pts. Q: Add the following fractions 2 over 3 plus 1 over 6 plus 1 over 6?. Submit. Still have questions?. Related questions. 0.9375. ### Add the following fractions 1 over 2 plus 1 over 4 plus 1 over 8 plus 1 over 16?. its going to be 4over 27 ----. ### What is 1 over 3 plus 1 over 6 plus 1 12?. Convert all the fractions to a common denominator. Then add.. ### What is 1 and 5 over 8 plus 1 and 5 over 8?. To do additions with mixed fractions, you should add the integer (whole) part, and the fractions, separately.. ### 1 over 9 plus 4 over 9?.	If you are using the word "over" to describe that these numbers are fractions, then 1/9 plus 4/9 is 5/9. Since the denominators are the same, you simply add the numerators of both fractions, and you have your answer, 5 over 9.. ### What is 2 over 3 plus 2 over 3?. 4 over 3. In adding fractions, you just add the numerators together if the denominators are the same.. ### How do you add fractions step by step?. you make fractions equivalent denominators, you add the numerators and put it over the denominator. ### How do you Add 6 one -half plus 6 one-half plus 6 three-fourth plus 6 three -fourth?. Carry out the following steps:Convert each of the mixed fractions into improper fractions.Rename these fractions so that their denominators are the same (= d).Add the numerators together (= n)The answer is n/d, which will be an improper fraction and you may wish to convert that to a mixed fraction.. 40. ### Is 1whole equal to 1third plus 1halve plus 1sixth?. Yes because the fractions add up to 1. ### How do you add and subtract by fractions?. If the fractions have the same denominator, add and subtract the numerators as if the denominators weren't there and put the result over that denominator. Reduce if possible. If the fractions have different denominators, find the LCM of the denominators and convert the fractions to equivalent fractions with like denominators. Then add and subtract the numerators as if the denominators weren't there and put the result over that denominator. Reduce if possible.. ### What is the answer to 5 over 6 plus 4 over 6?. For adding fractions, you need to make both denominators the same, then add the numerators. In this case 5/6 and 4/6 have the sum 9/6, which can be simplified to 1 1/2 (fractions are difficult in these answer windows).
http://metamath.tirix.org/mpests/issect2.html	1,716,500,343,000,000,000	text/html	crawl-data/CC-MAIN-2024-22/segments/1715971058671.46/warc/CC-MAIN-20240523204210-20240523234210-00582.warc.gz	18,035,719	2,640	Metamath Proof Explorer Theorem issect2 Description: Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017) Ref Expression Hypotheses issect.b ${⊢}{B}={\mathrm{Base}}_{{C}}$ issect.h ${⊢}{H}=\mathrm{Hom}\left({C}\right)$ issect.o issect.i issect.s ${⊢}{S}=\mathrm{Sect}\left({C}\right)$ issect.c ${⊢}{\phi }\to {C}\in \mathrm{Cat}$ issect.x ${⊢}{\phi }\to {X}\in {B}$ issect.y ${⊢}{\phi }\to {Y}\in {B}$ issect.f ${⊢}{\phi }\to {F}\in \left({X}{H}{Y}\right)$ issect.g ${⊢}{\phi }\to {G}\in \left({Y}{H}{X}\right)$ Assertion issect2 Proof Step Hyp Ref Expression 1 issect.b ${⊢}{B}={\mathrm{Base}}_{{C}}$ 2 issect.h ${⊢}{H}=\mathrm{Hom}\left({C}\right)$ 3 issect.o 4 issect.i 5 issect.s ${⊢}{S}=\mathrm{Sect}\left({C}\right)$ 6 issect.c ${⊢}{\phi }\to {C}\in \mathrm{Cat}$ 7 issect.x ${⊢}{\phi }\to {X}\in {B}$ 8 issect.y ${⊢}{\phi }\to {Y}\in {B}$ 9 issect.f ${⊢}{\phi }\to {F}\in \left({X}{H}{Y}\right)$ 10 issect.g ${⊢}{\phi }\to {G}\in \left({Y}{H}{X}\right)$ 11 9 10 jca ${⊢}{\phi }\to \left({F}\in \left({X}{H}{Y}\right)\wedge {G}\in \left({Y}{H}{X}\right)\right)$ 12 1 2 3 4 5 6 7 8 issect 13 df-3an 14 12 13 syl6bb 15 11 14 mpbirand	529	1,162	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.71875	4	CC-MAIN-2024-22	latest	en	0.345159	Metamath Proof Explorer. Theorem issect2. Description: Property of being a section. (Contributed by Mario Carneiro, 2-Jan-2017). Ref Expression. Hypotheses issect.b ${⊢}{B}={\mathrm{Base}}_{{C}}$. issect.h ${⊢}{H}=\mathrm{Hom}\left({C}\right)$. issect.o. issect.i. issect.s ${⊢}{S}=\mathrm{Sect}\left({C}\right)$. issect.c ${⊢}{\phi }\to {C}\in \mathrm{Cat}$. issect.x ${⊢}{\phi }\to {X}\in {B}$. issect.y ${⊢}{\phi }\to {Y}\in {B}$. issect.f ${⊢}{\phi }\to {F}\in \left({X}{H}{Y}\right)$. issect.g ${⊢}{\phi }\to {G}\in \left({Y}{H}{X}\right)$. Assertion issect2. Proof.	Step Hyp Ref Expression. 1 issect.b ${⊢}{B}={\mathrm{Base}}_{{C}}$. 2 issect.h ${⊢}{H}=\mathrm{Hom}\left({C}\right)$. 3 issect.o. 4 issect.i. 5 issect.s ${⊢}{S}=\mathrm{Sect}\left({C}\right)$. 6 issect.c ${⊢}{\phi }\to {C}\in \mathrm{Cat}$. 7 issect.x ${⊢}{\phi }\to {X}\in {B}$. 8 issect.y ${⊢}{\phi }\to {Y}\in {B}$. 9 issect.f ${⊢}{\phi }\to {F}\in \left({X}{H}{Y}\right)$. 10 issect.g ${⊢}{\phi }\to {G}\in \left({Y}{H}{X}\right)$. 11 9 10 jca ${⊢}{\phi }\to \left({F}\in \left({X}{H}{Y}\right)\wedge {G}\in \left({Y}{H}{X}\right)\right)$. 12 1 2 3 4 5 6 7 8 issect. 13 df-3an. 14 12 13 syl6bb. 15 11 14 mpbirand.
