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https://www.askiitians.com/forums/Integral-Calculus/what-are-indefinite-integrals_121692.htm | 1,708,643,565,000,000,000 | text/html | crawl-data/CC-MAIN-2024-10/segments/1707947473871.23/warc/CC-MAIN-20240222225655-20240223015655-00104.warc.gz | 680,634,556 | 42,725 | # what are indefinite integrals
Grade:12th pass
## 1 Answers
grenade
2061 Points
8 years ago
Indefinite integrals are also called general integrals. C is called constant of integration. All these integrals differ by a constant. From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.
## ASK QUESTION
Get your questions answered by the expert for free | 115 | 508 | {"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} | 2.546875 | 3 | CC-MAIN-2024-10 | latest | en | 0.906127 |
https://fr.slideserve.com/chill/wide-field-imaging-i-full-beam-imaging-surveys | 1,627,450,206,000,000,000 | text/html | crawl-data/CC-MAIN-2021-31/segments/1627046153521.1/warc/CC-MAIN-20210728025548-20210728055548-00085.warc.gz | 282,877,994 | 27,728 | Wide-Field Imaging I: Full-Beam Imaging & Surveys
# Wide-Field Imaging I: Full-Beam Imaging & Surveys
## Wide-Field Imaging I: Full-Beam Imaging & Surveys
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##### Presentation Transcript
1. Wide-Field Imaging I: Full-Beam Imaging & Surveys Steven T. Myers (NRAO-Socorro)
2. What is Wide-Field Imaging? • “Narrow-Field” Imaging: • imaging of a single field well within Primary Beam (PB) • able to ignore (orientation dependent) PB effects • able to ignore non-coplanar array (w-term) effects • this will get you far, but sometimes you need… • “Wide-Field” Imaging: • includes non-coplanar array (w-term) effects • includes orientation-dependent (polarized) PB effects • includes mosaciking outside PB • all these more complicated for wide-bandwidth imaging! • these effects limit dynamic range & fidelity even within PB Fourteenth Synthesis Imaging Workshop
3. Outline of this Lecture • The Imaging Equation revisited • The W-Term • The (Polarized) Primary Beam • Mosaicking 101 • Imaging Techniques for Linear Mosaics • Mosaic Sampling – Hex Grids & On-The-Fly • Practical Mosaicking – Observing Preparation • Setting up your Survey Fourteenth Synthesis Imaging Workshop
4. Interferometer Equation • Relates what we measure in a visibility k to what is on the sky and what our antennas and array are doing (no noise): • the visibility index k encapsulates the time, antenna pair, parallactic angle, pointing direction, phase center of the observation for A • To do 2D Fourier transforms between the (l,m) and (u,v) • Write with linear operators: (includes noise term) v = F-1WAs+ n Data Sky Primary Beam Geometry Fourier kernel Linear operator notation: can represent integrals or discrete matrix equations! Fourteenth Synthesis Imaging Workshop
5. From sky to uvplane Visibility vector v is over distinct integrations and channels. Sky image vector s is over pixels on sky. A is not square and is non-invertible! • Visibility equation (subsume W inside A for now): v = F-1As + n • Fourier transformation of vector and matrix operators: s=F-1 sA=F-1AF • Equivalent Fourier (uv) domain equation (insert FF-1): v = As+ n (this is just stating the Fourier convolution theorem) • This is saying that the Fourier transform of the sky (F-1s) is convolved in the uv-plane with the transform of the Primary Beam (including any geometric w-term) Fourteenth Synthesis Imaging Workshop
6. Image Reconstruction • Visibilities and the Sky v = As+ n • A known instrumental response, but is not invertible • true uv-plane convolved by aperture cross correlation A • has finite support (2x dish dia.) in uv-plane (not including w-term) • A has support only where there is data in v • incomplete sampling of uv-plane by visibilities • instrumental noise n is a random variable with covariance N = <nnT> • Maximum Likelihood Estimate (MLE) of sky: sMLE= ( ATN-1A)-1ATN-1v = R-1dd = Hv R =ATN-1AH = ATN-1 R singular (at best ill-conditioned) so no inversion practical Fourteenth Synthesis Imaging Workshop
7. The Dirty Map Use of N-1 here in gridding is “natural weighting”. Other choices give uniform or robust weighting! • Grid onto sampled uv-plane d = Hv = Hs + nd • H should be close to HMLE, e.g. H = ATN-1 : A ~ AN ~ N • AT should sample onto suitable grid in uv-plane • reminder: need only be approximate for gridding = “A-projection” • include w-term geometry A AG = “AW-projection” (later) • Invert onto sky “dirty image” d = Fd = R s + ndR = FRF-1 • image is “dirty” as it contains artifacts • convolution by “point spread function” (columns of R) = PSF dirty beam • multiplication by response function (diagonal of R) = Primary Beam • noise (& calibration errors, etc.) Fourteenth Synthesis Imaging Workshop
8. Imaging Equation Summary • Our best estimation (MLE) of the sky is sMLE= R-1ATN-1v • An intermediate estimate is the dirty image d = F Hv = F H ( F-1A s +n ) = R s + nd • all the data, regardless of where the antennas and array were pointed or phase, goes into this image through H • There is a single uv-plane • can choose the gridding kernel H to optimize image • e.g. H = ATN-1 • R is the known relation between d and s (PSF & PB) • iterative methods (e.g. Cotton-Schwab Clean) perform well You can store R and H for later use! Fourteenth Synthesis Imaging Workshop
9. Wide-field Imaging: W-term • 2D Fourier transform approximation of the imaging equation breaks down due to geometric term (“The W-term problem”) • Imaging dynamic range throughout the image is limited by deconvolution errors due to the sources away from the (phase) center. Without w-term correction With w-term correction (w-projection) For the Student: Hey, these look like diffraction patterns! Is there some sort of diffraction at work here? Fourteenth Synthesis Imaging Workshop
10. Wide-field Imaging: W-term • Deviation from 2D geometry increases with FoV and baseline length. • W-Term: convolution in the imaging equation (IE) by For the Student: Can W be treated as an extra Fourier transform (3D)? If so, is this a practical implementation? Measure of non-2D geometry Fourteenth Synthesis Imaging Workshop
11. W-term: when does it matter? • The geometric term in the IE is: • This is negligible when: • the Field-of-View (FoV) is small: • the array is nearly coplanar: • duration of observations is short • “snapshot” • Rule of thumb: ok when Remember: Earth rotation will cause non-coplanarity of baselines in 2D array! Ref: Chapter 19 D = Dish diameter Bmax = maximum baseline VLA: D=25m VLA configurations: (A) 36km (C) 3.4km (B) 11km (D) 1.0km Example: VLA A-configlmax < 2cm (15GHz) ! Fourteenth Synthesis Imaging Workshop
12. W-term: geometric picture • Phase j of the visibilities for offset angle q • For the interferometer in a plane (left): • For the interferometer not in a plane (right) : • W≈ 1 (coplanar) only when: (1) w «u, or (2) q≈ 0 w u u X X Fourteenth Synthesis Imaging Workshop
13. W-term: optics picture • We want to measure: (in wavefront tangent plane) • We actually measure: (in imaging tangent plane) • Propagate using Fresnel diffraction: • so For the Student: What is the functional form of transform W? Fourteenth Synthesis Imaging Workshop
14. W-term: image-plane faceting • Interpret S(l,m) as emission on the surface of the Celestial Sphere of unit radius: l2+m2+n2=1 • Approximate the celestial sphere by a set of tangent planes – a.k.a. “facets” – such that 2D geometry is valid per facet • Use 2D imaging on each facet • Re-project and stitch the facet-images to a single 2D plane • Number of facets required: Ref: Chapter 19 Example: VLA A-config1.0GHz N>34 f=1 for critical sampling. f<1 for high dynamic range Fourteenth Synthesis Imaging Workshop
15. W-term: uv-plane faceting • Since shifting (rotating axes on sky) and summing image facets is a linear operation, and our Imaging Equation is linear, there must be an equivalent in the uv-plane to faceting: • C = image-plane coordinate transformation • q and u are the image and uv plane coordinates respectively • uv-plane faceting vs. image plane faceting • errors same as in image plane faceting • produces a single image (no edge effects) • global (single plane) deconvolution straightforward • use of advanced algorithms for extended emission possible • can be faster in some implementations Note: uv-plane faceting used in CASA Fourteenth Synthesis Imaging Workshop
16. W-term: example • No correction • W-term introduces a phase error • dependent on distance from center of image • dependent on baseline length and frequency (uv radius) • characteristic arc shapes Fourteenth Synthesis Imaging Workshop
17. W-term: corrected! • Correction applied • using CASA w-projection • 256 w planes Fourteenth Synthesis Imaging Workshop
18. W-term: W-projection • Go back to our Imaging Equation (including W): v= F-1WAs + n • or in the uv domain: v = WAs + n • which implies that we image using: d=HvH =ATWTN-1 (W≈Wusing Hermitian transpose, WTW=1 unitary) • Gridding using the W-kernel (in the uv-plane) is called “W-projection” (see Cornwell, Golap, Bhatnagar EVLA Memo 67). This is an efficient alternative / augmentation to faceting. Ref: Cornwell et al. IEEE Special topics in SP, Vol. 2, No5, 2008 [arXiv:0807.4161] Fourteenth Synthesis Imaging Workshop
19. W-term: W-projection • As W is unitary and WTW≈1 WT and W can be used in the forward and inverse transforms to produce minimally distorted residual images as part of Cotton-Schwab (CS) clean Vo(u,v,w) = V(u,v,0) * W(u,v,w) • Model prediction during major cycle: • compute 2D FFT of model image V(u,v,0) • evaluate above convolution to get Vo(u,v,w) • subtract from visibilites to get residual • Compute dirty residual image for minor cycle: • use WT(u,v,w) on each Vo(u,v,w) on a grid in w, sum to get Vproj(u,v) • this is just a modification of normal gridding CF! • make dirty image with 2D FFT-1 of V(u,v) Radius of Fresnel zone: Fourteenth Synthesis Imaging Workshop
20. W-term: Performance In practice W-projection works well for modest (128 or 256) number of w-planes. Best to combine with faceting for larger problems! • Scaling • Faceted imaging: (N2fac + N2GCF) Nvis • W-projection: (N2Wplanes + N2GCF) Nvis • Ratio: • In practice W-Projection algorithm is about 10x faster • Size of G(u,v,w) increases with W Ref: Cornwell et al. IEEE Special topics in SP, Vol. 2, No5, 2008 [arXiv:0807.4161] Fourteenth Synthesis Imaging Workshop
21. Example: 2D imaging uncorrected Fourteenth Synthesis Imaging Workshop
22. Example: 2D imaging uncorrected Fourteenth Synthesis Imaging Workshop
23. Example: 2D imaging uncorrected Fourteenth Synthesis Imaging Workshop
24. Example: VLA 74MHz before correction Courtesy Kumar Golap Fourteenth Synthesis Imaging Workshop
25. Example: VLA 74MHz after correction Courtesy Kumar Golap Sub-image of an “outlier” field. This bright source should be peeled out! Fourteenth Synthesis Imaging Workshop
26. W-term: Practical Considerations • Faceted imaging: • number of facets in l and m q = FOV (radians) sw = rmswdA = max tolerable amp loss • Nfacets = nfacets x nfacets , q~ l/D sw~ √wmax ~ √Bmax/l • W-projection: • NWplanes~ Nfacets?? Choose 256? We are working on formula • space w-planes uniformly in √w Ref: Cornwell et al. IEEE Special topics in SP, Vol. 2, No5, 2008 [arXiv:0807.4161] Fourteenth Synthesis Imaging Workshop
27. Wide-field imaging: Primary Beam • The “Primary Beam” pattern for an interferometric array (visibility from the correlation of a pair of antennas) contains effects from: • amplitude fall-off (with characteristic FWHM or Gaussian dispersion) due to geometric mean of diffraction patterns from the antenna elements (including optics, blockage) • phase pattern due to diffraction and optics (focus, etc.) • polarization pattern due to optics (reflection from dish surface, feed legs, location of feed (off-axis) in focal plane, any secondary or tertiary mirrors) • large (angle) scale sidelobes and scattered power due to surface errors (e.g. misaligned panels) [T.Hunter talk] Fourteenth Synthesis Imaging Workshop
28. Primary Beam – responses • Example: response to a grid of point sources within the primary beam (courtesy T. Hunter): Fourteenth Synthesis Imaging Workshop
29. CASA tip: pbcor=True during clean or divide by .flux image. Uses Airy or Gaussian beam model Primary Beam – example 1 • Primary effect – apodization of image by PB amplitude pattern • suppression of emission far from pointing center PB applied: sensitive to center emission only Emission structure larger than PB PB sensitivity pattern on sky (circular symmetry assumed) Fourteenth Synthesis Imaging Workshop
30. Primary Beam – example 2 • another example… EVLA Special Issue, ApJ, 739, L20, 2011 Fourteenth Synthesis Imaging Workshop
31. Full-beam imaging: Antenna Primary Beam (PB) asymmetry cannot be ignored. “Sidelobe” pattern very sensitive to orientation. Primary Beam Time-dependence For the Student: What happens if there is a time-variable antenna-dependent pointing error? Antenna PB = autocorrelation of voltage patterns Ei*Ei* Visibility “vPB” = cross-correlation Ei*Ej* of voltage patterns. If not identical vPB will be complex Fourteenth Synthesis Imaging Workshop
32. Primary Beam: Polarized • The aperture response functions have polarization dependence • cross-correlation of complex antenna voltage patterns (R,L or X,Y) Parallel Hand Pattern: ARR Cross Hand Pattern: ARL Imperative – you MUST correct (at some level) for (polarized) primary beam effects during imaging if you want accurate wide-field images! Fourteenth Synthesis Imaging Workshop
33. For the Student: What do the leakages QI UI VI imply? Primary Beam Mapping Q I U V IV leakage showing R/L squint. Now underway for JVLA (Perley, Cotton, Jagannathan) This row: II IQ IU IV Q and U get quadrupolar patterns of induced cross-polarization. Note purity is good on-axis (after calibration). “Squint”: Due to VLA feeds off-axis on feed ring, R and L are displaced (phase gradients in X and Y). This causes apparent V signature in unpolarized sources! Direction-dependent. Must correct during imaging. Mueller matrix (IQUV) showing leakages in L-band. Data taken this past weekend! Fourteenth Synthesis Imaging Workshop
34. Primary Beam Summary • The primary beam response of the antennas in the array must be corrected for during imaging to get accurate intensities (and polarizations) for source outside the core of the beam. • Due to various optics effects, the primary beam is asymmetric and rotates with respect to the source as the sky rotates (in parallactic angle). • During imaging can be corrected approximately in gridding (A-projection) and accurately in de-gridding (major cycles). • Accurate time-dependent beam correction is expensive! • The primary beam structure depends on frequency, so wideband imaging is harder (see Wide Bandwidth Imaging lecture by Urvashi on Monday!) Fourteenth Synthesis Imaging Workshop
35. Mosaicking • Accurate imaging of emission covering areas larger than the primary beam requires the combination of multiple pointings. Example: Large area sky surveys • This is called “Mosaicking” • Simple mosaicking captures the distribution of compact structures (each << primary beam in size) = panorama • If you have good measurements on the shortest baselines (which probe the center of the uv-plane) then mosaicking can reconstruct spacing on sub-aperture scales! Fourteenth Synthesis Imaging Workshop