http://www.territorioscuola.com/wikipedia/en.wikipedia.php?title=Product_(mathematics)	1,397,964,861,000,000,000	text/html	crawl-data/CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00019-ip-10-147-4-33.ec2.internal.warc.gz	702,499,954	20,396	# Product (mathematics) Calculation results Subtraction (−) minuend − subtrahend = difference Multiplication (×) multiplicand × multiplier = product Division (÷) dividend ÷ divisor = quotient Exponentiation baseexponent = power nth root (√) Logarithm logbase(power) = exponent In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and $x\cdot (2+x)$ is the product of $x$ and $(2+x)$ (indicating that the two factors should be multiplied together). The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. ## Product of two numbers ### Product of two natural numbers 3 by 4 is 12 Placing several stones into a rectangular pattern with $r$ rows and $s$ columns gives $r \cdot s = \sum_{i=1}^s r = \sum_{j=1}^r s$ stones. ### Product of two integers Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs: $\begin{array}{\|c\|c c\|}\hline \cdot & - & + \\ \hline - & + & - \\ + & - & + \\ \hline \end{array}$ In words, we have: • Minus times Minus gives Plus • Minus times Plus gives Minus • Plus times Minus gives Minus • Plus times Plus gives Plus ### Product of two fractions Two fractions can be multiplied by multiplying their numerators and denominators: $\frac{z}{n} \cdot \frac{z'}{n'} = \frac{z\cdot z'}{n\cdot n'}$ ### Product of two real numbers The rigorous definition of the product of two real numbers is too complicated for this article. But the idea is that one takes a decimal approximation to each real and multiplies the approximations together, and then take better and better approximations. ### Product of two complex numbers Two complex numbers can be multiplied by the distributive law and the fact that $\mathrm i^2=-1$, as follows: \begin{align} (a + b\,\mathrm i)\cdot (c+d\,\mathrm i) & = a\cdot c + a \cdot d\,\mathrm i + b\cdot c \,\mathrm i + b\cdot d \cdot \mathrm i^2\\ & = (a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i \end{align} #### Geometric meaning of complex multiplication A complex number in polar coordinates. Complex numbers can be written in polar coordinates: $a + b\,\mathrm i = r \cdot ( \cos(\varphi) + \mathrm i \sin(\varphi) ) = r \cdot \mathrm e ^{\mathrm i \varphi}$ Furthermore, $c + d\,\mathrm i = s \cdot ( \cos(\psi) + \mathrm i \sin(\psi) ) = s \cdot \mathrm e ^{\mathrm i \psi}$, from which we obtain: $(a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i = r\cdot s \cdot ( \cos(\varphi+\psi) + \mathrm i \sin(\varphi+\psi) ) = r\cdot s \cdot \mathrm e ^{\mathrm i (\varphi+\psi)}$ The geometric meaning is that we multiply the magnitudes and add the angles. ### Product of two quaternions The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, $a\cdot b$ and $b\cdot a$ are different. ## Product of sequences The product operator for the product of a sequence is denoted by the capital Greek letter Pi (in analogy to the use of the capital Sigma as summation symbol). The product of a sequence consisting of only one number is just that number itself. The product of no factors at all is known as the empty product, and is equal to 1. ## Further examples for commutative rings ### Residue classes of integers Residue classes in the rings $\Z/N\Z$ can be added: $(a+N\Z) + (b+N\Z) = a+b + N\Z$ and multiplied: $(a+N\Z) \cdot (b+N\Z) = a\cdot b + N\Z$ ### Rings of functions Functions to the real numbers can be added or multiplied by adding or multiplying their outputs: $(f+g)(m) : = f(m) + g(m)$ $(f\cdot g) (m) := f(m) \cdot g(m)$ #### Convolution The convolution of the square wave with itself gives the triangular function Two functions from the reals to itself can be multiplied in another way, called the convolution. If :$\int\limits_{-\infty}^\infty \|f(t)\|\,\mathrm{d}\,t \;<\;\infty\quad\mbox{and } \int\limits_{-\infty}^\infty \|g(t)\|\,\mathrm{d}\,t \;<\; \infty$ then the integral $(f*g) (t) \;:= \int\limits_{-\infty}^\infty f(\tau)\cdot g(t-\tau)\,\mathrm{d}\tau$ is well defined and is called the convolution. Under the Fourier transform, convolution becomes multiplication. ### Polynomial rings The product of two polynomials is given by the following: $\left(\sum_{i=0}^n a_i X^i\right) \cdot \left(\sum_{j=0}^m b_j X^j\right) = \sum_{k=0}^{n+m} c_k X^k$ with $c_k = \sum_{i+j=k} a_i \cdot b_j$ ## Products in linear algebra ### Scalar multiplication By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map $\R \times V \rightarrow V$. ### Scalar product A scalar product is a bilinear map: $\cdot : V \times V \rightarrow \R$ with the following conditions, that $v\cdot v > 0$ for all $0 \not= v \in V$. From the scalar product, one can define a norm by letting $\\|v\\| := \sqrt{v\cdot v}$. The scalar product also allows one to define an angle between two vectors: $\cos \angle (v,w) = \frac{v\cdot w}{\\|v\\| \cdot \\|w\\|}$ In $n$-dimensional Euclidean space, the standard scalar product (called the dot product) is given by: $\left(\sum_{i=1}^n \alpha_i e_i \right) \cdot \left(\sum_{i=1}^n \beta_i e_i \right) = \sum_{i=1}^n \alpha_i\,\beta_i$ ### Cross product in 3-dimensional space The cross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors. The cross product can also be expressed as the formala determinant: $\mathbf{u\times v}=\begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_1&u_2&u_3\\ v_1&v_2&v_3\\ \end{vmatrix}$ ### Product of two matrices Given two matrices $A = (a_{i,j})_{i=1\ldots s;j=1\ldots r} \in \R^{s\times r}$ and $B = (b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \R^{r\times t}$ their product is given by $B \cdot A = \left( \sum_{j=1}^r a_{i,j} \cdot b_{j,k} \right)_{i=1\ldots s;k=1\ldots t} \;\in\R^{s\times t}$ ## Set theoretical product In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. ## Empty product The empty product has the value of 1 (the identity element of multiplication) just like the empty sum has the value of 0 (the identity element of addition). ## Other products Many different kinds of products are studied in mathematics:	2,029	6,933	{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 36, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.84375	5	CC-MAIN-2014-15	latest	en	0.837509	# Product (mathematics). Calculation results. Subtraction (−). minuend − subtrahend = difference. Multiplication (×). multiplicand × multiplier = product. Division (÷). dividend ÷ divisor = quotient. Exponentiation. baseexponent = power. nth root (√). Logarithm. logbase(power) = exponent. In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance, 6 is the product of 2 and 3 (the result of multiplication), and $x\cdot (2+x)$ is the product of $x$ and $(2+x)$ (indicating that the two factors should be multiplied together).. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, and multiplication in other algebras is in general non-commutative.. ## Product of two numbers. ### Product of two natural numbers. 3 by 4 is 12. Placing several stones into a rectangular pattern with $r$ rows and $s$ columns gives. $r \cdot s = \sum_{i=1}^s r = \sum_{j=1}^r s$. stones.. ### Product of two integers. Integers allow positive and negative numbers. The two numbers are multiplied just like natural numbers, except we need an additional rule for the signs:. $\begin{array}{\|c\|c c\|}\hline \cdot & - & + \\ \hline - & + & - \\ + & - & + \\ \hline \end{array}$. In words, we have:. • Minus times Minus gives Plus. • Minus times Plus gives Minus. • Plus times Minus gives Minus. • Plus times Plus gives Plus. ### Product of two fractions. Two fractions can be multiplied by multiplying their numerators and denominators:. $\frac{z}{n} \cdot \frac{z'}{n'} = \frac{z\cdot z'}{n\cdot n'}$. ### Product of two real numbers. The rigorous definition of the product of two real numbers is too complicated for this article. But the idea is that one takes a decimal approximation to each real and multiplies the approximations together, and then take better and better approximations.. ### Product of two complex numbers. Two complex numbers can be multiplied by the distributive law and the fact that $\mathrm i^2=-1$, as follows:. \begin{align} (a + b\,\mathrm i)\cdot (c+d\,\mathrm i) & = a\cdot c + a \cdot d\,\mathrm i + b\cdot c \,\mathrm i + b\cdot d \cdot \mathrm i^2\\ & = (a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i \end{align}. #### Geometric meaning of complex multiplication. A complex number in polar coordinates.. Complex numbers can be written in polar coordinates:. $a + b\,\mathrm i = r \cdot ( \cos(\varphi) + \mathrm i \sin(\varphi) ) = r \cdot \mathrm e ^{\mathrm i \varphi}$. Furthermore,. $c + d\,\mathrm i = s \cdot ( \cos(\psi) + \mathrm i \sin(\psi) ) = s \cdot \mathrm e ^{\mathrm i \psi}$, from which we obtain:. $(a \cdot c - b\cdot d) + (a\cdot d + b\cdot c) \,\mathrm i = r\cdot s \cdot ( \cos(\varphi+\psi) + \mathrm i \sin(\varphi+\psi) ) = r\cdot s \cdot \mathrm e ^{\mathrm i (\varphi+\psi)}$. The geometric meaning is that we multiply the magnitudes and add the angles.. ### Product of two quaternions. The product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, $a\cdot b$ and $b\cdot a$ are different.. ## Product of sequences.	The product operator for the product of a sequence is denoted by the capital Greek letter Pi (in analogy to the use of the capital Sigma as summation symbol). The product of a sequence consisting of only one number is just that number itself. The product of no factors at all is known as the empty product, and is equal to 1.. ## Further examples for commutative rings. ### Residue classes of integers. Residue classes in the rings $\Z/N\Z$ can be added:. $(a+N\Z) + (b+N\Z) = a+b + N\Z$. and multiplied:. $(a+N\Z) \cdot (b+N\Z) = a\cdot b + N\Z$. ### Rings of functions. Functions to the real numbers can be added or multiplied by adding or multiplying their outputs:. $(f+g)(m) : = f(m) + g(m)$. $(f\cdot g) (m) := f(m) \cdot g(m)$. #### Convolution. The convolution of the square wave with itself gives the triangular function. Two functions from the reals to itself can be multiplied in another way, called the convolution.. If :$\int\limits_{-\infty}^\infty \|f(t)\|\,\mathrm{d}\,t \;<\;\infty\quad\mbox{and } \int\limits_{-\infty}^\infty \|g(t)\|\,\mathrm{d}\,t \;<\; \infty$. then the integral. $(f*g) (t) \;:= \int\limits_{-\infty}^\infty f(\tau)\cdot g(t-\tau)\,\mathrm{d}\tau$. is well defined and is called the convolution.. Under the Fourier transform, convolution becomes multiplication.. ### Polynomial rings. The product of two polynomials is given by the following:. $\left(\sum_{i=0}^n a_i X^i\right) \cdot \left(\sum_{j=0}^m b_j X^j\right) = \sum_{k=0}^{n+m} c_k X^k$. with. $c_k = \sum_{i+j=k} a_i \cdot b_j$. ## Products in linear algebra. ### Scalar multiplication. By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map $\R \times V \rightarrow V$.. ### Scalar product. A scalar product is a bilinear map:. $\cdot : V \times V \rightarrow \R$. with the following conditions, that $v\cdot v > 0$ for all $0 \not= v \in V$.. From the scalar product, one can define a norm by letting $\\|v\\| := \sqrt{v\cdot v}$.. The scalar product also allows one to define an angle between two vectors:. $\cos \angle (v,w) = \frac{v\cdot w}{\\|v\\| \cdot \\|w\\|}$. In $n$-dimensional Euclidean space, the standard scalar product (called the dot product) is given by:. $\left(\sum_{i=1}^n \alpha_i e_i \right) \cdot \left(\sum_{i=1}^n \beta_i e_i \right) = \sum_{i=1}^n \alpha_i\,\beta_i$. ### Cross product in 3-dimensional space. The cross product of two vectors in 3-dimensions is a vector perpendicular to the two factors, with length equal to the area of the parallelogram spanned by the two factors.. The cross product can also be expressed as the formala determinant:. $\mathbf{u\times v}=\begin{vmatrix} \mathbf{i}&\mathbf{j}&\mathbf{k}\\ u_1&u_2&u_3\\ v_1&v_2&v_3\\ \end{vmatrix}$. ### Product of two matrices. Given two matrices. $A = (a_{i,j})_{i=1\ldots s;j=1\ldots r} \in \R^{s\times r}$ and $B = (b_{j,k})_{j=1\ldots r;k=1\ldots t}\in \R^{r\times t}$. their product is given by. $B \cdot A = \left( \sum_{j=1}^r a_{i,j} \cdot b_{j,k} \right)_{i=1\ldots s;k=1\ldots t} \;\in\R^{s\times t}$. ## Set theoretical product. In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.. ## Empty product. The empty product has the value of 1 (the identity element of multiplication) just like the empty sum has the value of 0 (the identity element of addition).. ## Other products. Many different kinds of products are studied in mathematics:.
http://www.battlerecharge.org/index-394.html	1,670,408,255,000,000,000	text/html	crawl-data/CC-MAIN-2022-49/segments/1669446711151.22/warc/CC-MAIN-20221207085208-20221207115208-00664.warc.gz	52,765,385	32,304	# Find a Mother Vertex in a Graph • Difficulty Level : Medium • Last Updated : 04 Sep, 2022 What is a Mother Vertex? A mother vertex in a graph G = (V, E) is a vertex v such that all other vertices in G can be reached by a path from v. Example: ```Input : Below Graph Output : 5``` There can be more than one mother vertices in a graph. We need to output anyone of them. For example, in the below graph, vertices 0, 1, and 2 are mother vertices. We strongly recommend you to minimize your browser and try this yourself first. How to find mother vertex? • Case 1:- Undirected Connected Graph : In this case, all the vertices are mother vertices as we can reach to all the other nodes in the graph. • Case 2:- Undirected/Directed Disconnected Graph : In this case, there is no mother vertices as we cannot reach to all the other nodes in the graph. • Case 3:- Directed Connected Graph : In this case, we have to find a vertex -v in the graph such that we can reach to all the other nodes in the graph through a directed path. A Naive approach : A trivial approach will be to perform a DFS/BFS on all the vertices and find whether we can reach all the vertices from that vertex. This approach takes O(V(E+V)) time, which is very inefficient for large graphs. Can we do better? We can find a mother vertex in O(V+E) time. The idea is based on Kosaraju’s Strongly Connected Component Algorithm. In a graph of strongly connected components, mother vertices are always vertices of source component in component graph. The idea is based on below fact. If there exist mother vertex (or vertices), then one of the mother vertices is the last finished vertex in DFS. (Or a mother vertex has the maximum finish time in DFS traversal). A vertex is said to be finished in DFS if a recursive call for its DFS is over, i.e., all descendants of the vertex have been visited. How does the above idea work? Let the last finished vertex be v. Basically, we need to prove that there cannot be an edge from another vertex u to v if u is not another mother vertex (Or there cannot exist a non-mother vertex u such that u-â†’v is an edge). There can be two possibilities. 