36. Mosaicking Options • Classic – Linear Image-plane Mosaicking • Form separate images from each pointing (using normal clean), then form weighted sum with PB correction to form larger image. Example: NVSS and FIRST • Modern – Joint Deconvolution • Linear: At minor cycles form linear mosaic of residual images. • Gridded: Use Fourier shift theorem to combine pointings in the uv-plane during gridding (A-projection) with application of the phase gradient from phase center offsets of each pointing. Example: CBI (Myers et al. 2003) of course there is a third path… • Post-modern – use Joint Mosaicking for subsets of nearby pointings, and Image plane Mosaicking to combine sub-mosaics. • Combine with “peeling” of sources outside of beams. Fourteenth Synthesis Imaging Workshop
37. Linear Mosaicking • Form a linear combination of the individual pointingspof the individual deconvolved images Ip on a pixel by pixel basis • Here σ-2p(diagonals of N-1) is the inverse noise variance of an individual pointing and A(m) is the primary response function of an antenna (primary beam) • W(m) is a weighting function that suppresses noise amplification at the edge of mosaic • This linear weighted mosaic can also be computed for the residual dirty image at minor cycles to carry out a joint linear mosaic deconvolution. Fourteenth Synthesis Imaging Workshop
38. Linear Mosaic – observe pointings Fourteenth Synthesis Imaging Workshop
39. Linear Mosaic – individual images • Treat each pointing separately • Image & deconvolveeach pointing • Stitch together linearly with optimal pointing weights from noise and primary beam Fourteenth Synthesis Imaging Workshop
40. Linear Mosaic – combine pointings Fourteenth Synthesis Imaging Workshop
41. Ekers & Rots (1979) – mosaicking can synthesize short spacings. Effectively is a FT of the mosaic grid. Mosaicking 101 • Mosaicking in the image and uv domains: offset & add phase gradients Mosaicking increases resolution in ALL parts of the uv-plane! Fourteenth Synthesis Imaging Workshop
42. Mosaicking and the uv-plane No Ekers-Rots Ekers-Rots Fourteenth Synthesis Imaging Workshop
43. each point on aperture gets correlated with each point on other aperture autocorrelations measure uvspacings inside D/l The Aperture Plane • See “Widefield Imaging II” tomorrow (Brian Mason) for more on recovery of short spacing information. visibility contains range of baselines from closest to furthest parts of apertures CBI interferometer cannot measure “zero-spacing” w/o autocorrelations Fourteenth Synthesis Imaging Workshop
44. Mosaicking Options – comparison • Individual Deconvolution, Linear Mosaic: • Disadvantages: • Deconvolutiononly possible to depth of individual pointing • Overlap regions rather noisy when not drastically oversampled • Advantage: • Each pointing can be treated and calibrated separately for best results. Can be an advantage for high-dynamic range imaging where calibration effects need to be treated with great care. Easy to integrate with “peeling”. • Joint (Gridded) Approach: • Advantages: • Uses all uv info per overlap better sensitivity & beam • More large-scale structure due to PB convolution in uv-plane • Disadvantages: • Requires a good model for the primary beam response • WA-projection can be expensive Fourteenth Synthesis Imaging Workshop
45. Mosaicking - equations • Use our Imaging Equation • Visibility k for the phase center f and pointing center p where A is the primary beam pattern (ignoring w) • express convolution integral in uv-plane (shift theorem) This tells you how to apply the phase offsets and gradients from the pointing and phase centers – these centers are not necessarily the same in a mosaic! Fourteenth Synthesis Imaging Workshop
46. Mosaicing – joint imaging example From JVLA casaguide for Supernova Remnant 3C391 at 4.6GHz: Stokes V image showing artifacts from R/L squint. Stokes I image showing extended SNR emission. Linear polarization vectors, on-axis leakage corrected. Hexagonal mosaic, 7 pointings. FWHM is 9.8’ at 4.6GHz. Fourteenth Synthesis Imaging Workshop
47. Mosaicking Limitations • Your ability to combine observations from different pointings is limited by your knowledge of the primary beam pattern(s) • Denser field sampling gives more uniform coverage, but requires more processing for imaging. • If you use fields taken with different uv-coverages (e.g. observed at different hour angles) then the PSF will vary over the mosaic (it will be a weighted sum of field PSFs). • Flagging can cause unanticipated gaps and variations across the mosaic. • Bright sources can cause problems for “nearby” fields, self-calibration can help, and possibly “peeling” • Pointing errors will induce errors into mosaic (via PB). Fourteenth Synthesis Imaging Workshop
48. Mosaicking in CASA • Calibrate as you would do for a single pointing (e.g. pipeline) • Use the clean task with your favorite parameters • In imagermode use ‘mosaic’ • Use ftmachine=‘ft’ for joint linear deconvolution, ‘mosaic’ for the joint gridded imaging (preferred, faster) • will always use Cotton-Schwab (major/minor cycle) algorithm • Use psfmode=‘clark’ (default) or ‘hogbom’ (for poor psf) • Fill in ‘multiscale’ parameters (scales) for MS Clean • Linear mosaicking of cleaned images only available from the CASA toolkit (im.linearmosaic) currently. [AIPS FLATN] • Contributed tasks for mosaicking field setup (makeschedule) • also check ALMA OT and JVLA OPT (e.g. for OTF) Fourteenth Synthesis Imaging Workshop
49. Practical Mosaicking • So you want to mosaic over a large area? • What field center pattern to use? • How often to come back to a individual pointing • Slew time of Antennas • Change of atmospheric conditions • For more information, see the Guide to VLA Observing section on mosaicking: https://science.nrao.edu/facilities/vla/docs/manuals/obsguide/modes/mosaicking Fourteenth Synthesis Imaging Workshop
50. Practical Mosaicking – pattern lower sensitivity at interstices • Different ways to lay out the field centers on the sky: • “Nyquist” sampling: Rectangular grid Hexagonal grid • Most efficient coverage with minimal non-uniformity (centers on equilateral triangles). Preferred! • Minimum “Nyquist” for structureinformation recovery Fourteenth Synthesis Imaging Workshop | 5,030 | 20,861 | {"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} | 2.609375 | 3 | CC-MAIN-2021-31 | latest | en | 0.78309 |
https://www.mrexcel.com/board/threads/combine-if-statements.276313/ | 1,680,031,090,000,000,000 | text/html | crawl-data/CC-MAIN-2023-14/segments/1679296948868.90/warc/CC-MAIN-20230328170730-20230328200730-00291.warc.gz | 1,000,674,708 | 18,348 | # Combine IF Statements
### Excel Facts
Excel Wisdom
Using a mouse in Excel is the work equivalent of wearing a lanyard when you first get to college
#### Scott Huish
##### MrExcel MVP
Perhaps:
=IF(M5,M5,(J5=0)*(I5*(1+K5))+L5)
Format cell as:
[=0]""
##### MrExcel MVP
Guessing that:
=IF(N(J5),MAX((I5*(1+K5)+L5),M5),"")
#### Scott Huish
##### MrExcel MVP
I think you might mean this:
=IF(N(J5),"",MAX(M5,I5*(1+K5)+L5))
I thought about using MAX. but then I thought what if they get a pay cut?
#### Manda
##### New Member
Thank you that worked perfectly
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Go back | 511 | 1,765 | {"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} | 2.65625 | 3 | CC-MAIN-2023-14 | latest | en | 0.746301 |
https://forum.poshenloh.com/topic/64/shouldn-t-the-x-2-at-7-52-be-positive-or-negative-as-well/2?lang=en-US | 1,623,810,197,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487621699.22/warc/CC-MAIN-20210616001810-20210616031810-00053.warc.gz | 256,002,865 | 12,527 | # Shouldn't the x/2 at 7:52 be positive or negative as well?
• This post is deleted!
• [Originally posted in the Discussions]
Shouldn't the x/2 at 7:52 be positive or negative as well?
• In this case, since x represents the length of the diagonal of the trapezoid, it wouldn't be possible for x/2 to be a negative number. In general, though, you are right; this equation falls in to the category of "quadratic equations," which have an x-squared term. It's possible to show that quadratic equations always have two solutions.
You can see Prof. Loh talk about a cool way of solving quadratic equations which he came up with while developing Module 4: Algebra Tools. You can also read more about him and his quadratic method in this NYT article.
Happy Learning!
The Daily Challenge Team | 185 | 791 | {"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} | 2.640625 | 3 | CC-MAIN-2021-25 | longest | en | 0.969119 |
http://www.medhelp.org/posts/Pregnancy-Ages-18-24-/If-im-26-weeks-what-month-Im-I-on-/show/1571078 | 1,511,488,684,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934807056.67/warc/CC-MAIN-20171124012912-20171124032912-00150.warc.gz | 446,672,403 | 30,835 | Aa
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If i'm 26 weeks what month I'm I on?
Just wondering I'm 26 weeks pregnant so does that mean I've completed my 6th month and I'm not starting my 7th month .... so i'm 6 months and 2 weeks into my 7th month? Or i'm not 7 months until I've hit 28 weeks?
As you can tell i'm confused lol
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Oh boy....you're going to get as many different answers as you get responses, lol.
It's VERY confusing....but maybe this will help a bit. your last menstrual period is the start of week 1...... 42 weeks is the completion of 10 months....(yes pregnancy is actually 9 1/2 or 10 months, not 9!)
So a woman who is 37 weeks is finished with her 8th month of pregnancy, and beginning her 9th month. moving backward from there, a woman who is 32 weeks is finished with her 7th month and beginning her 8th month. a woman who is 27 weeks is finishing her 6th month and beginning her 7th.....and so on backward....some of the "months" have 4 weeks, some have 5..it's very confusing.
So I would say you're "6, almost 7 months"....and you will be 7 months until 32 weeks at which point you will be 8 months...until 37 weeks when you will be 9...etc.
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Oh gosh Ashelen you confused me!!!
I would say you are 6 ½ months pregnant, gestational age of the baby is only 6 months
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I believe you would be 6.5 months pregnant, you are done your 6th month, and half way through your 7th...I think lol Congrats!
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To make it easier so you aren't confused….40 weeks of pregnancy, you are now 26 weeks pregnant which means 6 ½ months pregnant….when you reach 28 weeks pregnant consider it to be 7 months pregnant, when you get to 30 weeks pregnant you are 7 ½ months pregnant, when you are 32 weeks pregnant you are 8 months pregnant which gives you 8 weeks left of pregnancy…..when you get to 36 weeks you are considered 9 months pregnant, when you are 38 weeks you are 9 ½ months and when you reach 40 weeks you are obviously due…
Make sense?
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LOL thank you ladies!!! I think it got it down =)
Yesterday I ran into a friend and she's also expecting she asked how far along i was so i said 6 months and she said she was 7 months. WHen she asked for my due date I said Nov 13 and she said so I'm I, we are due the same day. That's where my confusion started lol
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You were right..she's a LITTLE off...at 26 weeks you are finishing your 6th month, at 27 weeks you will begin your 7th month.
this is my 3rd baby so I've had lots of experience trying to sort it all out....it's really confusing because pregnancy is actually counted to 42 weeks, which is 10 months. the 9th month has 5 weeks in it, which is from 37-42.
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This is also my 3rd pregnancy... but for some reason It always confuses me. When are you do? Do you know what your having?
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I'm due April 2-5th-ish...not sure exactly when my LMP was, lol! I get a dating ultrasound on the 24th.
And it's always been a thorn in my side too..I usually just tell people how many weeks I am, not months, and they can figure it out for themselves, LOL...I guess I should say this is my 4th pregnancy, my first was a m/c at about 8-10 weeks...but 3rd child.
I'm guessing you already know what you're having??
:-)
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My first was a little girl and I also lost my second baby (ectopic pregnancy) and my third was a little boy and this one we decided to wait and see what suprise we get on the special day. I thought it was going to be hard not knowing but its actually exciting =)
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im 30 weeks now and im moing towards the 7 1/2 months instead of 7..
When Im near the end of the "month" I just start saying almost 7 months or almost 6 months etc.. you are definitely in the almost 7 category
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Actually you only count to 40 weeks…technically if you counted to 42 weeks you are pregnant for 10 ½ months
Doctors go off your last menstrual for a due date to make it easier to count
If you find out when you are pregnant at 4 weeks you are technically only (gestational age) 2 weeks.
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I don't know if they'll let me post a link here. I've tried before and it doesn't work. Maybe if I separate it out. You should check out days to go ******* (without the spaces and obviously without dot spelled out). Then click on the blue "pregnancy countdown calendar" link. You will put in your due date and it will tell you your time along in months, weeks and days and also your time left. I am pretty much the same as you. Due 11/11. So I'm also 26 weeks along and the site tells me I'm 6 months. :)
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See? It totally starred out what I put. Just put in daystogo and then the dot and then the c-o-m at the end. (See if it lets me write THAT! Ha ha!)
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Thank you! I"m going to check the website out. And congrats on your pregnancy... that would be awsome if you deliver on 11/11/1
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Comment | 1,426 | 5,291 | {"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} | 2.53125 | 3 | CC-MAIN-2017-47 | latest | en | 0.951256 |
https://www.enotes.com/homework-help/solve-indefinite-integral-f-x-2x-5-x-2-5x-6-then-195987 | 1,521,811,457,000,000,000 | text/html | crawl-data/CC-MAIN-2018-13/segments/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00236.warc.gz | 781,132,859 | 10,803 | # Solve the indefinite integral of f(x)=(2x-5)/(x^2-5x+6) . Then solve the definite integral of f from x=0 to x=1.
hala718 | Certified Educator
f(x) = (2x-5) /(x^2 -5x + 6)
F(x)= intg f(x) = intg (2x-5)/(x^2 - 5x + 6)
Let u = x^2 - 5x + 6
==> du = (2x - 5) dx
F(x) = intg du/u
= ln u
= ln (x^2 -5x +6)
F(1) = ln (1-5+6) = ln 2
F(0) = ln (0-0+6) = ln 6
Then the definite intergral = ln 2 - ln 6
= ln 2/6
= ln 1/3
= -ln3
giorgiana1976 | Student
To calculate the indefinite integral of the given function, we'll use the substitution method.
We'll calculate Integral of f(x) = (2x-5)/(x^2-5x+6)
We notice that if we'll differentiate the denominator, x^2-5x+6, we'll get 2x-5.
So, we'll note x^2-5x+6 = t
(x^2-5x+6)'dx = dt
(2x-5)dx = dt
We'll re-write the integral in the variable t:
Int (2x-5)dx/(x^2-5x+6) = Int dt / t
Int dt / t = ln t + C
But x^2-5x+6 = t.
Int (2x-5)dx/(x^2-5x+6) = ln(x^2-5x+6) + C, where C is a family of constants.
Now, we'll evaluate the definite integral, using Leibniz-Newton formula:
Int (2x-5)dx/(x^2-5x+6) = F(1) - F(0)
F(1) = ln(1^2-5*1+6)
F(1) = ln (1-5+6)
F(1) = ln 2
F(0) = ln(0^2-5*0+6)
F(0) = ln 6
Int (2x-5)dx/(x^2-5x+6) = ln 2 - ln 6
We'll apply the quotient rule to the difference of logarithms:
Int (2x-5)dx/(x^2-5x+6) = ln (2/6)
Int (2x-5)dx/(x^2-5x+6)=ln (1/3)=ln 1 - ln 3=0-ln 3=- ln 3
neela | Student
To evaluate the indefinit Int (2x-5)/(x^2-5x-6) dx .(ii) find the definite integral fro x= 0 to x=1.