1. Recursive DFS call is made for u before v. If an edge u-â†’v exists, then v must have finished before u because v is reachable through u and a vertex finishes after all its descendants. 2. Recursive DFS call is made for v before u. In this case also, if an edge u-â†’v exists, then either v must finish before u (which contradicts our assumption that v is finished at the end) OR u should be reachable from v (which means u is another mother vertex). Algorithm : 1. Do DFS traversal of the given graph. While doing traversal keep track of last finished vertex ‘v’. This step takes O(V+E) time. 2. If there exist mother vertex (or vertices), then v must be one (or one of them). Check if v is a mother vertex by doing DFS/BFS from v. This step also takes O(V+E) time. Below is the implementation of above algorithm. ## C++ `// C++ program to find a mother vertex in O(V+E) time``#include ``using` `namespace` `std;` `class` `Graph``{`` ``int` `V; ``// No. of vertices`` ``list<``int``> adj; ``// adjacency lists` ` ``// A recursive function to print DFS starting from v`` ``void` `DFSUtil(``int` `v, vector<``bool``> &visited);``public``:`` ``Graph(``int` `V);`` ``void` `addEdge(``int` `v, ``int` `w);`` ``int` `findMother();``};` `Graph::Graph(``int` `V)``{`` ``this``->V = V;`` ``adj = ``new` `list<``int``>[V];``}` `// A recursive function to print DFS starting from v``void` `Graph::DFSUtil(``int` `v, vector<``bool``> &visited)``{`` ``// Mark the current node as visited and print it`` ``visited[v] = ``true``;` ` ``// Recur for all the vertices adjacent to this vertex`` ``list<``int``>::iterator i;`` ``for` `(i = adj[v].begin(); i != adj[v].end(); ++i)`` ``if` `(!visited[i])`` ``DFSUtil(i, visited);``}` `void` `Graph::addEdge(``int` `v, ``int` `w)``{`` ``adj[v].push_back(w); ``// Add w to vâ€™s list.``}` `// Returns a mother vertex if exists. Otherwise returns -1``int` `Graph::findMother()``{`` ``// visited[] is used for DFS. Initially all are`` ``// initialized as not visited`` ``vector <``bool``> visited(V, ``false``);` ` ``// To store last finished vertex (or mother vertex)`` ``int` `v = 0;` ` ``// Do a DFS traversal and find the last finished`` ``// vertex `` ``for` `(``int` `i = 0; i < V; i++)`` ``{`` ``if` `(visited[i] == ``false``)`` ``{`` ``DFSUtil(i, visited);`` ``v = i;`` ``}`` ``}` ` ``// If there exist mother vertex (or vertices) in given`` ``// graph, then v must be one (or one of them)` ` ``// Now check if v is actually a mother vertex (or graph`` ``// has a mother vertex). We basically check if every vertex`` ``// is reachable from v or not.` ` ``// Reset all values in visited[] as false and do`` ``// DFS beginning from v to check if all vertices are`` ``// reachable from it or not.`` ``fill(visited.begin(), visited.end(), ``false``);`` ``DFSUtil(v, visited);`` ``for` `(``int` `i=0; i ## Java `// Java program to find a mother``// vertex in O(V+E) time``import` `java.util.;` `class` `GFG{`` ` `static` `void` `addEdge(``int` `u, ``int` `v,`` ``ArrayList> adj)``{`` ``adj.get(u).add(v);``}` `// A recursive function to print DFS starting from v``static` `void` `DFSUtil(ArrayList> g,`` ``int` `v, ``boolean``[] visited)``{`` ``// Mark the current node as`` ``// visited and print it`` ``visited[v] = ``true``;`` ` ` ``// Recur for all the vertices`` ``// adjacent to this vertex`` ``for``(``int` `x : g.get(v))`` ``{`` ``if` `(!visited[x])`` ``{`` ``DFSUtil(g, x, visited);`` ``}`` ``}``}` `// Returns a mother vertex if exists.``// Otherwise returns -1``static` `int` `motherVertex(ArrayList>g,`` ``int` `V)``{`` ` ` ``// visited[] is used for DFS. Initially`` ``// all are initialized as not visited`` ``boolean``[] visited = ``new` `boolean``[V];`` ` ` ``// To store last finished vertex`` ``// (or mother vertex)`` ``int` `v = -``1``;`` ` ` ``for``(``int` `i = ``0``; i < V; i++)`` ``{`` ``if` `(!visited[i])`` ``{`` ``DFSUtil(g, i, visited);`` ``v = i;`` ``}`` ``}`` ` ` ``// If there exist mother vertex (or vertices)`` ``// in given graph, then v must be one`` ``// (or one of them)`` ` ` ``// Now check if v is actually a mother`` ``// vertex (or graph has a mother vertex).`` ``// We basically check if every vertex`` ``// is reachable from v or not.`` ` ` ``// Reset all values in visited[] as false`` ``// and do DFS beginning from v to check`` ``// if all vertices are reachable from`` ``// it or not.`` ``boolean``[] check = ``new` `boolean``[V];`` ``DFSUtil(g, v, check);`` ``for``(``boolean` `val : check)`` ``{`` ``if` `(!val)`` ``{`` ``return` `-``1``;`` ``}`` ``}`` ``return` `v;``}` `// Driver code`` ``public` `static` `void` `main(String[] args)`` ``{`` ``int` `V = ``7``;`` ``int` `E = ``8``;`` ` ` ``ArrayList<`` ``ArrayList> adj = ``new` `ArrayList<`` ``ArrayList>();`` ``for``(``int` `i = ``0``; i < V; i++)`` ``{`` ``adj.add(``new` `ArrayList());`` ``}`` ``addEdge(``0``, ``1``,adj);`` ``addEdge(``0``, ``2``,adj);`` ``addEdge(``1``, ``3``,adj);`` ``addEdge(``4``, ``1``,adj);`` ``addEdge(``6``, ``4``,adj);`` ``addEdge(``5``, ``6``,adj);`` ``addEdge(``5``, ``2``,adj);`` ``addEdge(``6``, ``0``,adj);`` ` ` ``System.out.println(``"The mother vertex is "` `+`` ``motherVertex(adj, V));``}``}` `// This code is contributed by Tanay Shah` ## Python3 `# program to find a mother vertex in O(V+E) time``from` `collections ``import` `defaultdict` `# This class represents a directed graph using adjacency list``# representation``class` `Graph:` ` ``def` `__init__(``self``,vertices):`` ``self``.V ``=` `vertices ``#No. of vertices`` ``self``.graph ``=` `defaultdict(``list``) ``# default dictionary` ` ``# A recursive function to print DFS starting from v`` ``def` `DFSUtil(``self``, v, visited):` ` ``# Mark the current node as visited and print it`` ``visited[v] ``=` `True` ` ``# Recur for all the vertices adjacent to this vertex`` ``for` `i ``in` `self``.graph[v]:`` ``if` `visited[i] ``=``=` `False``:`` ``self``.DFSUtil(i, visited)` ` ``# Add w to the list of v`` ``def` `addEdge(``self``, v, w):`` ``self``.graph[v].append(w)` ` ``# Returns a mother vertex if exists. Otherwise returns -1`` ``def` `findMother(``self``):` ` ``# visited[] is used for DFS. Initially all are`` ``# initialized as not visited`` ``visited ``=``[``False``]````(``self``.V)` ` ``# To store last finished vertex (or mother vertex)`` ``v``=``0` ` ``# Do a DFS traversal and find the last finished`` ``# vertex`` ``for` `i ``in` `range``(``self``.V):`` ``if` `visited[i]``=``=``False``:`` ``self``.DFSUtil(i,visited)`` ``v ``=` `i` ` ``# If there exist mother vertex (or vertices) in given`` ``# graph, then v must be one (or one of them)` ` ``# Now check if v is actually a mother vertex (or graph`` ``# has a mother vertex). We basically check if every vertex`` ``# is reachable from v or not.` ` ``# Reset all values in visited[] as false and do`` ``# DFS beginning from v to check if all vertices are`` ``# reachable from it or not.`` ``visited ``=` `[``False``]````(``self``.V)`` ``self``.DFSUtil(v, visited)`` ``if` `any``(i ``=``=` `False` `for` `i ``in` `visited):`` ``return` `-``1`` ``else``:`` ``return` `v` `# Create a graph given in the above diagram``g ``=` `Graph(``7``)``g.addEdge(``0``, ``1``)``g.addEdge(``0``, ``2``)``g.addEdge(``1``, ``3``)``g.addEdge(``4``, ``1``)``g.addEdge(``6``, ``4``)``g.addEdge(``5``, ``6``)``g.addEdge(``5``, ``2``)``g.addEdge(``6``, ``0``)``print` `(``"A mother vertex is "` `+` `str``(g.findMother()))` `# This code is contributed by Neelam Yadav` ## C# `// C# program to find a mother``// vertex in O(V+E) time``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{`` ` `static` `void` `addEdge(``int` `u, ``int` `v,`` ``List> adj)``{`` ``adj[u].Add(v);``}` `// A recursive function to print DFS starting from v``static` `void` `DFSUtil(List> g,`` ``int` `v, ``bool``[] visited)``{`` ``// Mark the current node as`` ``// visited and print it`` ``visited[v] = ``true``;`` ` ` ``// Recur for all the vertices`` ``// adjacent to this vertex`` ``foreach``(``int` `x ``in` `g[v])`` ``{`` ``if` `(!visited[x])`` ``{`` ``DFSUtil(g, x, visited);`` ``}`` ``}``}` `// Returns a mother vertex if exists.``// Otherwise returns -1``static` `int` `motherVertex(List>g,`` ``int` `V)``{`` ` ` ``// visited[] is used for DFS. Initially`` ``// all are initialized as not visited`` ``bool``[] visited = ``new` `bool``[V];`` ` ` ``// To store last finished vertex`` ``// (or mother vertex)`` ``int` `v = -1; `` ``for``(``int` `i = 0; i < V; i++)`` ``{`` ``if` `(!visited[i])`` ``{`` ``DFSUtil(g, i, visited);`` ``v = i;`` ``}`` ``}`` ` ` ``// If there exist mother vertex (or vertices)`` ``// in given graph, then v must be one`` ``// (or one of them)`` ` ` ``// Now check if v is actually a mother`` ``// vertex (or graph has a mother vertex).`` ``// We basically check if every vertex`` ``// is reachable from v or not.`` ` ` ``// Reset all values in visited[] as false`` ``// and do DFS beginning from v to check`` ``// if all vertices are reachable from`` ``// it or not.`` ``bool``[] check = ``new` `bool``[V];`` ``DFSUtil(g, v, check);`` ``foreach``(``bool` `val ``in` `check)`` ``{`` ``if` `(!val)`` ``{`` ``return` `-1;`` ``}`` ``}`` ``return` `v;``}` `// Driver code`` ``public` `static` `void` `Main(String[] args)`` ``{`` ``int` `V = 7;`` ``int` `E = 8; `` ``List<`` ``List<``int``>> adj = ``new` `List>();`` ``for``(``int` `i = 0; i < V; i++)`` ``{`` ``adj.Add(``new` `List<``int``>());`` ``}`` ``addEdge(0, 1,adj);`` ``addEdge(0, 2,adj);`` ``addEdge(1, 3,adj);`` ``addEdge(4, 1,adj);`` ``addEdge(6, 4,adj);`` ``addEdge(5, 6,adj);`` ``addEdge(5, 2,adj);`` ``addEdge(6, 0,adj);`` ` ` ``Console.WriteLine(``"The mother vertex is "` `+`` ``motherVertex(adj, V));``}``}` `// This code is contributed by Rajput-Ji` ## Javascript `` Output `A mother vertex is 5` Time Complexity : O(V + E)^2	4,308	13,199	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	4.09375	4	CC-MAIN-2022-49	latest	en	0.945077	# Find a Mother Vertex in a Graph. • Difficulty Level : Medium. • Last Updated : 04 Sep, 2022. What is a Mother Vertex?. A mother vertex in a graph G = (V, E) is a vertex v such that all other vertices in G can be reached by a path from v.. Example:. ```Input : Below Graph. Output : 5```. There can be more than one mother vertices in a graph. We need to output anyone of them. For example, in the below graph, vertices 0, 1, and 2 are mother vertices.. We strongly recommend you to minimize your browser and try this yourself first.. How to find mother vertex?. • Case 1:- Undirected Connected Graph : In this case, all the vertices are mother vertices as we can reach to all the other nodes in the graph.. • Case 2:- Undirected/Directed Disconnected Graph : In this case, there is no mother vertices as we cannot reach to all the other nodes in the graph.. • Case 3:- Directed Connected Graph : In this case, we have to find a vertex -v in the graph such that we can reach to all the other nodes in the graph through a directed path.. A Naive approach :. A trivial approach will be to perform a DFS/BFS on all the vertices and find whether we can reach all the vertices from that vertex. This approach takes O(V(E+V)) time, which is very inefficient for large graphs.. Can we do better?. We can find a mother vertex in O(V+E) time. The idea is based on Kosaraju’s Strongly Connected Component Algorithm. In a graph of strongly connected components, mother vertices are always vertices of source component in component graph. The idea is based on below fact.. If there exist mother vertex (or vertices), then one of the mother vertices is the last finished vertex in DFS. (Or a mother vertex has the maximum finish time in DFS traversal).. A vertex is said to be finished in DFS if a recursive call for its DFS is over, i.e., all descendants of the vertex have been visited.. How does the above idea work?. Let the last finished vertex be v. Basically, we need to prove that there cannot be an edge from another vertex u to v if u is not another mother vertex (Or there cannot exist a non-mother vertex u such that u-â†’v is an edge). There can be two possibilities.. 1. Recursive DFS call is made for u before v. If an edge u-â†’v exists, then v must have finished before u because v is reachable through u and a vertex finishes after all its descendants.	2. Recursive DFS call is made for v before u. In this case also, if an edge u-â†’v exists, then either v must finish before u (which contradicts our assumption that v is finished at the end) OR u should be reachable from v (which means u is another mother vertex).. Algorithm :. 1. Do DFS traversal of the given graph. While doing traversal keep track of last finished vertex ‘v’. This step takes O(V+E) time.. 2. If there exist mother vertex (or vertices), then v must be one (or one of them). Check if v is a mother vertex by doing DFS/BFS from v. This step also takes O(V+E) time.. Below is the implementation of above algorithm.. ## C++. `// C++ program to find a mother vertex in O(V+E) time``#include ``using` `namespace` `std;` `class` `Graph``{`` ``int` `V; ``// No. of vertices`` ``list<``int``> adj; ``// adjacency lists` ` ``// A recursive function to print DFS starting from v`` ``void` `DFSUtil(``int` `v, vector<``bool``> &visited);``public``:`` ``Graph(``int` `V);`` ``void` `addEdge(``int` `v, ``int` `w);`` ``int` `findMother();``};` `Graph::Graph(``int` `V)``{`` ``this``->V = V;`` ``adj = ``new` `list<``int``>[V];``}` `// A recursive function to print DFS starting from v``void` `Graph::DFSUtil(``int` `v, vector<``bool``> &visited)``{`` ``// Mark the current node as visited and print it`` ``visited[v] = ``true``;` ` ``// Recur for all the vertices adjacent to this vertex`` ``list<``int``>::iterator i;`` ``for` `(i = adj[v].begin(); i != adj[v].end(); ++i)`` ``if` `(!visited[i])`` ``DFSUtil(i, visited);``}` `void` `Graph::addEdge(``int` `v, ``int` `w)``{`` ``adj[v].push_back(w); ``// Add w to vâ€™s list.``}` `// Returns a mother vertex if exists. Otherwise returns -1``int` `Graph::findMother()``{`` ``// visited[] is used for DFS. Initially all are`` ``// initialized as not visited`` ``vector <``bool``> visited(V, ``false``);` ` ``// To store last finished vertex (or mother vertex)`` ``int` `v = 0;` ` ``// Do a DFS traversal and find the last finished`` ``// vertex `` ``for` `(``int` `i = 0; i < V; i++)`` ``{`` ``if` `(visited[i] == ``false``)`` ``{`` ``DFSUtil(i, visited);`` ``v = i;`` ``}`` ``}` ` ``// If there exist mother vertex (or vertices) in given`` ``// graph, then v must be one (or one of them)` ` ``// Now check if v is actually a mother vertex (or graph`` ``// has a mother vertex). We basically check if every vertex`` ``// is reachable from v or not.