Solution:
method (i) :
Pbviously numerator is diffrential coefficient of dinominator or
(x^2-5x-6)' = 2x-5.
So Int d(x^2-5x-6)/ (x^2-5x-6) = log(x^2-5x -6) + constant.
2nd method:
The denominator x^2-5x-6 = (x-6)(x+1)
So we split (2x-5)/(x^2-5x-6) into A/(x-6) +B/(x+1).
So (2x-5) = A(x+1) +B(x-6)....(1) Put x = 6, then
2*6-5 = A(6+1) + 0 . So 7 = 7A, So A = 1.
Put x = -1 in (1): 2*-1-5 = 0 + B(-1-6) . -7 = -7B. S B = a.
So 2x-5 = 1/(x-6) +1/(x+1)
Therefore Int (2x-5)/(x^2-5x-6) = Int dx/(x-6) +Int dx/(x-5)
Int (2x-5)/(x^2-5x-6) = log(x-6)+log(x+1) + Constant = log(x^2-5x-6) + C = F(x) say
To find the definite integral from x= 0 to x=1
F(1) - F(0) = log (1-5-6) - log (0-0-6) = ln(-10)/(-6) = ln(5/3) = 0.510825623. | 1,113 | 2,230 | {"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-2018-13 | latest | en | 0.541546 |
https://www.airmilescalculator.com/distance/pmi-to-brq/ | 1,674,772,505,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764494826.88/warc/CC-MAIN-20230126210844-20230127000844-00516.warc.gz | 651,177,586 | 69,999 | # How far is Brno from Palma de Mallorca?
The distance between Palma de Mallorca (Palma de Mallorca Airport) and Brno (Brno–Tuřany Airport) is 955 miles / 1537 kilometers / 830 nautical miles. The estimated flight time is 2 hours and 18 minutes.
Driving distance from Palma de Mallorca (PMI) to Brno (BRQ) is 1356 miles / 2182 kilometers and travel time by car is about 28 hours 23 minutes.
955
Miles
1537
Kilometers
830
Nautical miles
2 h 18 min
148 kg
## Distance from Palma de Mallorca to Brno
There are several ways to calculate the distance from Palma de Mallorca to Brno. Here are two standard methods:
Vincenty's formula (applied above)
• 954.914 miles
• 1536.785 kilometers
• 829.797 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth's surface using an ellipsoidal model of the planet.
Haversine formula
• 953.871 miles
• 1535.106 kilometers
• 828.891 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## How long does it take to fly from Palma de Mallorca to Brno?
The estimated flight time from Palma de Mallorca Airport to Brno–Tuřany Airport is 2 hours and 18 minutes.
## What is the time difference between Palma de Mallorca and Brno?
There is no time difference between Palma de Mallorca and Brno.
## Flight carbon footprint between Palma de Mallorca Airport (PMI) and Brno–Tuřany Airport (BRQ)
On average, flying from Palma de Mallorca to Brno generates about 148 kg of CO2 per passenger, and 148 kilograms equals 326 pounds (lbs). The figures are estimates and include only the CO2 generated by burning jet fuel.
## Map of flight path and driving directions from Palma de Mallorca to Brno
Shortest flight path between Palma de Mallorca Airport (PMI) and Brno–Tuřany Airport (BRQ).
## Airport information
Origin Palma de Mallorca Airport
City: Palma de Mallorca
Country: Spain
IATA Code: PMI
ICAO Code: LEPA
Coordinates: 39°33′6″N, 2°44′19″E
Destination Brno–Tuřany Airport
City: Brno
Country: Czech Republic
IATA Code: BRQ
ICAO Code: LKTB
Coordinates: 49°9′4″N, 16°41′39″E | 589 | 2,202 | {"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} | 2.671875 | 3 | CC-MAIN-2023-06 | latest | en | 0.803151 |
http://www.macs.hw.ac.uk/~pjbk/pathways/cpp1/node139.html | 1,386,908,929,000,000,000 | text/html | crawl-data/CC-MAIN-2013-48/segments/1386164858282/warc/CC-MAIN-20131204134738-00007-ip-10-33-133-15.ec2.internal.warc.gz | 426,226,910 | 2,048 | Next: Example Program: Student Mark Up: The for statement Previous: Example for statement: Print
# Example for statement: Print table of sine function
The following loop tabulates the sin function from `x = 0.0` to `x = 1.6` in steps of `0.1`.
```int i;
float x;
for (i = 0; i <= 16; i++)
{
x = 0.1 * i;
cout << x << " " << sin(x) << endl;
}
```
Note how an integer variable `i` is used to control the loop while inside the loop the corresponding value of `x` is calculated as a function of `i`. This is preferable to the following:
```float x;
for (x = 0.0; x <= 1.6; x += 0.1)
cout << x << " " << sin(x) << endl;
```
The problem with the above is that floating point variables are not held exactly, thus 0.1 might be represented by something slightly larger than 0.1. Then after continually adding 0.1 to `x` the final value that should be 1.6 may actually be something like 1.60001. Thus the test `x <= 1.6` would fail prematurely and the last line of the table would not be printed. This could be corrected by making the test `x <= 1.605` say.
In general it is probably best to use integer variables as control variables in for loops.
Next: Example Program: Student Mark Up: The for statement Previous: Example for statement: Print
Peter JB King
1999-08-31 | 355 | 1,267 | {"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} | 2.953125 | 3 | CC-MAIN-2013-48 | latest | en | 0.87391 |
http://www.onlinemathlearning.com/indices-exam-questions.html | 1,513,422,184,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948587577.92/warc/CC-MAIN-20171216104016-20171216130016-00089.warc.gz | 422,546,897 | 9,486 | # Exam Questions on Indices
Related Topics:
More Lessons for GCSE Maths
Math Worksheets
Videos, examples, and solutions to help GCSE Maths students learn about indices by working through some examination questions.
All you need to know about indices (Revision and Summary)
x1 means x multiplied repeatedly by itself n times.
xmxn = xm + n
(xm)n = xmn
(xmyn)p = xmpynp
xm/xn = xm-n
x0 = 1
x-n = 1/xn
Indices GCSE Maths revision Higher level worked exam questions (include fractional and negative powers)
Examples:
1. Work out 561 - 560
2. Explain why 271/3 = 3
3. Write 27-1/3 as a fraction.
4. Work out the value of 642/3
5. Work out all solutions of the equation: 8m = 2m2
6. Show clearly that 43/2 = 8
. Hence, or otherwise, work out the value of y if 4y = 86
7. Write down the value of 20.
8. Simplify 718 ÷ (73)2. Give your answer as a power of 7.
9. A patient has a disease. She has 43 body cells on day 1. The number of affected cells doubles every day. The disease becomes serious when 210 body cells are affected. On which day does the disease become serious? You must show your working.
10. Work out 10-2 × 641/2.
11. Write down the square of the cube root of x using index notation.
12. Given x = 8, what is the value of the square of the cube root of x?
13. Work out the value of 9-3/2.
14. Evaluate 27-2/3.
15. Explain why 254 = 58.
16. Write 4-1 as a fraction.
17. Work out 84/3. Indices (Exponents) : C1 Edexcel June 2013 Q3
(a) Find the value 85/3
(b) Simplify fully (2x1/2)3/(4x2)
Indices Tricky - GCSE Further Maths revision Exam paper practice & help
Example:
x3/2 = 8 where x > 0 and y-2 = 25/4 where y > 0.
Work out the value of x/y.
Indices beastie - Dare you try this difficult question
Indices and Surds - GCSE Maths Higher revision Exam paper practice & help
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. | 623 | 2,187 | {"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-2017-51 | latest | en | 0.870883 |
https://www.experts-exchange.com/questions/20163322/why-x-3-y-x-x-then-x-3-and-y-9.html | 1,513,185,182,000,000,000 | text/html | crawl-data/CC-MAIN-2017-51/segments/1512948529738.38/warc/CC-MAIN-20171213162804-20171213182804-00475.warc.gz | 731,327,771 | 40,300 | [Last Call] Learn how to a build a cloud-first strategyRegister Now
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# why x = 3; y = x++ * x++ then x=3 and y =9 ?
Posted on 2001-08-04
Medium Priority
1,179 Views
Look at the following code :
int x = 3;
int y = x++ * x++;
I though the result is x = 5 and y = 12, because
x++ * x++ is evaluated as
push x value on stack ; now stack has ... 3
x = x + 1; // x= 4
push x value on stack again ; now stack has ... 3 4
x = x + 1; // x = 5
// mul two nubers on stack
y = 3 * 4 ; ---> 12
However, when I run the above code in VC++ and C++ Builder,
the result is x = 5 and y = 9. Because the VC++, C++ Builder
evaluates x++ * x++ as
y = x+ x;
x++;
x++;
Could someone points out why the result is not x = 5 and y = 12 ?
0
Question by:thienpnguyen
• 11
• 4
• 3
• +7
LVL 33
Expert Comment
ID: 6351679
Simple.
x++ is different from ++x.
The ++ at the back means postfix while the other one is prefix. Simply to say, postfix is telling the computer to take the current value of x first before doing an increment.
So
int x = 3;
int y = x++ * x++;
would be equivalent to this
int x = 3;
int y = x * x;
x = x + 1;
x = x + 1;
So result in
int x = 3;
int y = 3 * 3; // 9
x = 3 + 1; // 4
x = 4 + 1; // 5
So x = 5 while y = 9.
hongjun
0
LVL 6
Author Comment
ID: 6351718
Hi hongjun,
Why compiler doesn't not
y = evaluate(x++) * evaluate(x++)
and after evaluate(x++) then x = x++.
1. evaluate x++ ( x left of x++ * x++ )
result_1 = evaluate x++ ==> result_1 = 3 and x = 4
2. evaluate x++ ( x right of x++ * x++ ) // note x = 4
result_2 = evaluate x++ ==> result_1 =4 and x = 5
3. y = result_1 * result_2;
I want to know what the C Ansi rule for processing this situation.
0
LVL 6
Author Comment
ID: 6351731
Hi hongjun,
Why compiler doesn't not
y = evaluate(x++) * evaluate(x++)
and after evaluate(x++) then x = x+1.
1. evaluate x++ ( x left of x++ * x++ )
result_1 = evaluate x++ ==> result_1 = 3 and x = 4
2. evaluate x++ ( x right of x++ * x++ ) // note x = 4
result_2 = evaluate x++ ==> result_1 =4 and x = 5
3. y = result_1 * result_2;
want to know what the C Ansi rule for processing this situation.
0
LVL 22
Expert Comment
ID: 6351737
>> nt y = x++ * x++;
>> would be equivalent to this
No. there is no sequence in this expression. So while your explnation of the prefix vs postfix part is correct, there is no telling what this expressiom might do.
Y could be set to x*x or (x+1)*x. (Possibly other values).
>> I want to know what the C Ansi rule
>> for processing this situation.
The rule is that there is no rule. The effect of the postincrement must take place at the following sequence point, but it does not have to take effecit earlier, but it can.
I'll see if I can find you the description form the standard. But suffice to say, the expression is ambigious any value you get might be right or might be wrong, so using it is a mistake.
0
LVL 22
Accepted Solution
nietod earned 80 total points
ID: 6351747
From the C++ standard 5.4
4 Except where noted, the order of evaluation of operands of individual
operators and subexpressions of individual expressions, and the order
in which side effects take place, is unspecified. Between the previ-
ous and next sequence point a scalar object shall have its stored
value modified at most once by the evaluation of an expression. Fur-
thermore, the prior value shall be accessed only to determine the
value to be stored. The requirements of this paragraph shall be met
for each allowable ordering of the subexpressions of a full expres-
sion; otherwise the behavior is undefined. [Example:
i = v[i++]; // the behavior is undefined
i = 7, i++, i++; // `i' becomes 9
i = ++i + 1; // the behavior is undefined
i = i + 1; // the value of 'i' is incremented
--end example]
As you can see your expression midifies the value of x twice between sequence points (the next sequence point is the ";" and the previous one is the ";" of the preceeding statement) and thus the behavior is undefined.
0
Expert Comment
ID: 6352929
my dear friend
x++ is post increment operator
so when u write x++; in your statement 1st value of x is used then x is intremented
so as ur question says that
y=x++*x++;
means
y=3*3
then
x++
means x=4
then
x++
means x=5
i hope now U can understand pre-increment operator ++x and post increment operator x++ both r different.
bhavin4@rediffmail.com
0
LVL 22
Expert Comment
ID: 6352962
bhawin, I'm not surprised that you don't understand the answer to the problem. But I'm surprised that after reading the answer you still don't understand. Read the answer before posting a comment!
>> y=x++*x++;
>> means
>> y=3*3
>> then
>> x++
>> means x=4
>> then
>> x++
>> means x=5
Absolutely wrong!
There is no sequence point within that expression. x is modified twice withinn that expression. The behavior is undefined.
0
LVL 3
Expert Comment
ID: 6353806
sorry, nietod but I think someone else has to repeat it.
The behaviour of x++ * x++; is *undefined*.
That doesn't mean that it will not work in any compiler, but that ANSI C does not specify an answer for this, and that the result cannot be relied upon in C compilers, or even in multiple versions of a C compiler.
After all, there is nothing in that expression that cannot be done in other ways, to make it a must for this to work..
0
LVL 3
Expert Comment
ID: 6354097
I think the main reason for this is more to do with the precedence of operator
In the expression
y = x++ * x++;
Multiplication (*) has precedence over the operator ++ and that?s the only reason for the final output.
0
LVL 22
Expert Comment
ID: 6354173
What? Are you guys unable to read or something?
I explained the problem. I posted a section of the C++ standard that describes the problem.
Why do you see the need to propose alternate explanations that are simply guesses? Guesses that conflict with the C++ standard's official discription of the problem? The description that I posted so there could be no doubt that it was correct!
>> Multiplication (*) has precedence over
>> the operator ++
No. It has NOTHING to do with operator precedence. I'm not guessing that it has nothing to do with operator precedence. I know it has nothing to do with operator precedence because the standard says
Between the previous and next sequence point a
scalar object shall have its stored value modified
at most once by the evaluation of an expression. ...
The requirements of this paragraph shall be met
...otherwise the behavior is undefined.
This is the only correct explanation. Yes, tThere are lots of other explanatiosn for this, but they are incorrect!
Do you even understand the fact that different compilers will look at that statement and produce code with different behavior? Trying to describe that as operator preecedenc--which has a precisely defined behavor--is ludicrious.
0
LVL 30
Expert Comment
ID: 6354779
Hi (nietod),
If you read the following link, you'll see why this is the preferred method for many of our valued experts, including myself.
Hi (thienpnguyen):
Feel free to click the [Reject Answer] button near (Answer-poster's) response, even if it seems like a good answer.
Doing so will increase your chance of obtaining additional input from other experts. Later, you can click the [Select Comment as Answer] button on any response.
0
LVL 9
Expert Comment
ID: 6355064
> An answer should not be posted as an answer, if other
> experts have previously posted possible answers
oh, come on...not again...
in this case, nietod was justified in posting an answer, what went before him was just plain wrong...nietod is completely correct in what he says (i.e. he has provided a complete answer)
perhaps you should keep your opinions on things like this to yourself...
0
LVL 22
Expert Comment
ID: 6355288
You don't understand, jason. There are other answers to this question. Admitidly they are all the wrong ones. But that apparently escapes Axter.
thienpnguyen, if you want the wrong answer, you should reject my answer. Although it seems unnecessary since others are willing ot provide them even with it locked.
0
LVL 3
Expert Comment
ID: 6355477
Now it completely overwhelms me..