` ` ``// Reset all values in visited[] as false and do`` ``// DFS beginning from v to check if all vertices are`` ``// reachable from it or not.`` ``fill(visited.begin(), visited.end(), ``false``);`` ``DFSUtil(v, visited);`` ``for` `(``int` `i=0; i. ## Java. `// Java program to find a mother``// vertex in O(V+E) time``import` `java.util.;` `class` `GFG{`` ` `static` `void` `addEdge(``int` `u, ``int` `v,`` ``ArrayList> adj)``{`` ``adj.get(u).add(v);``}` `// A recursive function to print DFS starting from v``static` `void` `DFSUtil(ArrayList> g,`` ``int` `v, ``boolean``[] visited)``{`` ``// Mark the current node as`` ``// visited and print it`` ``visited[v] = ``true``;`` ` ` ``// Recur for all the vertices`` ``// adjacent to this vertex`` ``for``(``int` `x : g.get(v))`` ``{`` ``if` `(!visited[x])`` ``{`` ``DFSUtil(g, x, visited);`` ``}`` ``}``}` `// Returns a mother vertex if exists.``// Otherwise returns -1``static` `int` `motherVertex(ArrayList>g,`` ``int` `V)``{`` ` ` ``// visited[] is used for DFS. Initially`` ``// all are initialized as not visited`` ``boolean``[] visited = ``new` `boolean``[V];`` ` ` ``// To store last finished vertex`` ``// (or mother vertex)`` ``int` `v = -``1``;`` ` ` ``for``(``int` `i = ``0``; i < V; i++)`` ``{`` ``if` `(!visited[i])`` ``{`` ``DFSUtil(g, i, visited);`` ``v = i;`` ``}`` ``}`` ` ` ``// If there exist mother vertex (or vertices)`` ``// in given graph, then v must be one`` ``// (or one of them)`` ` ` ``// Now check if v is actually a mother`` ``// vertex (or graph has a mother vertex).`` ``// We basically check if every vertex`` ``// is reachable from v or not.`` ` ` ``// Reset all values in visited[] as false`` ``// and do DFS beginning from v to check`` ``// if all vertices are reachable from`` ``// it or not.`` ``boolean``[] check = ``new` `boolean``[V];`` ``DFSUtil(g, v, check);`` ``for``(``boolean` `val : check)`` ``{`` ``if` `(!val)`` ``{`` ``return` `-``1``;`` ``}`` ``}`` ``return` `v;``}` `// Driver code`` ``public` `static` `void` `main(String[] args)`` ``{`` ``int` `V = ``7``;`` ``int` `E = ``8``;`` ` ` ``ArrayList<`` ``ArrayList> adj = ``new` `ArrayList<`` ``ArrayList>();`` ``for``(``int` `i = ``0``; i < V; i++)`` ``{`` ``adj.add(``new` `ArrayList());`` ``}`` ``addEdge(``0``, ``1``,adj);`` ``addEdge(``0``, ``2``,adj);`` ``addEdge(``1``, ``3``,adj);`` ``addEdge(``4``, ``1``,adj);`` ``addEdge(``6``, ``4``,adj);`` ``addEdge(``5``, ``6``,adj);`` ``addEdge(``5``, ``2``,adj);`` ``addEdge(``6``, ``0``,adj);`` ` ` ``System.out.println(``"The mother vertex is "` `+`` ``motherVertex(adj, V));``}``}` `// This code is contributed by Tanay Shah`. ## Python3. `# program to find a mother vertex in O(V+E) time``from` `collections ``import` `defaultdict` `# This class represents a directed graph using adjacency list``# representation``class` `Graph:` ` ``def` `__init__(``self``,vertices):`` ``self``.V ``=` `vertices ``#No. of vertices`` ``self``.graph ``=` `defaultdict(``list``) ``# default dictionary` ` ``# A recursive function to print DFS starting from v`` ``def` `DFSUtil(``self``, v, visited):` ` ``# Mark the current node as visited and print it`` ``visited[v] ``=` `True` ` ``# Recur for all the vertices adjacent to this vertex`` ``for` `i ``in` `self``.graph[v]:`` ``if` `visited[i] ``=``=` `False``:`` ``self``.DFSUtil(i, visited)` ` ``# Add w to the list of v`` ``def` `addEdge(``self``, v, w):`` ``self``.graph[v].append(w)` ` ``# Returns a mother vertex if exists. Otherwise returns -1`` ``def` `findMother(``self``):` ` ``# visited[] is used for DFS. Initially all are`` ``# initialized as not visited`` ``visited ``=``[``False``]````(``self``.V)` ` ``# To store last finished vertex (or mother vertex)`` ``v``=``0` ` ``# Do a DFS traversal and find the last finished`` ``# vertex`` ``for` `i ``in` `range``(``self``.V):`` ``if` `visited[i]``=``=``False``:`` ``self``.DFSUtil(i,visited)`` ``v ``=` `i` ` ``# If there exist mother vertex (or vertices) in given`` ``# graph, then v must be one (or one of them)` ` ``# Now check if v is actually a mother vertex (or graph`` ``# has a mother vertex). We basically check if every vertex`` ``# is reachable from v or not.` ` ``# Reset all values in visited[] as false and do`` ``# DFS beginning from v to check if all vertices are`` ``# reachable from it or not.`` ``visited ``=` `[``False``]````(``self``.V)`` ``self``.DFSUtil(v, visited)`` ``if` `any``(i ``=``=` `False` `for` `i ``in` `visited):`` ``return` `-``1`` ``else``:`` ``return` `v` `# Create a graph given in the above diagram``g ``=` `Graph(``7``)``g.addEdge(``0``, ``1``)``g.addEdge(``0``, ``2``)``g.addEdge(``1``, ``3``)``g.addEdge(``4``, ``1``)``g.addEdge(``6``, ``4``)``g.addEdge(``5``, ``6``)``g.addEdge(``5``, ``2``)``g.addEdge(``6``, ``0``)``print` `(``"A mother vertex is "` `+` `str``(g.findMother()))` `# This code is contributed by Neelam Yadav`. ## C#. `// C# program to find a mother``// vertex in O(V+E) time``using` `System;``using` `System.Collections.Generic;` `class` `GFG``{`` ` `static` `void` `addEdge(``int` `u, ``int` `v,`` ``List> adj)``{`` ``adj[u].Add(v);``}` `// A recursive function to print DFS starting from v``static` `void` `DFSUtil(List> g,`` ``int` `v, ``bool``[] visited)``{`` ``// Mark the current node as`` ``// visited and print it`` ``visited[v] = ``true``;`` ` ` ``// Recur for all the vertices`` ``// adjacent to this vertex`` ``foreach``(``int` `x ``in` `g[v])`` ``{`` ``if` `(!visited[x])`` ``{`` ``DFSUtil(g, x, visited);`` ``}`` ``}``}` `// Returns a mother vertex if exists.``// Otherwise returns -1``static` `int` `motherVertex(List>g,`` ``int` `V)``{`` ` ` ``// visited[] is used for DFS. Initially`` ``// all are initialized as not visited`` ``bool``[] visited = ``new` `bool``[V];`` ` ` ``// To store last finished vertex`` ``// (or mother vertex)`` ``int` `v = -1; `` ``for``(``int` `i = 0; i < V; i++)`` ``{`` ``if` `(!visited[i])`` ``{`` ``DFSUtil(g, i, visited);`` ``v = i;`` ``}`` ``}`` ` ` ``// If there exist mother vertex (or vertices)`` ``// in given graph, then v must be one`` ``// (or one of them)`` ` ` ``// Now check if v is actually a mother`` ``// vertex (or graph has a mother vertex).`` ``// We basically check if every vertex`` ``// is reachable from v or not.`` ` ` ``// Reset all values in visited[] as false`` ``// and do DFS beginning from v to check`` ``// if all vertices are reachable from`` ``// it or not.`` ``bool``[] check = ``new` `bool``[V];`` ``DFSUtil(g, v, check);`` ``foreach``(``bool` `val ``in` `check)`` ``{`` ``if` `(!val)`` ``{`` ``return` `-1;`` ``}`` ``}`` ``return` `v;``}` `// Driver code`` ``public` `static` `void` `Main(String[] args)`` ``{`` ``int` `V = 7;`` ``int` `E = 8; `` ``List<`` ``List<``int``>> adj = ``new` `List>();`` ``for``(``int` `i = 0; i < V; i++)`` ``{`` ``adj.Add(``new` `List<``int``>());`` ``}`` ``addEdge(0, 1,adj);`` ``addEdge(0, 2,adj);`` ``addEdge(1, 3,adj);`` ``addEdge(4, 1,adj);`` ``addEdge(6, 4,adj);`` ``addEdge(5, 6,adj);`` ``addEdge(5, 2,adj);`` ``addEdge(6, 0,adj);`` ` ` ``Console.WriteLine(``"The mother vertex is "` `+`` ``motherVertex(adj, V));``}``}` `// This code is contributed by Rajput-Ji`. ## Javascript. ``. Output. `A mother vertex is 5`. Time Complexity : O(V + E)^2.
https://web2.0calc.com/questions/help_19510	1,618,816,058,000,000,000	text/html	crawl-data/CC-MAIN-2021-17/segments/1618038878326.67/warc/CC-MAIN-20210419045820-20210419075820-00327.warc.gz	653,506,398	5,581	+0 # Help 0 37 1 Consider the following. f(x) = sqrt(x + 5), (−1, 2) (a) Find the slope of the graph of f at the given point. Mar 23, 2021 #1 +117787 +1 f(x) = ( x +5)^(1/2) f ' (x) = ( 1/2) (x + 5)^(-1/2) Slope at (-1,2) = (1/2) (-1 + 5)^(-1/2) = (1/2) (4)^(-1/2) = (1/2) (1/2) = 1 / 4 Mar 23, 2021	186	328	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.796875	4	CC-MAIN-2021-17	longest	en	0.35318	+0. # Help. 0. 37. 1. Consider the following.. f(x) = sqrt(x + 5), (−1, 2). (a) Find the slope of the graph of f at the given point.. Mar 23, 2021.	#1. +117787. +1. f(x) = ( x +5)^(1/2). f ' (x) = ( 1/2) (x + 5)^(-1/2). Slope at (-1,2) = (1/2) (-1 + 5)^(-1/2) = (1/2) (4)^(-1/2) = (1/2) (1/2) = 1 / 4. Mar 23, 2021.
https://wiki.haskell.org/index.php?title=Euler_problems/11_to_20&diff=43329&oldid=12281	1,498,726,245,000,000,000	text/html	crawl-data/CC-MAIN-2017-26/segments/1498128323889.3/warc/CC-MAIN-20170629070237-20170629090237-00213.warc.gz	840,207,315	14,576	# Euler problems/11 to 20 (Difference between revisions) ## 1 Problem 11 What is the greatest product of four numbers on the same straight line in the 20 by 20 grid? Solution: using Array and Arrows, for fun : ```import Control.Arrow import Data.Array input :: String -> Array (Int,Int) Int input = listArray ((1,1),(20,20)) . map read . words senses = [(+1) * id,(+1) * (+1), id * (+1), (+1) * (\n -> n - 1)] inArray a i = inRange (bounds a) i prods :: Array (Int, Int) Int -> [Int] prods a = [product xs \| i <- range \$ bounds a, s <- senses, let is = take 4 \$ iterate s i, all (inArray a) is, let xs = map (a!) is] main = print . maximum . prods . input =<< getContents``` ## 2 Problem 12 What is the first triangle number to have over five-hundred divisors? Solution: ```--primeFactors in problem_3 problem_12 = head \$ filter ((> 500) . nDivisors) triangleNumbers where nDivisors n = product \$ map ((+1) . length) (group (primeFactors n)) triangleNumbers = scanl1 (+) [1..]``` ## 3 Problem 13 Find the first ten digits of the sum of one-hundred 50-digit numbers. Solution: ```main = do xs <- fmap (map read . lines) (readFile "p13.log") print . take 10 . show . sum \$ xs``` ## 4 Problem 14 Find the longest sequence using a starting number under one million. Solution: ``` import Data.List problem_14 = j 1000000 where f :: Int -> Integer -> Int f k 1 = k f k n = f (k+1) \$ if even n then div n 2 else 3n + 1 g x y = if snd x < snd y then y else x h x n = g x (n, f 1 n) j n = fst \$ foldl' h (1,1) [2..n-1]``` Faster solution, using unboxed types and parallel computation: ```import Control.Parallel import Data.Word collatzLen :: Int -> Word32 -> Int collatzLen c 1 = c collatzLen c n = collatzLen (c+1) \$ if n `mod` 2 == 0 then n `div` 2 else 3n+1 pmax x n = x `max` (collatzLen 1 n, n) solve xs = foldl pmax (1,1) xs main = print soln where s1 = solve [2..500000] s2 = solve [500001..1000000] soln = s2 `par` (s1 `pseq` max s1 s2)``` Even faster solution, using an Array to memoize length of sequences : ```import Data.Array import Data.List import Data.Ord (comparing) syrs n = a where a = listArray (1,n) \$ 0:[1 + syr n x \| x <- [2..n]] syr n x = if x' <= n then a ! x' else 1 + syr n x' where x' = if even x then x `div` 2 else 3 * x + 1 main = print \$ maximumBy (comparing snd) \$ assocs \$ syrs 1000000``` ## 5 Problem 15 Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner? Solution: A direct computation: ``` problem_15 = iterate (scanl1 (+)) (repeat 1) !! 20 !! 20``` Thinking about it as a problem in combinatorics: Each route has exactly 40 steps, with 20 of them horizontal and 20 of them vertical. We need to count how many different ways there are of choosing which steps are horizontal and which are vertical. So we have: `problem_15 = product [21..40] `div` product [2..20]` ## 6 Problem 16 What is the sum of the digits of the number 21000? Solution: ```import Data.Char problem_16 = sum k where s = show (2^1000) k = map digitToInt s``` ## 7 Problem 17 How many letters would be needed to write all the numbers in words from 1 to 1000? Solution: ```import Char one = ["one","two","three","four","five","six","seven","eight", "nine","ten","eleven","twelve","thirteen","fourteen","fifteen", "sixteen","seventeen","eighteen", "nineteen"] ty = ["twenty","thirty","forty","fifty","sixty","seventy","eighty","ninety"] decompose x \| x == 0 = [] \| x < 20 = one !! (x-1) \| x >= 20 && x < 100 = ty !! (firstDigit (x) - 2) ++ decompose ( x - firstDigit (x) * 10) \| x < 1000 && x `mod` 100 ==0 = one !! (firstDigit (x)-1) ++ "hundred" \| x > 100 && x <= 999 = one !! (firstDigit (x)-1) ++ "hundredand" ++decompose ( x - firstDigit (x) * 100) \| x == 1000 = "onethousand" where firstDigit x = digitToInt . head . show \$ x problem_17 = length . concatMap decompose \$ [1..1000]``` ## 8 Problem 18 Find the maximum sum travelling from the top of the triangle to the base. Solution: ```problem_18 = head \$ foldr1 g tri where f x y z = x + max y z g xs ys = zipWith3 f xs ys \$ tail ys tri = [ [75], [95,64], [17,47,82], [18,35,87,10], [20,04,82,47,65], [19,01,23,75,03,34], [88,02,77,73,07,63,67], [99,65,04,28,06,16,70,92], [41,41,26,56,83,40,80,70,33], [41,48,72,33,47,32,37,16,94,29], [53,71,44,65,25,43,91,52,97,51,14], [70,11,33,28,77,73,17,78,39,68,17,57], [91,71,52,38,17,14,91,43,58,50,27,29,48], [63,66,04,68,89,53,67,30,73,16,69,87,40,31], [04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]]``` ## 9 Problem 19 You are given the following information, but you may prefer to do some research for yourself. • 1 Jan 1900 was a Monday. • Thirty days has September, • April, June and November. • All the rest have thirty-one, • Saving February alone, Which has twenty-eight, rain or shine. And on leap years, twenty-nine. • A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400. How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)? Solution: ```problem_19 = length . filter (== sunday) . drop 12 . take 1212 \$ since1900 since1900 = scanl nextMonth monday . concat \$ replicate 4 nonLeap ++ cycle (leap : replicate 3 nonLeap) nonLeap = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31] leap = 31 : 29 : drop 2 nonLeap nextMonth x y = (x + y) `mod` 7 sunday = 0 monday = 1``` Here