Nietod had an excellent complete answer, then some one must explain it, then someone must say it is right, then nietod has to justify it is an answer, not a comment..
Respects to EE and to Axter, some times the answer is so supported by evidence that it is *the* answer, as is the case here (withh ANSI C standard referenced). I feel it should be right to propose an answer, or why the EE had put an answer option at all?
That does not mean that posting answers as comments is not the preferred method to me, just there are times at which it is valid to do it the other way.
0
Expert Comment
ID: 6356474
Dear friend,
According to the operators precedence * is having more priority than ++ so 3*3 will be executed first then x++ will be executed i.e, x=5,y=9;
0
LVL 22
Expert Comment
ID: 6356531
>> According to the operators precedence *
>> is having more priority than ++
That is wrong. Both the prefix and the postfix ++ operators have higher priorioty than the * operator (Or any other mathematical or bitwise operators).
But even if you were right. Say for one instance that * does have higher precendence than ++. How does that explain the fact that one compiler calculates 9 for the expression and one compiler calculates 12? Please explain that!
Here is a hint. If you want to explain it, you could read the posted answer. The answer is correct.
And for your further reference Here is a C++ operator procedence table. You might want to consult it the next time you try to explain things in therms of operator precendence.
Operator Name Associativity
:: Scope resolution None
:: Global None
[ ] Array subscript Left to right
( ) Function call Left to right
( ) Conversion None
. Member selection (object) Left to right
-> Member selection (pointer) Left to right
++ Postfix increment None
-- Postfix decrement None
new Allocate object None
delete Deallocate object None
delete[ ] Deallocate object None
++ Prefix increment None
-- Prefix decrement None
* Dereference None
+ Unary plus None
- Arithmetic negation (unary) None
! Logical NOT None
~ Bitwise complement None
sizeof Size of object None
sizeof ( ) Size of type None
typeid( ) type name None
(type) Type cast (conversion) Right to left
const_cast Type cast (conversion) None
dynamic_cast Type cast (conversion) None
reinterpret_cast Type cast (conversion) None
static_cast Type cast (conversion) None
.* Apply pointer to class member (objects) Left to right
->* Dereference pointer to class member Left to right
* Multiplication Left to right
/ Division Left to right
% Remainder (modulus) Left to right
- Subtraction Left to right
<< Left shift Left to right
>> Right shift Left to right
< Less than Left to right
> Greater than Left to right
<= Less than or equal to Left to right
>= Greater than or equal to Left to right
== Equality Left to right
!= Inequality Left to right
& Bitwise AND Left to right
^ Bitwise exclusive OR Left to right
| Bitwise OR Left to right
&& Logical AND Left to right
|| Logical OR Left to right
e1?e2:e3 Conditional Right to left
= Assignment Right to left
*= Multiplication assignment Right to left
/= Division assignment Right to left
%= Modulus assignment Right to left
+= Addition assignment Right to left
-= Subtraction assignment Right to left
<<= Left-shift assignment Right to left
>>= Right-shift assignment Right to left
&= Bitwise AND assignment Right to left
|= Bitwise inclusive OR assignment Right to left
^= Bitwise exclusive OR assignment Right to left
, Comma Left to right
0
LVL 2
Expert Comment
ID: 6356696
So then what is the difference between
y = ++x * x++;
and
y = THAT HAS NO DEFINED MEANING AT ALL, DEAL WITH IT!
~Lockias
0
LVL 22
Expert Comment
ID: 6356775
there is a difference. A program with the former is properly defined (unless there are problems elsewhere) yet has undefined behavior at this point. a program with the later is ill-formed and will not compile.
:-)
0
LVL 2
Expert Comment
ID: 6356837
But let me elaborate...
#define THAT ++
#define HAS
#define NO x
#define DEFINED *
#define MENAING x
#define AT ++
#define ALL
#define DEAL ;
#define WITH
#define IT
#define THE 9
#define STANDARD 12
y = THAT HAS NO DEFINED MEANING AT ALL, DEAL WITH IT
Shoule y be READ, THE, or STANDARD?
~Lockias
0
LVL 22
Expert Comment
ID: 6356924
:-)
0
LVL 30
Expert Comment
ID: 6357206
jasonclarke,
>>perhaps you should keep your opinions on things like
>>this to yourself...
I do not try to oppress your opinion. Please do not try to oppress mine.
Just as you are free to express your opinion, so am I.
0
LVL 22
Expert Comment
ID: 6357240
When you suggest that someone reject a correct answer, you are not unduely influencing others. An unsoliceted option like this raises doubts about an answer needlessly.
Or do you think I should begin posting cooments on all questiosn that suggest that the correct answer not be rejected? Or even comments that suggest that the a comment be accepted asn an answer so that the person posting the comment could then work witht he client to reach a solution?
Obviously not.
maybe your time would be better spent answering questions, rather than policing the actions of others.
0
LVL 30
Expert Comment
ID: 6357265
>>questions, rather than policing the actions of others.
This is not the proper place for this.
If you have problems with my posting, you and your partner need to address it to the Community Support topic area.
0
LVL 2
Expert Comment
ID: 6357301
Does anyone know what which PAQ has the most comments with least amount of useful information. I believe I am going to paste in the lastest source code I have for LINUX to help this one along.
~Lockias
0
LVL 22
Expert Comment
ID: 6357448
>> This is not the proper place for this.
That is my opinion. What is this a propper place for you opinions an not mine?
If you don't think I should post comment like that, I won't. Just stoping your and if by magic mine will stop.
>> to the Community Support topic area.
I did notify CS. My partner is my wife and I think we should leave her out of this.
0
LVL 2
Expert Comment
ID: 6357522
>> I think Lockias is great.
>> She should have really got the points for this.
Yes, I agree.
0
LVL 33
Expert Comment
ID: 6366988
nietod, just spot this in another topic area. Go there and points is yours.
http://www.experts-exchange.com/jsp/qShow.jsp?ta=progsoftgen&qid=20161844
hongjun
0
LVL 22
Expert Comment
ID: 6367634
thanks. I never would have seen it.
0
LVL 33
Expert Comment
ID: 6368675
It's ok. I myself also never really go to that topic area. Just happen to be too free and so went there.
hongjun
0
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## Kilogram-force meter into kcal/s
numbers in scientific notation
Direct link to this calculator:
https://www.convert-measurement-units.com/convert+Kilogram-force+meter+to+kcal+s.php
## How many kcal/s make 1 Kilogram-force meter?
1 Kilogram-force meter [kgf·m] = 0.002 342 278 112 162 kcal/s - Measurement calculator that can be used to convert Kilogram-force meter to kcal/s, among others.
# Convert Kilogram-force meter to kcal/s (kgf·m to kcal/s):
1. Choose the right category from the selection list, in this case 'Power'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Kilogram-force meter [kgf·m]'.
4. Finally choose the unit you want the value to be converted to, in this case 'kcal/s'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '233 Kilogram-force meter'. In so doing, either the full name of the unit or its abbreviation can be usedas an example, either 'Kilogram-force meter' or 'kgf·m'. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Power'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '45 kgf·m to kcal/s' or '59 kgf·m into kcal/s' or '12 Kilogram-force meter -> kcal/s' or '98 kgf·m = kcal/s' or '50 Kilogram-force meter to kcal/s' or '10 Kilogram-force meter into kcal/s'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(58 * 77) kgf·m'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '233 Kilogram-force meter + 699 kcal/s' or '78mm x 47cm x 81dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
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https://m.wikihow.com/Simplify-a-Square-Root | 1,580,128,322,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251700675.78/warc/CC-MAIN-20200127112805-20200127142805-00319.warc.gz | 532,419,166 | 66,235 | # How to Simplify a Square Root
Co-authored by wikiHow Staff
Updated: January 22, 2020
Simplifying a square root isn't as hard as it looks. To simplify a square root, you just have to factor the number and pull the roots of any perfect squares you find out of the radical sign. Once you've memorized a few common perfect squares and know how to factor a number, you'll be well on your way to simplifying the square root.
### Method 1 of 3: Simplifying a Square Root by Factoring
1. 1
Understand factoring. The goal of simplifying a square root is to rewrite it in a form that is easy to understand and to use in math problems. Factoring breaks down a large number into two or more smaller factors, for instance turning 9 into 3 x 3. Once we find these factors, we can rewrite the square root in simpler form, sometimes even turning it into a normal integer. For example, √9 = √(3x3) = 3. Follow the steps below to learn this process for more complicated square roots.[1]
2. 2
Divide by the smallest prime number possible. If the number under the square root is even, divide it by 2. If your number is odd, try dividing it by 3 instead. If neither of these gives you a whole number, move down this list, testing the other primes until you get a whole number result. You only need to test the prime numbers, since all other numbers have prime numbers as their factors. For example, you don't need to test 4, because any number divisible by 4 is also divisible by 2, which you already tried.[2]
• 2
• 3
• 5
• 7
• 11
• 13
• 17
3. 3
Rewrite the square root as a multiplication problem. Keep everything underneath the square root sign, and don't forget to include both factors. For example, if you're trying to simplify √98, follow the step above to discover that 98 ÷ 2 = 49, so 98 = 2 x 49. Rewrite the "98" in the original square root using this information: √98 = √(2 x 49).[3]
4. 4
Repeat with one of the remaining numbers. Before we can simplify the square root, we keep factoring it until we've broken it down into two identical parts. This makes sense if you think about what a square root means: the term √(2 x 2) means "the number you can multiply with itself to equal 2 x 2." Obviously, this number is 2! With this goal in mind, let's repeat the steps above for our example problem, √(2 x 49):
• 2 is already factored as low as it will go. (In other words, it's one of those prime numbers on the list above.) We'll ignore this for now and try to divide 49 instead.
• 49 can't be evenly divided by 2, or by 3, or by 5. You can test this yourself using a calculator or long division. Because these don't give us nice, whole number results, we'll ignore them and keep trying.
• 49 can be evenly divided by seven. 49 ÷ 7 = 7, so 49 = 7 x 7.
• Rewrite the problem: √(2 x 49) = √(2 x 7 x 7).
5. 5
Finish simplifying by "pulling out" an integer. Once you've broken the problem down into two identical factors, you can turn that into a regular integer outside the square root. Leave all other factors inside the square root. For example, √(2 x 7 x 7) = √(2)√(7 x 7) = √(2) x 7 = 7√(2).[4]
• Even if it's possible to keep factoring, you don't need to once you've found two identical factors. For example, √(16) = √(4 x 4) = 4. If we kept on factoring, we'd end up with the same answer but have to do more work: √(16) = √(4 x 4) = √(2 x 2 x 2 x 2) = √(2 x 2)√(2 x 2) = 2 x 2 = 4.
6. 6
Multiply integers together if there are more than one. With some large square roots, you can simplify more than once. If this happens, multiply the integers together to get your final problem. Here's an example:
• √180 = √(2 x 90)
• √180 = √(2 x 2 x 45)
• √180 = 2√45, but this can still be simplified further.
• √180 = 2√(3 x 15)
• √180 = 2√(3 x 3 x 5)
• √180 = (2)(3√5)
• √180 = 6√5
7. 7
Write "cannot be simplified" if there are no two identical factors. Some square roots are already in simplest form. If you keep factoring until every term under the square root is a prime number (listed in one of the steps above), and no two are the same, then there's nothing you can do. You might have been given a trick question! For example, let's try to simplify √70:[5]
• 70 = 35 x 2, so √70 = √(35 x 2)
• 35 = 7 x 5, so √(35 x 2) = √(7 x 5 x 2)
• All three of these numbers are prime, so they cannot be factored further. They're all different, so there's no way to "pull out" an integer. √70 cannot be simplified.
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### Method 2 of 3: Knowing the Perfect Squares
1. 1
Memorize a few perfect squares. Squaring a number, or multiplying it by itself, creates a perfect square. For example, 25 is a perfect square because 5 x 5, or 52, equals 25. Memorizing at least the first ten perfect squares can help you recognize and quickly simplify perfect square roots. Here are the first ten perfect squares:
• 12 = 1
• 22 = 4
• 32 = 9
• 42 = 16
• 52 = 25
• 62 = 36
• 72 = 49
• 82 = 64
• 92 = 81
• 102 = 100
2. 2
Find the square root of a perfect square. If you recognize a perfect square under a square root symbol, you can immediately turn it into its square root and get rid of the radical sign (√). For example, if you see the number 25 under the square root sign, you know that the answer is 5 because 25 is a perfect square. Here's the same list as above, going from the square root to the answer:
• √1 = 1
• √4 = 2
• √9 = 3
• √16 = 4
• √25 = 5
• √36 = 6
• √49 = 7
• √64 = 8
• √81 = 9
• √100 = 10
3. 3
Factor numbers into perfect squares. Use the perfect squares to your advantage when following the factor method of simplifying square roots. If you notice a way to factor out a perfect square, it can save you time and effort. Here are some tips:[6]
• √50 = √(25 x 2) = 5√2. If the last two digits of a number end in 25, 50, or 75, you can always factor out 25.
• √1700 = √(100 x 17) = 10√17. If the last two digits end in 00, you can always factor out 100.
• √72 = √(9 x 8) = 3√8. Recognizing multiples of nine is often helpful. There's a trick to it: if all digits in a number add up to nine, then nine is always a factor.
• √12 = √(4 x 3) = 2√3. There's no special trick here, but it's usually easy to check whether a small number is divisible by 4. Keep this in mind when looking for factors.
4. 4
Factor a number with more than one perfect square. If the number's factors contain more than one perfect square, move them all outside the radical symbol. If you found multiple perfect squares during your simplification process, move all of their square roots to the outside of the √ symbol and multiply them together. For example, let's simplify √72:
• √72 = √(9 x 8)
• √72 = √(9 x 4 x 2)
• √72 = √(9) x √(4) x √(2)
• √72 = 3 x 2 x √2
• √72 = 6√2
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### Method 3 of 3: Knowing the Terminology
1. 1
Know that the radical symbol (√) is the square root symbol. For example, in the problem, √25, "√" is the radical symbol.[7]
2. 2
Know that the radicand is the number inside the radical symbol. You will need to find the square root of this number. For example, in the problem √25, "25" is the radicand.[8]
3. 3
Know that the coefficient is the number outside the radical symbol. This is the number that the square root is being multiplied by; this sits to the left of the √ symbol. For example, in the problem, 7√2, "7" is the coefficient.
4. 4
Know that a factor is a number that can be evenly divided out of another number. For example, 2 is a factor of 8 because 8 ÷ 4 = 2, but 3 is not a factor of 8 because 8÷3 doesn’t result in a whole number. As another example, 5 is a factor of 25 because 5 x 5 = 25.
5. 5
Understand the meaning of simplifying a square root. Simplifying a square root just means factoring out any perfect squares from the radicand, moving them to the left of the radical symbol, and leaving the other factor inside the radical symbol. If the number is a perfect square, then the radical sign will disappear once you write down its root. For example, √98 can be simplified to 7√2.
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## Community Q&A
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Add New Question
• Question
How do you simplify square roots with variables?
wikiHow Staff Editor
Staff Answer
Use the same procedure you would for simplifying square roots with numerical radicands. Bring any identical pairs of factors out of the radical to become the coefficient. For example, if you need to simplify √25a^3, change it to √5×5×a×a×a. Factor out 5 and a^2 to get 5a^2√a.