is an alternative that is simpler, but it is cheating a bit: ```import Data.Time.Calendar import Data.Time.Calendar.WeekDate problem_19_v2 = length [() \| y <- [1901..2000], m <- [1..12], let (_, _, d) = toWeekDate \$ fromGregorian y m 1, d == 7]``` ## 10 Problem 20 Find the sum of digits in 100! Solution: `problem_20 = sum \$ map Char.digitToInt \$ show \$ product [1..100]`	2,027	5,898	{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}	3.65625	4	CC-MAIN-2017-26	latest	en	0.67316	# Euler problems/11 to 20. (Difference between revisions). ## 1 Problem 11. What is the greatest product of four numbers on the same straight line in the 20 by 20 grid?. Solution: using Array and Arrows, for fun :. ```import Control.Arrow. import Data.Array. input :: String -> Array (Int,Int) Int. input = listArray ((1,1),(20,20)) . map read . words. senses = [(+1) * id,(+1) * (+1), id * (+1), (+1) * (\n -> n - 1)]. inArray a i = inRange (bounds a) i. prods :: Array (Int, Int) Int -> [Int]. prods a = [product xs \| i <- range \$ bounds a,. s <- senses,. let is = take 4 \$ iterate s i,. all (inArray a) is,. let xs = map (a!) is]. main = print . maximum . prods . input =<< getContents```. ## 2 Problem 12. What is the first triangle number to have over five-hundred divisors?. Solution:. ```--primeFactors in problem_3. problem_12 = head \$ filter ((> 500) . nDivisors) triangleNumbers. where nDivisors n = product \$ map ((+1) . length) (group (primeFactors n)). triangleNumbers = scanl1 (+) [1..]```. ## 3 Problem 13. Find the first ten digits of the sum of one-hundred 50-digit numbers.. Solution:. ```main = do xs <- fmap (map read . lines) (readFile "p13.log"). print . take 10 . show . sum \$ xs```. ## 4 Problem 14. Find the longest sequence using a starting number under one million.. Solution:. ```. import Data.List. problem_14 = j 1000000 where. f :: Int -> Integer -> Int. f k 1 = k. f k n = f (k+1) \$ if even n then div n 2 else 3n + 1. g x y = if snd x < snd y then y else x. h x n = g x (n, f 1 n). j n = fst \$ foldl' h (1,1) [2..n-1]```. Faster solution, using unboxed types and parallel computation:. ```import Control.Parallel. import Data.Word. collatzLen :: Int -> Word32 -> Int. collatzLen c 1 = c. collatzLen c n = collatzLen (c+1) \$ if n `mod` 2 == 0 then n `div` 2 else 3n+1. pmax x n = x `max` (collatzLen 1 n, n). solve xs = foldl pmax (1,1) xs. main = print soln. where. s1 = solve [2..500000]. s2 = solve [500001..1000000]. soln = s2 `par` (s1 `pseq` max s1 s2)```. Even faster solution, using an Array to memoize length of sequences :. ```import Data.Array. import Data.List. import Data.Ord (comparing). syrs n =. a. where. a = listArray (1,n) \$ 0:[1 + syr n x \| x <- [2..n]]. syr n x =. if x' <= n then a ! x' else 1 + syr n x'. where. x' = if even x then x `div` 2 else 3 * x + 1. main =. print \$ maximumBy (comparing snd) \$ assocs \$ syrs 1000000```. ## 5 Problem 15. Starting in the top left corner in a 20 by 20 grid, how many routes are there to the bottom right corner?. Solution: A direct computation:. ```. problem_15 = iterate (scanl1 (+)) (repeat 1) !! 20 !! 20```. Thinking about it as a problem in combinatorics:. Each route has exactly 40 steps, with 20 of them horizontal and 20 of them vertical. We need to count how many different ways there are of choosing which steps are horizontal and which are vertical. So we have:. `problem_15 = product [21..40] `div` product [2..20]`. ## 6 Problem 16. What is the sum of the digits of the number 21000?.	Solution:. ```import Data.Char. problem_16 = sum k. where s = show (2^1000). k = map digitToInt s```. ## 7 Problem 17. How many letters would be needed to write all the numbers in words from 1 to 1000?. Solution:. ```import Char. one = ["one","two","three","four","five","six","seven","eight",. "nine","ten","eleven","twelve","thirteen","fourteen","fifteen",. "sixteen","seventeen","eighteen", "nineteen"]. ty = ["twenty","thirty","forty","fifty","sixty","seventy","eighty","ninety"]. decompose x. \| x == 0 = []. \| x < 20 = one !! (x-1). \| x >= 20 && x < 100 =. ty !! (firstDigit (x) - 2) ++ decompose ( x - firstDigit (x) * 10). \| x < 1000 && x `mod` 100 ==0 =. one !! (firstDigit (x)-1) ++ "hundred". \| x > 100 && x <= 999 =. one !! (firstDigit (x)-1) ++ "hundredand" ++decompose ( x - firstDigit (x) * 100). \| x == 1000 = "onethousand". where firstDigit x = digitToInt . head . show \$ x. problem_17 = length . concatMap decompose \$ [1..1000]```. ## 8 Problem 18. Find the maximum sum travelling from the top of the triangle to the base.. Solution:. ```problem_18 = head \$ foldr1 g tri. where. f x y z = x + max y z. g xs ys = zipWith3 f xs ys \$ tail ys. tri = [. [75],. [95,64],. [17,47,82],. [18,35,87,10],. [20,04,82,47,65],. [19,01,23,75,03,34],. [88,02,77,73,07,63,67],. [99,65,04,28,06,16,70,92],. [41,41,26,56,83,40,80,70,33],. [41,48,72,33,47,32,37,16,94,29],. [53,71,44,65,25,43,91,52,97,51,14],. [70,11,33,28,77,73,17,78,39,68,17,57],. [91,71,52,38,17,14,91,43,58,50,27,29,48],. [63,66,04,68,89,53,67,30,73,16,69,87,40,31],. [04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]]```. ## 9 Problem 19. You are given the following information, but you may prefer to do some research for yourself.. • 1 Jan 1900 was a Monday.. • Thirty days has September,. • April, June and November.. • All the rest have thirty-one,. • Saving February alone,. Which has twenty-eight, rain or shine. And on leap years, twenty-nine.. • A leap year occurs on any year evenly divisible by 4, but not on a century unless it is divisible by 400.. How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?. Solution:. ```problem_19 = length . filter (== sunday) . drop 12 . take 1212 \$ since1900. since1900 = scanl nextMonth monday . concat \$. replicate 4 nonLeap ++ cycle (leap : replicate 3 nonLeap). nonLeap = [31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31]. leap = 31 : 29 : drop 2 nonLeap. nextMonth x y = (x + y) `mod` 7. sunday = 0. monday = 1```. Here is an alternative that is simpler, but it is cheating a bit:. ```import Data.Time.Calendar. import Data.Time.Calendar.WeekDate. problem_19_v2 = length [() \| y <- [1901..2000],. m <- [1..12],. let (_, _, d) = toWeekDate \$ fromGregorian y m 1,. d == 7]```. ## 10 Problem 20. Find the sum of digits in 100!. Solution:. `problem_20 = sum \$ map Char.digitToInt \$ show \$ product [1..100]`.

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