• Question
What is the simplest radical form?
wikiHow Staff Editor
Staff Answer
If something is written in its simplest radical form, that means that you have already found all possible roots and eliminated any radicals from the denominator of a fraction. A square root can’t be simplified any further if there are no 2 identical factors remaining and every term under the radical symbol is a prime number.
• Question
How do you simplify radical fractions?
wikiHow Staff Editor
Staff Answer
Start by multiplying the numerator and denominator by a radical that will eliminate the radical in the denominator. For example, if your fraction is 2/√7, multiply by √7/√7 to get 2√7/√49. This will simplify to 2√7/7. If possible, simplify the fraction by dividing out any common factors in the numerator and denominator.
• Question
What is the square root of 18 minus the square root of 8?
Donagan
Top Answerer
√18 = √(2x3x3) = 3√2. √8 = √(2x2x2) = 2√2. Then (3√2) - (2√2) = 1√2 = √2.
• Question
What is the square root of 841?
Community Answer
The square root of 841 is 29.
• Question
How do you simplify 4sqrt2 - 2 sqrt 8 + 3 sqrt 2?
Community Answer
You have to factor 2sqrt8 further. To simplify the numbers you listed, the numbers in the root have to be equivalent so you can perform addition/subtraction. Try this: sqrt8 is 2 root 2. 2*2=4 plus the root. Your equation now is 4sqrt2- 4sqrt 2 + 3sqrt 2. 4sqrt2-4sqrt2=0. Your answer is 3sqrt 2.
• Question
How do you simplify the square root of 25?
Community Answer
25 is a whole number square root. X*X=25 What is X? 5*5=25. That means the square root of 25 is 5.
• Question
Why is 48 not a perfect square?
Donagan
Top Answerer
A "perfect" square is the square of a whole number, but 48 is the square of 6.928.
• Question
How can I write a simplest form of root 20?
Community Answer
You have to make a factor tree. Move from top to bottom like this: 20 2-10 *-5-2 * means no further splits. what number(s) are repeated? 2 is repeated twice, 5 is only written once. Since 2 is repeated twice, the answer is 2sqrt5. However, if you have an odd number of repeats of one number, go up the tree once. EX: 40 2-20 *-2-10 *-*-2-5 2 repeated 3 times, 5 repeated once. That means your answer is 2sqrt10
• Question
How do I solve √16n^2?
Donagan
Top Answerer
√(16n²) = √[(4²)(n²)] = (4)(n) = 4n.
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## Tips
• One way to find perfect squares that factor into a number is to look through the list of perfect squares, beginning with the one that is the next smallest compared to your radicand, or the number under the square root sign. For example, when looking for the perfect square that goes into 27, you might start at 25 and go down the list to 16 and stop at 9, when you found the one that divides into 27.
Thanks!
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## Warnings
• Calculators can be useful for large numbers, but the more you practice working this out on your own, the easier this will get.
Thanks!
• Simplifying is not the same as evaluating. At no point in this process should you get a number with a decimal point in it!
Thanks!
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## About This Article
Co-Authored By:
wikiHow Staff Editor
This article was co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness. Together, they cited information from 8 references. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article meets our high standards.
292 votes - 63%
Co-authors: 76
Updated: January 22, 2020
Views: 1,118,470
Article SummaryX
To simplify a square root, start by dividing the square root by the smallest prime number possible. For example, if you're trying to find the square root of 98, the smallest prime number possible is 2. If you divide 98 by 2, you get 49. Then, rewrite the square root as a multiplication problem under the square root sign. In this case, you'd rewrite the square root as 2 × 49 under the square root sign. From there, keep factoring the numbers until you have 2 identical factors. In this example, 49 divided by 7 is 7. Rewrite the square root as 2 × 7 × 7. Finally, once you have two identical numbers, move them outside of the square root to make them a regular integer. So the simplified square root of 98 is 7 × the square root of 2. However, if you factor the numbers inside the square root as much as you can without getting two identical numbers, then your square root can't be simplified! To learn other ways you can simplify a square root, keep reading!
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### Definition/Introduction
A common definition of aerospace medicine is the medicine of normal physiology in an abnormal environment. A subspecialty within occupational medicine, the challenge of aerospace medicine lies in safeguarding the health of individuals who work in a dynamic atmosphere of lowering pressure and oxygen content, coupled with increasing altitude. It is essential to comprehend the physical changes that occur as we ascend from sea level to address the medical needs of the pilot, aircrew, and aircraft passengers. Most notably, an understanding of the chemical gas laws that describe the characteristics of atmospheric gases as well as those gases that reside within our bodies is critical to the practice of aerospace medicine. Henry’s law is one of the most pertinent of these gas laws to the practice of aerospace medicine. Named after the English physician William Henry, this law defines the relationship between the partial pressure of gases overlying a solution and the gases’ ability to dissolve in that solution.[1]
### Issues of Concern
Henry's law states that when a gaseous mixture (e.g., the atmosphere) is in contact with a solution, the amount of any gas in that mixture that dissolves in the solution is in direct proportion to the partial pressure of that gas. The partial pressure of a gas is the amount of pressure that the gas contributes to the total pressure of that gas mixture. Per Henry's law, if the pressure of a gas over liquid increases, the amount of gas dissolved in the liquid will increase proportionally. Conversely, as the gas pressure decreases, the amount of gas dissolved in the solution drops.
A person experiences Henry's law in action when they open a new bottle of soda pop. Upon removing the cap, the carbon dioxide gas "atmosphere" in contact with the soda rushes out, and the gas pressure drops precipitously. In turn, less of the gas in the soda stays dissolved; the gas comes out of solution as bubbles and foam. So long as adequate gas pressure is maintained over the liquid, the dissolved gases will remain in the solution.[2]
Presented as a formula, Henry's law can be written as the following:
• P1 / A1 = P2 / A2
The left side shows the ratio of P1, the partial pressure of gas overlying a solution initially, to A1, the corresponding amount of gas dissolved in solution at that pressure. Likewise, the right side is the ratio of the same gas at different pressure, P2, and its corresponding amount of dissolved gas, A2, at this new pressure. Since the two sides are equal to each other, a change in the size of P2 accompanies a corresponding change in A2.
For example, your blood is a solution containing multiple gases. At sea level, those gases remain in solution (i.e., blood). This is because, at sea level, air and arterial blood contain approximately the same partial pressure of gases, primarily nitrogen. As one rises in the atmosphere, the partial pressure decreases, and the amount of these gases held in solution (i.e., blood) necessarily must decrease. This results in gas evolving in the bloodstream. As a result, the otherwise inert nitrogen supersaturates in the bloodstream or "bubbles out" like the soda pop example. While some oxygen can potentially evolve out of solution, the overwhelming majority of oxygen in the bloodstream is bound to hemoglobin, which prevents this occurrence.[3][4]
The unbound nitrogen bubbles in the vascular system result in various forms of decompression illness, a catchall term referring to both the discomfort associated with decompression sickness as well as more severe conditions like an arterial gas embolism. This problem can be partly controlled with the pressurization of the cabin. Additionally, so long as the rate of ascent is relatively slow, the risk of a supersaturation resulting in bubble formation is relatively low. It is largely because of this ascent at a controlled rate and cabin pressurization that bubbles normally do not form during a commercial airline flight. In most commercial and military fixed-wing aircraft, pressurization typically is performed to maintain the partial pressure of a gas in the cabin to the equivalent of the pressure at 8000 feet (2438 meters). It is considered an acceptable risk to fly at elevations up to 12000 feet (3658 meters) unpressurized, though this practice is discouraged.[1]
The applications Henry'sy's law, however, do have their limitations. At high concentrations of gas in the liquid phase, Henry's law is highly inaccurate. Low concentrations of gas in a solution can be accurately captured wiHenry'sy's law, as shown in the figure.[3]
### Clinical Significance
Henry’s law is significant to aerospace medicine in several ways. The most apparent is that Henry’s law accurately describes the phenomena that result in decompression illness as well as affording the framework for the measures used to treat these conditions.
While the concept of Henry’s law can logically apply to both aerospace and undersea medicine, the manifestations of Henry’s law in flight do not perfectly correspond to those underwater. Specifically, manifestations such as decompression sickness directly tie to the absolute altitude achieved. While pressurization reduces the effects of decompression illness and can allow the pilot and other crew members to gradually acclimate to decreased air pressure, at higher elevations, even pressurized aircraft cannot fly safely due to the risk of supersaturation of inert gases in the bloodstream.
Current data suggests that a flyer in a pressurized cabin is less likely to experience decompression illness at an exposure equivalent of fewer than 18000 feet (5486 meters). In civilian and military aviation, the challenge of avoiding decompression illness tends to be higher in rotor wing, or helicopter, pilots, and crew. These aircraft have unique requirements that usually preclude cabin pressurization and thus are forced to fly at lower altitudes to avoid the effects of Henry’s law on human physiology.
Several techniques are effective in both the prevention and treatment of conditions at altitude related to Henry’s law. Breathing gases used in high altitude aircraft usually have a higher concentration of oxygen than at sea level to both prevent hypoxia and help promote denitrogenation; by removing nitrogen, the hope is to avoid supersaturation effects at altitude. This approach is also the premise behind placing oxygen masks on oneself during an airline emergency where the cabin pressure is compromised. Cabin pressurization is the other significant preventive measure to reduce the risk of decompression.
The ideal option for resolving symptoms of bubble formation while in flight is threefold. First, drop in altitude, ideally to sea level if possible. Second, oxygen should be used therapeutically to “blow off” nitrous gas. Finally, landing to permit evaluation by a flight surgeon is optimal to determine if further interventions are necessary. If symptoms persist or significant central nervous system manifestations occur, it is essential to immediately transfer the patient to a hyperbaric chamber to pressurize the gas surrounding the patient and dissolve nitrogen gas back into solution in the bloodstream.[5]
### Nursing, Allied Health, and Interprofessional Team Interventions
Nursing staff and other ancillary health team members who treat patients with diving and/or aerospace medical concerns would be well advised to have at least a basic understanding of the concepts surrounding Henry's law, so they can effectively contribute to patient care and evaluation and communicate appropriately on the physiological ramifications with clinicians.
(Click Image to Enlarge)
Henry's Law use at low solution composition. At higher concentrations, Henry's Law is not a valid assumption.
Contributed by Kevin M. Tenny
###### Article Details
Article Author
Dor M. Avishay
Article Editor:
Kevin M. Tenny
Updated:
1/29/2023 9:24:03 AM
### References
[1]
Ferraro G,Jadhav AJ,Barigou M, A Henry's law method for generating bulk nanobubbles. Nanoscale. 2020 Aug 7; [PubMed PMID: 32696779]
[2]
Jones MW,Brett K,Han N,Wyatt HA, Hyperbaric Physics StatPearls. 2021 Jan; [PubMed PMID: 28846268]
[3]
Dukes Iii AD, Measuring the Henry's Law Constant for Carbon Dioxide and Water with UV-visible Absorption Spectroscopy. Analytical sciences : the international journal of the Japan Society for Analytical Chemistry. 2020 Aug 10; [PubMed PMID: 32092731]
[4]
Goldman S,Solano-Altamirano JM, An explicitly multi-component arterial gas embolus dissolves much more slowly than its one-component approximation. Mathematical biosciences. 2020 Aug; [PubMed PMID: 32497622]
[5]
Van Poucke S,Hans G,Hens P, The release of dissolved gases from solution during decompression after hyperbaric treatment: effervescent tables and Henry's law. Anesthesiology. 2001 Sep; [PubMed PMID: 11575571] | 1,851 | 8,970 | {"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} | 2.578125 | 3 | CC-MAIN-2023-23 | latest | en | 0.934277 |
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### Appendix: Explanation of the Math behind the Individual Shock Model of the Interbank Rate
This appendix explains the mathematical foundations for the "individual shock model of liquidity management.” All that's required to understand this a basic familiarity with probability and calculus.
Here's the basic mathematical representation of the model:
The integral in the equation is an expected value calculation. We can understand it by first thinking of an expected value calculation without calculus.
Assume a function, f(x), of a random variable, x, equals 0 or 1 with 50% probability each. If I asked you to calculate the expected value of f(x), you’d probably be able to figure out intuitively that the answer is: 0*.5 + 1*.5 = .5. We simply multiple each possible value of f(x) by its respective probability of occurring. We can represent this in mathematical terms as E[f(x)] = ∑fipi, where E[] represents the expected value of whatever we’re interested in, fi represents each possible value of f(x), and pi represents the probability of a given value of f(x), and ∑ represents the summation of each fipi pair.
In the example above, the random variable is discrete, as there were a finite number of values f(x) could take on. Thus the ∑fipi calculation works. However, if we’re thinking about an infinite number of values, the random variable is continuous, and we need to use calculus. The expected value calculation analog for a continuous random variable is
E[f(x)] = ∫f(x)*Φ(x)dx, integrated from negative infinity to positive infinity
Φ(x) is the probability density function of the continuous random variable x. You can see this is very similar to the discrete calculation. We’ve pretty much just substituted an integral for the ∑ sign, and a probability density function for pi.
To address the equation in the post specifically, let’s match up the notation of the individual shock model to the general example E[f(x)] = ∫f(x)*Φ(x)dx.
C(q) corresponds to E[f(x)]. Our f(x) is the following step function:
f(x) = iB*(q+x) , if x < -q
f(x) = 0 , if x > -q
So now we just need to explain the limits on the integral. Before I said that the expected value integral integrates from negative to positive infinity. Why is it from negative infinity to –q in this equation? The answer is that you don’t have to integrate to positive infinity, because once the random liquidity shock x > -q, then C(q) = 0 by definition. In other words, at all values of x where x > -q, the liquidity shock is not negative enough to drive the bank’s reserves below zero, in which case the bank will face 0 borrowing costs. C(q) will thus be 0, and taking the integral of 0 across any range of values is 0. Thus, it would be unnecessary to continuing integrating, but you could if you wanted.
And that’s it! | 652 | 2,824 | {"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} | 4.34375 | 4 | CC-MAIN-2017-34 | longest | en | 0.887809 |
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Home | Contact | Blog | About | Terms | Privacy | © Purple Inc. | 681 | 2,217 | {"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} | 2.53125 | 3 | CC-MAIN-2020-45 | latest | en | 0.888533 |
http://www.homeofbob.com/pedagogy/plan/reflectDecisions/lightLCLesnPln.htm | 1,580,291,075,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579251789055.93/warc/CC-MAIN-20200129071944-20200129101944-00369.warc.gz | 220,004,650 | 5,902 | # Initial Planning Reflection for - An Activity Sequence on Light
Generalization
Light is a form of energy that travels in a straight line until it strikes or interacts with an object. Interactions can cause light to be reflected such as with a mirror, refracted by objects such as lens, and reflected or absorbed by any object it strikes such as a block.
Possible Related Concepts
• Light interacts with matter by transmission (including refraction), absorption, or scattering (including reflection).
• To see an object, light is emitted by the object or reflected from it and enters the eye.
• Light is reflected at an angle proportional to the angle it strikes an object.
• Light is refracted at an angle related to the angle that it enters or leaves a medium and the density of the mediums.
• Color is a property of light.
• White light has all colors.
• Black is the absence of light and/or color.
• White light can be refracted to view a spectrum.
• Light is a form of energy that can be transfered or changed to other forms of energy.
What instructional theory and learning theory should be used to begin to facilitate student learning for these ideas?
I would want to begin with exploration so the students will learn through their own actions and reactions in a new situation. If they explore new materials and new ideas with minimal guidance or expectation of specific accomplishments, the new experience should raise questions they cannot answer with their present ideas and patterns of reasoning. Having made an effort that most likely will not be completely successful, they should be motivated to ask questions and begin to look for ideas that will lead them to self-regulation. Meanwhile, I will have collected enough assessment data to begin to know what they understand and where I might begin to facilitate their learning in the second phase: invention.
What activities could be used for the first activity with students?
Activities Review a resource file or a list of possible activities that would fit the concepts and generalization.
List of possible activities
1. Listing the assumption of light as a particle that travels in a straight line?
2. Doing an activity in which students could assemble light sources, mirrors, plastic blocks, colored filters, and glasses of water to observe how it interacts with these objects?
3. Asking students what everyday experiences with light they have and describe the properties of light that those observations indicate.
4. Describing the transfer of electromagnetic radiation at various frequencies, and then focusing on the visible spectrum of light?
5. Doing an activity as in two, but making certain that your students could work with a laser pointer as source with a good projection of a line?
6. Doing an activity where students are assigned to measure accurately the focal lengths of plane, convergent, and divergent mirrors, and lenses (diagram their results).
Making a decision
Review what resources are needed for each activity and the preparation of students needed for each activity. Eliminate activities that wouldn't fit the availability of resources and the readiness of students. Then think about how each would or would not be good to use as the first activity.choice. When you have done that, compare the ideas below with yours, and if possible, with those of others.
Reflect on the positives and negatives for each and make a decision as to what you believe might be the best before continuing.
Compare your ideas with the following:
1. This procedure is frequently used because of its conciseness but it is likely to be difficult for students, especially students using concrete reasoning patterns, to assimilate. They do not know the basis of the assumptions and therefore cannot evaluate when and how the new ideas are to be used.
2. We would prefer an approach of this kind, where the student has a great deal of freedom to use his or her own judgment and try out his or her own ideas as he or she gains practical experience with the objects he will study theoretically later. See also five.
3. In the absence of exploratory materials, this approach can be attempted to connect any new ideas about how light interacts with objects with the students previous experience; demonstrations with student participation would help.
4. This theoretical approach is inappropriate as one of the activities, because it highlights the wave nature of the light which is not helpful for constructing a model to describe lights interactions with these objects.
5. Since light “rays” play an important part in creating a model of how light interacts, it should be very helpful. An ordinary comb with coarse teeth can be used very effectively to make a bundle of light “rays” whose behavior can be followed.
6. This type of activity prevents the student from asking his or her own questions and satisfying his own curiosity. The concept of refraction and reflection needs to be operationally defined and understood before this lab can be worthwhile. At a later time in 7-8 grade middle school, it might be quite appropriate, though a more open approach would be preferred.
The preferred approach in two or five is an example of the “exploration” phase in the learning cycle which is recommend for the planning at the beginning of a learning sequence. The entire learning cycle consists of three phases: exploration, invention, and discovery. During exploration students learn through their own more or less spontaneous reactions to a new situation. In this phase, they explore new materials or ideas with minimal guidance or expectation of specific achievements. Their patterns of reasoning may be inadequate to cope with the new data, and they may begin self-regulation. During this phase of the learning sequence the teacher is collecting assessment information to facilitate students' learning in the invention phase of the learning cycle.
During the “invention” phase, it is appropriate to define a new concept, introduce a new principle, or explain a new kind of application to expand the students’ knowledge, skills, or reasoning. This step should always follow exploration and relate to the exploration activities. It will assist in students’ self-regulation. In the example above, for instance, alternative one represents a possible “invention” phase, perhaps introduced via activity three as an intermediate step to relate exploration and invention. Encourage all students in the invention to “invent” part or all of a new concept (idea) for themselves, before presenting it to the class. The invention of some concepts will require several activities to develop students’ understanding.
Multiple activities sequenced in the invention phase of the learning cycle, allows students to find the limits of the concepts or skills he or she has conceptualized (learned) earlier. This provides additional time and experiences for self-regulation. It also gives opportunities to introduce the new concept repeatedly to help students whose conceptual re-organization proceeds more slowly, or who did not adequately construct accurate representations from their observations in the earlier experiences. Individual conferences with these students are helpful to identify their difficulties, By reaquainting them with observable data that is contrary to their explanations, change in their understanding is possible. Without previous experiences to relate to the new ideas students will likely continue with their alternative nonscientific or less accurate understandings.
Generalization of concepts is encouraged in the third phase, expansion, where concepts are combined to create generalizations or experienced in unique or unusual ways. Which provides a measure of the students' depth and breadth of understanding. | 1,481 | 7,838 | {"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} | 2.578125 | 3 | CC-MAIN-2020-05 | latest | en | 0.955019 |
https://pinoybix.org/2015/01/mcqs-in-engineering-mechanics-part3.html | 1,582,974,157,000,000,000 | text/html | crawl-data/CC-MAIN-2020-10/segments/1581875148850.96/warc/CC-MAIN-20200229083813-20200229113813-00108.warc.gz | 498,912,394 | 23,088 | # MCQ in Engineering Mechanics Part 3 | ECE Board Exam
(Last Updated On: February 22, 2020)
This is the Multiples Choice Questions Part 3 of the Series in Engineering Mechanics as one of the General Engineering and Applied Sciences (GEAS) topic. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including past Board Questions in General Engineering and Applied Sciences (GEAS), Engineering Mechanics Books, Journals and other Engineering Mechanics References.
### Online Questions and Answers in Engineering Mechanics Series
Following is the list of multiple choice questions in this brand new series:
Engineering Mechanics MCQs
PART 1: MCQs from Number 1 – 50 Answer key: PART I
PART 2: MCQs from Number 51 – 100 Answer key: PART II
PART 3: MCQs from Number 101 – 150 Answer key: PART III
PART 4: MCQs from Number 151 – 200 Answer key: PART IV
PART 5: MCQs from Number 201 – 250 Answer key: PART V
PART 6: MCQs from Number 251 – 300 Answer key: PART VI
PART 7: MCQs from Number 301 – 350 Answer key: PART VII
PART 8: MCQs from Number 351 – 400 Answer key: PART VIII
### Continue Practice Exam Test Questions Part III of the Series
Choose the letter of the best answer in each questions.
101. A physical quantity that is completely described by a real number is called ___________.
• A. scalar
• B. vector
• C. tensor
• D. none of the above
102. A convenient means of representing physical quantities that have magnitude and direction.
• A. scalars
• B. vectors
• C. tensors
• D. none of the above
103. The product of a scalar and a vector is a
• A. scalar
• B. vector
• C. tensor
• D. none of the above
104. It is simply a vector whose magnitude is 1
• A. moment vector
• B. tensor
• C. unit vector
• D. vector unity
105. It is sometimes called the scalar product.
• A. dot product
• B. vector product
• C. cross product
• D. unit scalar
106. To calculate for the force exerted on a charged particle by a magnetic field, _________ is used.
• A. vector product
• B. scalar product
• C. dot product
• D. vector sum
107. It is sometimes called the vector product.
• A. dot product
• B. cross product
• C. tensor product
• D. unit vector
108. Which of the following statements is false?
• A. The cross product is commutative.
• B. The cross product is associative with respect to scalar multiplication.
• C. The cross product is distributive with respect to vector addition.
• D. The angle between two identical vectors placed tail to tail is zero.
109. When a force is represented by a vector, the straight line collinear with the vector is called the ________.
• A. line of apsides
• B. line of reaction
• C. line of vector
• D. line of action
110. A system of forces is __________ if the lines of action of the forces intersect at a point.
• A. parallel
• B. coplanar
• C. concurrent
• D. two-dimensional
111. A system of two forces is ___________ if the lines of action of the forces lie in a plane.
• A. coplanar
• B. two-dimensional
• C. A or B
• D. none of the above
112. Force acting on an object is called a ___________ if its acts on the volume of the object.
• A. internal force
• B. external force
• C. body force
• D. surface force
113. If each point on the object has the same constant velocity, this is referred to as __________.
• A. continuum translation
• B. discrete translation
• C. finite translation
114. The moment of a force about a point P is equal to the sum of the moments of its components about P.
• A. Cavalieri’s Theorem
• B. Pascal’s Theorem
• C. Varignon’s Theorem
• D. Torricelli’s Theorem
115. It is the measure of the tendency of a force to cause rotation about a line or axis.
• A. moment
• B. momentum
• C. impulse
• D. torsion
116. A couple is composed of two forces that are
• A. equal
• B. equal and opposite
• C. equal and different lines of action
• D. equal, opposite and different lines of action
117. Which of the following statements is true about a couple?
• A. A couple does not tend to cause a rotation of an object.
• B. The vector sum of the force couple always has a value.
• C. A couple tends to cause a rotation of an object.
• D. The moment it exerts is not the same about any point.
118. If an object is on an inclined plane having an angle θ, the component of weight (w) parallel to incline is __________.
• A. w sinθ
• B. w cosθ
• C. w tanθ
• D. w cotθ
119. A type of force acting on a body due to the acceleration of gravity.
• B. shear
• C. bear
• D. mass
120. A type of force acting on a body caused by the friction between the body and the ground.
• B. shear
• C. bear
• D. mass
121. The unit of force, Newton, is equivalent to
• A. lbm-ft/s2
• B. g-cm/s2
• C. kg-m/s2
• D. kgf
122. When a body is in contact with the ground, the force that is reflected back to the body is called
• A. ground reflected force
• B. gravity reflected force
• C. ground reaction force
• D. gravity reaction force
123. The gravity in the moon is about
• A. 1.6 m/s2
• B. 2.6 m/s2
• C. 3.6 m/s2
• D. 0.6 m/s2
124. The ground reaction force on a body can be represented by a single force acting on a point called
• A. center of force
• B. center of reaction
• C. center of reflection
• D. center of pressure
125. The tuning effect on a body is dependent on which of the following?
• A. mass of the load
• B. acceleration of gravity
• C. moment arm
• D. all of the above
126. The perpendicular distance of the force from the point about which the body will turn is called
• A. moment arm
• B. moment distance
• C. lever arm
• D. A or C
127. If the force is moved in the direction parallel to the direction of the force, the moment exerted by the force ___________.
• A. increases
• B. decreases
• C. is unchanged
• D. becomes zero
128. The moment of force is zero when
• A. the applied force is zero.
• B. the force is applied at the moment axis.
• C. the line of action of the force is parallel to the axis.
• D. all of the above
129. __________ is finding a single force which shall be equal to two or more given forces when acting in given directions.
• A. resolution of forces
• B. integration of forces
• C. composition of forces
• D. quantization of forces
130. He is the father of the modern engineering mechanics
• A. Gilbert Lewis
• B. Stephen Timoshenko
• C. J. Gordon
• D. A. Cotrell
131. It is a method of applying mechanics that assumes all objects are continuous.
• A. Discrete Mechanics
• B. Finite Element Method
• C. Continuum Mechanics
• D. Contact Mechanics
132. Which of the following is an example of contact force?
• A. gravitational force
• B. magnetic force
• C. air resistance force
• D. electric force
133. It occurs when an object is moving across a surface.
• A. dynamic friction
• B. static friction
• C. kinetic friction
• D. sliding friction
134. Given µ = 0.35 between the object of mass 400 g and the floor, the object will __________ if pulled with a force of 3 N.
• A. remain at rest
• B. move
• C. accelerate
• D. B and C
135. Which of the following statements is correct?
• A. The coefficient of static friction is always less than 1.
• B. The coefficient of static friction is typically greater than the coefficient of kinetic friction.
• C. The coefficient of kinetic friction is typically greater than the coefficient of static friction.
• D. The coefficient of static and kinetic friction are always equal.
136. The equations that apply to bodies moving linearly (that is, one dimension) with uniform accelerations are often referred to as
• A. UVATS
• B. SUVAT
• C. UVATS
• D. Either of the above
137. “Observed from an internal reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum”. This is known as Newton’s ___________ of motion.
• A. Zeroth Law
• B. First Law
• C. Second Law
• D. Third Law
138. It is also known as quantity of motion.
• A. momentum
• B. force
• C. mass
• D. acceleration
139. This concept assumes that the substance of the body is distributed throughout and completely fills the space it occupies.
• A. Finite Element
• B. Contact
• C. Discrete
• D. Continuum
140. In fluids, _________ is used to assess to what extent the approximation of continuity can be made.
• A. Brayton Number
• B. Knudsen Number
• C. Reynolds Number
• D. Prandtl Number
141. It is the time rate of change of any property of a continuum for a specified group of particles of the moving continuum body.
• A. Material Derivative
• B. Continual Derivative
• C. Particle Derivative
• D. Quantum Derivative
142. Material derivative is also known as __________.
• A. substantial derivative
• B. commoving derivative
• C. convective derivative
• D. all of the above
143. The vector connecting the positions of a particle in the undeformed and deformed configuration is called the ________.
• A. displacement vector
• B. position vector
• C. displacement field
• D. position field
144. A __________ is a vector field of all displacement vectors for all particles in the body.
• A. position field
• B. action field
• C. displacement field
• D. path field
145. _________ is the study of the physics of continuous solids with a defined rest shape.
• A. Continuum Mechanics
• B. Solid Mechanics
• C. Fluid Mechanics
• D. Discrete Mechanics
146. It is an experimental method for visualizing and analyzing fluid flow.
• A. Particle Image Velocimetry
• B. Particle Image Accelerometry
• C. Particle Image Flowmeter
• D. Particle Image Viscosimetry
147. A fluid at rest has no
• A. longitudinal stress
• B. shear stress
• C. tensile stress
• D. compressive stress
148. A property of fluids which is the force generated by a fluid in response to a velocity gradient.
• A. compressibility
• B. plasticity
• C. elasticity
• D. viscosity
149. These equations state that changes in momentum of fluid particles depend only on the external pressure and internal viscous forces acting on the fluid.
• A. Navier – Stokes Equations
• B. Torricelli Equations
• C. Reynolds Equations
• D. Lagrangian Equations
150. It is defined as, regardless of the forces acting on a fluid, the fluid continues to flow
• A. Newtonian fluid
• B. non-Newtonian fluid
• C. Lagrangian fluid
• D. non-Lagrangian fluid
### Complete List of MCQs in General Engineering and Applied Science per topic
Help Me Makes a Difference!
P inoyBIX educates thousands of reviewers/students a day in preparation for their board examinations. Also provides professionals with materials for their lectures and practice exams. Help me go forward with the same spirit. “Will you make a small \$5 gift today?” Option 1 : \$1 USD Option 2 : \$3 USD Option 3 : \$5 USD Option 4 : \$10 USD Option 5 : \$25 USD Option 6 : \$50 USD Option 7 : \$100 USD Option 8 : Other Amount | 2,862 | 10,987 | {"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} | 3.1875 | 3 | CC-MAIN-2020-10 | longest | en | 0.820136 |
http://mathhelpforum.com/trigonometry/27387-algebra-iii-trig-2-problems.html | 1,527,385,009,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00589.warc.gz | 193,416,700 | 9,727 | # Thread: Algebra III/trig 2 problems
1. ## Algebra III/trig 2 problems
I need help on these two problems my teacher gave us. I'm already failing the class but I need at least one good grade. Directions: A) f+g(x)
B) f-g(x) C) fg(x) D) F/g(x) Find the domain
1. f(x)=2x-5 g(x)=2-x
A=
B=
C=
D=
2. f(x)=2x-5 g(x)=4
2. Originally Posted by hatemath101
I need help on these two problems my teacher gave us. I'm already failing the class but I need at least one good grade. Directions: A) f+g(x)
B) f-g(x) C) fg(x) D) F/g(x) Find the domain
1. f(x)=2x-5 g(x)=2-x
A=
B=
C=
D=
2. f(x)=2x-5 g(x)=4
$\displaystyle (f + g)(x) = f(x) + g(x)$
$\displaystyle (f - g)(x) = f(x) - g(x)$
$\displaystyle fg(x) = f(x)g(x)$
$\displaystyle \frac fg(x) = \frac {f(x)}{g(x)}$
we are dealing with polynomials here, so the domains for A, B, and C are all real x, that is $\displaystyle x \in (- \infty, \infty)$
for D, obviously the denominator cannot be zero, so the domain is all real x such that $\displaystyle g(x) \ne 0$ | 364 | 1,013 | {"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 | 4 | CC-MAIN-2018-22 | latest | en | 0.799509 |
https://fabulousfrocksofatlanta.com/what-is-real-valued-function-in-complex-analysis/ | 1,721,155,985,000,000,000 | text/html | crawl-data/CC-MAIN-2024-30/segments/1720763514789.44/warc/CC-MAIN-20240716183855-20240716213855-00767.warc.gz | 212,952,570 | 10,179 | A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values.
## What is real-valued function in complex analysis?
A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values.
## What is complex function theory?
Complex Function Theory is a concise and rigorous introduction to the theory of functions of a complex variable. Written in a classical style, it is in the spirit of the books by Ahlfors and by Saks and Zygmund. Being designed for a one-semester course, it is much shorter than many of the standard texts.
What is meant by a real-valued function?
A real-valued function of a real variable is a mapping of a subset of the set R of all real numbers into R. For example, a function f(n) = 2n, n = 0, ±1, ±2, …, is a mapping of the set R’ of all integers into R’, or more precisely a one-to-one mapping of R’ onto the set R″ of all even numbers, which shows R’ ∼ R″’.
How a complex function is related with real function?
Complex functions are all real-valued. Similarly, any complex-valued function f on an arbitrary set X can be considered as an ordered pair of two real-valued functions: (Re f, Im f) or, alternatively, as a vector-valued function from X into.
### What is real to real function?
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
### What is the difference between real function and real-valued function?
According to my textbook: A function which has either R or one of its subsets as its range is called a real valued function. Further, if its domain is also either R or a subset of R, it is called a real function.
Where is complex analysis used in real life?
Complex analysis is used in analog electronic design. Filters are characterized by singularities of a complex transfer function. Impedance is modeled as a complex value in AC circuits such as audio amplifiers. The wave function of quantum mechanics and quantum field theory is complex-valued.
What is the real life application of complex analysis?
The application of these methods to real world problems include propagation of acoustic waves relevant for the design of jet engines, development of boundary-integral techniques useful for solution of many problems arising in solid and fluid mechanics as well as conformal geometry in imaging, shape analysis and …
## What is the difference between real functions and real-valued function?
What is meant by real analysis?
Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.
Is real function and real-valued function same?
### What is a real-valued function?
A function whose range is a set of real numbers is called a real-valued function. Example 1: Consider the sets D and Y related to each other as shown below. Can we consider this relation as a real-valued function?
### Is the relation K 6 8 and 9 a function?
So, the given relation is a function. The range of this function consists of the elements K, 6, 8, and 9. Since ‘K’ is not a real number, the range is not the set of real numbers and therefore, this function is not a real-valued function.
Is every continuous real-valued function upper and lower semicontinuous?
Indeed, each continuous real-valued function on a topological space is upper and lower semicontinuous. A real-valued function ∥ · ∥ defined on a linear space V over F is called a norm if for all x, y ∈ V and a ∈ F: | 860 | 3,962 | {"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-2024-30 | latest | en | 0.931549 |
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Topic: possible explanation for Doppler redshift as tired light Chapt13.4091
Summary of DTW theory #1063 New Physics #1183 ATOM TOTALITY 5th ed
Replies: 0
plutonium.archimedes@gmail.com Posts: 7,382 Registered: 3/31/08
possible explanation for Doppler redshift as tired light Chapt13.4091
Summary of DTW theory #1063 New Physics #1183 ATOM TOTALITY 5th ed
Posted: Dec 5, 2012 5:18 PM
The lowest electron-neutrino is
M- M+
taking up just 2 poles of the 4 poles possible
The next lowest neutrino is
M-
M- M+
taking up 3 of the 4 poles
and the next lowest is
M-
M- M+
M+
The lowest of low photons, a radio wave photon
? 0.5*10^6M-
1M- 1M+
? 0.5*10^6M+
next lowest photon
? 0.5*10^6M-
2M- 2M+
? 0.5*10^6M+
next lowest photon
? 0.5*10^6M-
3M- 3M+
? 0.5*10^6M+
As can be seen I am still battling the symmetry asymmetry problem of
how
many poles involved.
It occurred to me that we possibly can have a easy explanation of
Doppler redshift of light from far distant galaxies in that as the
photon is travelling through space it
picks up M charges of magnetic monopoles and causes the photon to so
to speak "tired light in travel".
So if we had a photon of this:
0.5*10^6M-
4M- 3M+
? 0.5*10^6M+
where it picked up a extra M- causing a slight imbalance and asymmetry
that it becomes "tired light". But I do not have time to explore this
issue now for I have more pressing problems to fix.
Google's New-Newsgroups censors AP posts and halted a proper
archiving of author, but Drexel's Math Forum does not and my posts?in
archive form is seen here:
http://mathforum.org/kb/profile.jspa?userID=499986
Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies | 602 | 2,002 | {"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} | 2.5625 | 3 | CC-MAIN-2013-20 | latest | en | 0.802907 |
https://www.studymode.com/essays/Allele-And-Probability-53240129.html | 1,544,977,390,000,000,000 | text/html | crawl-data/CC-MAIN-2018-51/segments/1544376827769.75/warc/CC-MAIN-20181216143418-20181216165418-00309.warc.gz | 1,059,513,440 | 25,451 | # Allele and Probability
Topics: Allele, Doctorate, Academic degree Pages: 5 (1288 words) Published: May 31, 2014
5.1 #12 , #34a. and b, #40, 48
#12. Which of the following numbers could be the probability of an event? 1.5, 0, = ,0
#34 More Genetics In Problem 33, we learned that for some diseases, such as sickle-cell anemia, an individual will get the disease only if he or she receives both recessive alleles. This is not always the case. For example, Huntington’s disease only requires one dominant gene for an individual to contract the disease. Suppose that a husband and wife, who both have a dominant Huntington’s disease allele (S) and a normal recessive allele (s), decide to have a child. (a) List the possible genotypes of their offspring. (a) Sample space is {SS,Ss,sS,ss} where S=dominant disease allele and s=normal recessive allele (b) What is the probability that the offspring will not have Huntington’s disease? In other words, what is the probability that the offspring will have genotype ss? Interpret this probability(b) Since P(S)=P(s)=1/2 and S,s are independent, P(offspring will not have Huntington’s disease)=P(SS)=P(S)*P(S)=1/4
#40. Which of the assignments of probabilities should be used if the coin is known to be fair? If coin is fair, then assignment A is used because P(H)=P(T)=1/2 #48. Classifying Probability Determine whether the probabilities on the following page are computed using classical methods, empirical methods, or subjective methods. a) The probability of having eight girls in an eight-child family is 0.390625%. Empirical method
(b) On the basis of a survey of 1000 families with eight children, the probability of a family having eight girls is 0.54%. Classical method
(c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about 30%. Subjective method
(d) On the basis of clinical trials, the probability of efficacy of a new drug is 75%. Empirical method
5.2 #26 a-d #32
#26 . Doctorates Conferred The following probability model shows the distribution of doctoral degrees from U.S. universities in 2009 by area of study. Area of Study Probability
Engineering 0.154
Physical sciences0.087
Life sciences 0.203
Mathematics 0.031
Computer sciences0.033
Social sciences0.168
Humanities0.094
Education0.132
Professional and other fields0.056
Health0.042
(a) Verify that this is a probability model. ) First, all probabilities range from 0 to 1 inclusive. Second, the sum of probabilities=0.154+…+0.042=1. So it is a probability model.
(b) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science or life science? Interpret this probability. P=0.087+0.203=0.29. Interpretation: if there are 100 candidates, 29 of them studies physical science or life science. (c) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 studied physical science, life science, mathematics, or computer science? Interpret this probability. P=0.087+0.203+0.031+0.033=0.354 Interpretation: if there are 1000 candidates, 354 of them studies physical science, life science, mathematics or compute science (d) What is the probability that a randomly selected doctoral candidate who earned a degree in 2009 did not study mathematics? Interpret this probability. d) P=1-0.031=0.969. Interpretation: if there are 1000 candidates, 969 of them do not study mathematics #32. A Deck of Cards A standard deck of cards contains 52 cards, as shown in Figure 9. One card is randomly selected from the deck. (a) Compute the probability of randomly selecting a two or three from a deck of cards. Since there are 4 “two’s” and 4 “three’s” in a deck of 52 cards, P( a two or a three)=P(a two)+P(a three)=4/52+4/52=2/13 (b) Compute the probability of randomly selecting a two or three or four from a deck of cards. Since there are 4 “two’s”, 4 “three’s” and 4 “four’s”in a deck of 52 cards, P( a two or a three or a four)=P(a two)+P(a three)+P(a... | 1,037 | 4,013 | {"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-2018-51 | longest | en | 0.929118 |
http://www.rabcpafirm.com/calculators/lease-vs-buy/ | 1,537,971,995,000,000,000 | text/html | crawl-data/CC-MAIN-2018-39/segments/1537267165261.94/warc/CC-MAIN-20180926140948-20180926161348-00051.warc.gz | 385,981,927 | 7,726 | # Lease vs. Buy Calculator
Should you lease or buy your car? Use this calculator to find out! We calculate your monthly payments and your total net cost. By comparing these amounts, you can determine which is the better value for you.
## Definitions
• Term in Months. Term in months for your auto lease or your auto loan.
• Down Payment. Amount paid as a down payment, which for leases is often called a capital reduction.
• Other Fees. Any fee, other than a capital reduction or down payment, required to be paid at the close of the lease or loan. This may include license, title transfer fees, etc.
• Purchase Price. Total purchase price. Price should be after any manufacturer's rebate.
• Interest Rate. Annual interest rate for your loan or your lease.
• Sales Tax Rate. Percentage sales tax to be charged on this purchase. Sales tax is included in each lease payment. Sales tax for buying is charged on the total sale amount.
• Rate of Depreciation. The rate of depreciation gauges how fast your new automobile will lose its market value. A high depreciation rate is about 20% per year, medium is 15% per year and low is 10% per year.
• Residual Percent. For leases, this is remaining value after the lease term expires. The higher this amount, the lower your lease payment will be.
• Market Value of Vehicle. Value of your auto after the lease term is over.
• Investment Rate of Return. Rate of return on investments. This is the return that you would make if you were to invest your down payment or security deposit instead of using it in your auto purchase or lease.
The actual rate of return is largely dependent on the type of investments you select. The S&P 500 for the ten years ending on December 31st, 2011 had an annual compounded rate of return of 2.92%, including reinvestment of dividends. From January 1970 through the end of 2011, the average annual compounded rate of return for the S&P 500, including reinvestment of dividends, was approximately 10.01% (source: www.standardandpoors.com). Since 1970, the highest 12-month return was 61% (June 1982 through June 1983). The lowest 12-month return was -43% (March 2008 to March 2009). Savings accounts at a bank may pay as little as 0.25% or less but carry significantly lower risk of loss of principal balances.
It is important to remember that these scenarios are hypothetical and that future rates of return can't be predicted with certainty and that investments that pay higher rates of return are generally subject to higher risk and volatility. The actual rate of return on investments can vary widely over time, especially for long-term investments. This includes the potential loss of principal on your investment. It is not possible to invest directly in an index and the compounded rate of return noted above does not reflect sales charges and other fees that funds and/or investment companies may charge.
• Lost Interest on Buy Option. This includes any interest you would have earned at your investment rate of return on the buy option's down payment and other fees. If the monthly payment for leasing is less than the monthly payment for buying, this also includes any lost interest due to the higher monthly payments. If leasing is more expensive than buying, your interest costs for buying are reduced by the amount of interest you would earn on the difference.
• Lost Interest on Lease Option. This includes any interest you would have earned at your investment rate of return on the lease option's down payment, security deposit and other fees. Please see the definition for "Lost interest on buy option" for an explanation on how we account for any interest you might earn by having a lower monthly lease payment. | 779 | 3,705 | {"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} | 2.546875 | 3 | CC-MAIN-2018-39 | latest | en | 0.955102 |
https://www.teacherspayteachers.com/Product/CCSS-4th-Grade-Numbers-Operations-in-Base-10-Word-Wall-Posters-Flash-Cards-819365 | 1,485,295,592,000,000,000 | text/html | crawl-data/CC-MAIN-2017-04/segments/1484560285289.45/warc/CC-MAIN-20170116095125-00089-ip-10-171-10-70.ec2.internal.warc.gz | 989,306,208 | 52,152 | # CCSS 4th Grade Numbers & Operations in Base 10 Word Wall Posters & Flash Cards
Subjects
Resource Types
Common Core Standards
Product Rating
4.0
File Type
PDF (Acrobat) Document File
15.42 MB | 75 pages
### PRODUCT DESCRIPTION
This set covers 27 Common Core vocabulary words from the 4th grade Numbers and Operations in Base 10 domain. For each word there is a 8.5 x 11" color poster, 8.5 x 11" black and white poster, 3 x 5" color flash card, and 3 x 5" black and white flash card. Each word includes a short definition as well as a picture and/or example.
The visual support makes these a great resource for English Language Learners. The posters can be used for a word wall or as a teaching aide. I use the flash cards for ELL students as well as struggling students. These flash cards go on a binder ring that they keep in their binders and can use when completing work.
Words:
Area model
Base-ten
Compare
Digit
Dividend
Division
Divisor
Equal to
Equation
Estimate
Expanded form
Greater than
Less than
Multi-digit
Multiplication
Number name
Numeral
Operation
Place value
Product
Quotient
Rectangular array
Remainder
Rounding
Standard subtraction algorithm
Whole number
Common Core State Standards
Cluster: Generalize place value understanding for multi-digit whole numbers.
Standards:
CCSS.Math.Content.4.NBT.A.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
CCSS.Math.Content.4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
CCSS.Math.Content.4.NBT.A.3 Use place value understanding to round multi-digit whole numbers to any place.
Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic.
Standards:
CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
CCSS.Math.Content.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Also available:
Common Core 4th Grade Mathematics Vocabulary Posters & Flash Cards - ALL Domains
Common Core 4th Grade Numbers and Operations Fractions Vocabulary
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16 ratings | 802 | 3,405 | {"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} | 3.390625 | 3 | CC-MAIN-2017-04 | longest | en | 0.825405 |
https://numbermatics.com/n/3125/ | 1,723,347,421,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640843545.63/warc/CC-MAIN-20240811013030-20240811043030-00367.warc.gz | 329,115,013 | 6,523 | 3125
3,125 is an odd composite number. It is composed of a single prime number multiplied by itself four times.
What does the number 3125 look like?
This visualization shows the relationship between its 1 prime factors (large circles) and 6 divisors.
3125 is an odd composite number. It is composed of one distinct prime number multiplied by itself four times. It has a total of six divisors.
Prime factorization of 3125:
55
(5 × 5 × 5 × 5 × 5)
See below for interesting mathematical facts about the number 3125 from the Numbermatics database.
Names of 3125
• Cardinal: 3125 can be written as Three thousand, one hundred twenty-five.
Scientific notation
• Scientific notation: 3.125 × 103
Factors of 3125
• Number of distinct prime factors ω(n): 1
• Total number of prime factors Ω(n): 5
• Sum of prime factors: 5
Divisors of 3125
• Number of divisors d(n): 6
• Complete list of divisors:
• Sum of all divisors σ(n): 3906
• Sum of proper divisors (its aliquot sum) s(n): 781
• 3125 is a deficient number, because the sum of its proper divisors (781) is less than itself. Its deficiency is 2344
Bases of 3125
• Binary: 1100001101012
• Hexadecimal: 0xC35
• Base-36: 2ET
Squares and roots of 3125
• 3125 squared (31252) is 9765625
• 3125 cubed (31253) is 30517578125
• The square root of 3125 is 55.9016994375
• The cube root of 3125 is 14.6200886911
Scales and comparisons
How big is 3125?
• 3,125 seconds is equal to 52 minutes, 5 seconds.
• To count from 1 to 3,125 would take you about fifty-two minutes.
This is a very rough estimate, based on a speaking rate of half a second every third order of magnitude. If you speak quickly, you could probably say any randomly-chosen number between one and a thousand in around half a second. Very big numbers obviously take longer to say, so we add half a second for every extra x1000. (We do not count involuntary pauses, bathroom breaks or the necessity of sleep in our calculation!)
• A cube with a volume of 3125 cubic inches would be around 1.2 feet tall.
Recreational maths with 3125
• 3125 backwards is 5213
• The number of decimal digits it has is: 4
• The sum of 3125's digits is 11
• More coming soon!
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The information we have on file for 3125 includes mathematical data and numerical statistics calculated using standard algorithms and methods. We are adding more all the time. If there are any features you would like to see, please contact us. Information provided for educational use, intellectual curiosity and fun!
Keywords: Divisors of 3125, math, Factors of 3125, curriculum, school, college, exams, university, Prime factorization of 3125, STEM, science, technology, engineering, physics, economics, calculator, three thousand, one hundred twenty-five.
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Some bits of this website may not work unless you switch it on. | 923 | 3,439 | {"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-2024-33 | latest | en | 0.884276 |
https://the-algorithms.com/algorithm/srtf-scheduling?lang=java | 1,726,891,128,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725701427996.97/warc/CC-MAIN-20240921015054-20240921045054-00594.warc.gz | 517,079,830 | 19,427 | #### SRTF Scheduling
A
```package com.thealgorithms.scheduling;
import com.thealgorithms.devutils.entities.ProcessDetails;
import java.util.ArrayList;
import java.util.List;
/**
* Implementation of Shortest Remaining Time First Scheduling Algorithm.
* In the SRTF scheduling algorithm, the process with the smallest amount of time remaining until completion is selected to execute.
* Example:
* Consider the processes p1, p2 and the following table with info about their arrival and burst time:
* Process | Burst Time | Arrival Time
* P1 | 6 ms | 0 ms
* P2 | 2 ms | 1 ms
* In this example, P1 will be executed at time = 0 until time = 1 when P2 arrives. At time = 2, P2 will be executed until time = 4. At time 4, P2 is done, and P1 is executed again to be done.
* That's a simple example of how the algorithm works.
*/
public class SRTFScheduling {
protected List<ProcessDetails> processes;
/**
* Constructor
* @param processes ArrayList of ProcessDetails given as input
*/
public SRTFScheduling(ArrayList<ProcessDetails> processes) {
this.processes = new ArrayList<>();
this.processes = processes;
}
public void evaluateScheduling() {
int time = 0;
int cr = 0; // cr=current running process, time= units of time
int n = processes.size();
int[] remainingTime = new int[n];
/* calculating remaining time of every process and total units of time */
for (int i = 0; i < n; i++) {
remainingTime[i] = processes.get(i).getBurstTime();
time += processes.get(i).getBurstTime();
}
/* if the first process doesn't arrive at 0, we have more units of time */
if (processes.get(0).getArrivalTime() != 0) {
time += processes.get(0).getArrivalTime();
}
/* printing id of the process which is executed at every unit of time */
// if the first process doesn't arrive at 0, we print only \n until it arrives
if (processes.get(0).getArrivalTime() != 0) {
for (int i = 0; i < processes.get(0).getArrivalTime(); i++) {
}
}
for (int i = processes.get(0).getArrivalTime(); i < time; i++) {
/* checking if there's a process with remaining time less than current running process.
If we find it, then it executes. */
for (int j = 0; j < n; j++) {
if (processes.get(j).getArrivalTime() <= i && (remainingTime[j] < remainingTime[cr] && remainingTime[j] > 0 || remainingTime[cr] == 0)) {
cr = j;
}
} | 617 | 2,304 | {"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} | 2.75 | 3 | CC-MAIN-2024-38 | latest | en | 0.747647 |
https://cs.stackexchange.com/questions/66368/is-it-mandatory-to-define-transitions-on-every-possible-alphabet-in-deterministi/66390 | 1,723,081,629,000,000,000 | text/html | crawl-data/CC-MAIN-2024-33/segments/1722640713903.39/warc/CC-MAIN-20240808000606-20240808030606-00431.warc.gz | 148,435,247 | 46,033 | Is it mandatory to define transitions on every possible alphabet in Deterministic Finite Automata?
Tomorrow is my presentation and I want to clear my concepts…
I've read that in DFA, "For each state, transition on all possible symbols (alphabet) should be defined."
Is for each state, defining transition on all possible symbols mandatory in DFA? If its not, then please give any examples?
• Welcome to CS.SE! We prefer that you ask only one question per post. This looks like two separate questions. It would be better to post the second one (about NFA's) separately. Also, have you searched thoroughly on this site, and checked the formal definition in your textbook? If not, you should do that before asking; and you should show us in the question what you found when you did that.
– D.W.
Commented Nov 23, 2016 at 19:24
• Thank you for warm welcome, I did actually searched on this site and on google as well, but I am getting contrary views which is actually confusing me.. Commented Nov 23, 2016 at 19:39
• The second question has been removed, but you can find it in the edit history and post it separately as a separate question using the 'Ask Question' button in the upper-right. However, before asking, please do make sure to do the research suggested and tell us in the question what research you've done, including telling us which textbook(s) you've read. As far as this question, you could still edit this question to address the feedback I gave here by looking up the formal definition in your textbook, including it in the question, and showing your interpretation of that definition.
– D.W.
Commented Nov 23, 2016 at 20:51
• Anyway, this seems covered by cs.stackexchange.com/q/12587/755. Community votes, please: is this a duplicate?
– D.W.
Commented Nov 23, 2016 at 20:54
• I don't really understand your question. It seems to be "I've read that the definition is X. Is the definition X?" Commented Nov 24, 2016 at 0:24
A DFA is specified by the following data:
• An alphabet $\Sigma$.
• A set of states $Q$.
• An initial state $q_0 \in Q$.
• A set of final states $F \subseteq Q$.
• A transition function $\delta\colon Q \times \Sigma \to Q$.
As you can see from the signature of $\delta$, it specifies a transition at every state for every symbol.
• Except that DFAs are sometimes defined with a partial transition function. Commented Nov 23, 2016 at 21:31
• You're right, there is no "official" definition of a DFA. But the OP's reading betrays influence of this particular definition. Commented Nov 23, 2016 at 21:37
• Should explicitly say that the transition function is total. Commented Nov 24, 2016 at 17:51
Suppose a DFA was allowed to have missing transitions. What happens if you encounter a symbol which has no transtion defined for it? The result is undefined. That would seem to violate the "deterministic" characteristic of a DFA.
However, it's trivial to transform such an incomplete DFA into a complete DFA. Simply add a new state, illegal, and map any undefined transitions to the illegal state. Finally, add transitions for every symbol from the illegal state back to itself. This illegal state is often called a sink state, because once data falls into the sink there's no way to get out.
So, from a practical perspective, it's kind of moot, as long as you have a well-defined way to handle missing transitions.
• Careful: A transition being undefined doesn't make the automaton non-deterministic, just incomplete. There are some definitions of DFA that allow such undefined transitions precisely because it's trivial to complete it systematically. Commented Nov 23, 2016 at 22:58
• @Darkhogg, I don't necessarily disagree, but wouldn't determinism of an incomplete DFA be dependent on how a particular implementation handles these undefined / missing transitions? And wouldn't such an implementation implicitly complete the DFA? Commented Nov 24, 2016 at 1:08
• No, it doesn't depend on the implementation, it depends on the definition. If you define DFAs as having a total transition function and then use a partial function sure, you have undefined behaviour and might end up with non-determinism, but that's not a given. However, DFAs are sometimes explicitly defined to use a partial function, and when encountering an undefined transition the behaviour is "not accept", period. No non-determinism or anything funky there, for any implementation because the result is defined even if the transition is not. Commented Nov 24, 2016 at 8:50
• BTW: you can also make the converse transformation. Take a "total automaton" and remove a sink state to obtain a "incomplete automaton". In the end the only difference is that a total automaton is always able to read a word to the end, and after that it decides whether it accepts the word or not, while a partial automaton is able to reject some words before reading all of their characters. Commented Nov 24, 2016 at 13:37
A DFA is often defined as a restricted type of NFA. If $\Sigma$ is the input alphabet and $Q$ is the set of states, the transition structure of an NFA is specified as either a relation $\rho \subseteq Q \times \Sigma \times Q$, or as a function $\delta : (Q \times \Sigma) \to 2^Q$. If we adopt the latter definition, then we can say that an NFA is deterministic if $|\delta(q,\sigma)| \leq 1$ for all $q\in Q$ and $\sigma \in \Sigma$, and complete if $\delta(q,\sigma) \neq \emptyset$, again, for all $q \in Q$ and $\sigma \in \Sigma$.
A word is accepted by an NFA if it has an accepting run. A deterministic automaton has at most one run. A complete automaton has at least one run.
Some authors define trim automata as those in which each state is on some path from an initial state to a final state. For certain languages, you cannot have automata that are both trim and complete. In those cases, it is convenient to keep the completeness requirement out of the definition of deterministic automaton. | 1,401 | 5,927 | {"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": 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} | 2.546875 | 3 | CC-MAIN-2024-33 | latest | en | 0.955271 |
https://oeis.org/A205341 | 1,639,053,018,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964364169.99/warc/CC-MAIN-20211209122503-20211209152503-00017.warc.gz | 478,379,708 | 5,208 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
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A205341 T(n,k)=Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than k 10
0, 0, 1, 0, 2, 0, 0, 3, 2, 2, 0, 4, 6, 11, 0, 0, 5, 12, 35, 24, 5, 0, 6, 20, 82, 138, 93, 0, 0, 7, 30, 160, 454, 689, 272, 14, 0, 8, 42, 277, 1130, 2912, 3272, 971, 0, 0, 9, 56, 441, 2370, 8927, 18652, 16522, 3194, 42, 0, 10, 72, 660, 4424, 22297, 71630, 124299, 83792, 11293, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET 1,5 COMMENTS Table starts ..0...0.....0......0......0.......0.......0........0........0........0 ..1...2.....3......4......5.......6.......7........8........9.......10 ..0...2.....6.....12.....20......30......42.......56.......72.......90 ..2..11....35.....82....160.....277.....441......660......942.....1295 ..0..24...138....454...1130....2370....4424.....7588....12204....18660 ..5..93...689...2912...8927...22297...48335....94456...170529...289229 ..0.272..3272..18652..71630..214724..542850..1211784..2459988..4633800 .14.971.16522.124299.594405.2133784.6285127.16018970.36557640.76469705 LINKS R. H. Hardin, Table of n, a(n) for n = 1..9999 FORMULA Empirical for row n: n=2: T(2,k) = k n=3: T(3,k) = k^2 - k n=4: T(4,k) = (4/3)*k^3 - (1/2)*k^2 + (7/6)*k n=5: T(5,k) = (23/12)*k^4 - (1/2)*k^3 + (1/12)*k^2 - (3/2)*k n=6: T(6,k) = (44/15)*k^5 - (5/12)*k^4 + (5/12)*k^2 + (31/15)*k n=7: T(7,k) = (841/180)*k^6 - (1/3)*k^5 - (19/36)*k^4 + (1/3)*k^3 - (103/90)*k^2 - 3*k T(n,m) = 1/n*Sum_{i=1..n} (Sum_{,l,0,i} (binomial(i,l)*(-1)^l *Sum_{j=0..(i-l)* m/(2*m+1)}((-1)^j*binomial(i-l,j)*binomial((-l-2*j+i)*m-l-j+i-1,(-l-2*j+i)*m-j)))*T(n-i,m)), T(0,m)=1. - Vladimir Kruchinin, Apr 07 2017 EXAMPLE Some solutions for n=5, k=3: ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0 ..2....2....2....2....3....2....1....2....2....2....2....2....1....3....2....3 ..4....5....4....0....2....4....4....4....1....4....3....1....2....5....5....5 ..6....4....3....1....4....1....2....2....0....1....0....2....4....4....4....4 ..3....3....2....3....1....2....1....3....3....3....2....3....2....2....2....1 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0 MATHEMATICA T[n_, m_] := T[n, m] = If[n == 0, 1, 1/(n)*Sum[Sum[Binomial[i, l]*(-1)^l* Sum[(-1)^j*Binomial[i-l, j]*Binomial[(-l - 2*j + i)*m - l - j + i - 1, (-l - 2*j + i)*m-j], {j, 0, (i-l)*m/(2*m+1)}], {l, 0, i}]*T[n-i, m], {i, 1, n}]]; Table[T[n-m+1, m], {n, 1, 11}, {m, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *) PROG (Maxima) T(n, m):=if n=0 then 1 else 1/(n)*sum(sum(binomial(i, l)*(-1)^l*sum((-1)^j*binomial(i-l, j)*binomial((-l-2*j+i)*m-l-j+i-1, (-l-2*j+i)*m-j), j, 0, (i-l)*m/(2*m+1)), l, 0, i)*T(n-i, m), i, 1, n); /* Vladimir Kruchinin, Apr 07 2017 */ CROSSREFS Column 1 odd n is A000108((n+5)/2). Column 2 is A187430. Row 3 is A002378(n-1). Sequence in context: A142886 A099026 A341410 * A195664 A053202 A188122 Adjacent sequences: A205338 A205339 A205340 * A205342 A205343 A205344 KEYWORD nonn,tabl,look AUTHOR R. H. Hardin, Jan 26 2012 STATUS approved
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Last modified December 9 07:21 EST 2021. Contains 349627 sequences. (Running on oeis4.) | 1,651 | 3,922 | {"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} | 2.6875 | 3 | CC-MAIN-2021-49 | latest | en | 0.721575 |
https://www.javacodegeeks.com/2020/04/sql-group-by-and-having-example-write-sql-query-to-find-duplicate-emails-leetcode-solution.html | 1,718,514,057,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861643.92/warc/CC-MAIN-20240616043719-20240616073719-00501.warc.gz | 734,965,173 | 47,937 | Software Development
# SQL GROUP BY and HAVING Example – Write SQL Query to find Duplicate Emails – LeetCode Solution
Write a SQL query to find all duplicate emails in a table named Person.
```+----+---------+
| Id | Email |
+----+---------+
| 1 | a@b.com |
| 2 | c@d.com |
| 3 | a@b.com |
+----+---------+
```
For example, your query should return the following for the above table:
```+---------+
| Email |
+---------+
| a@b.com |
+---------+
```
Note: All emails are in lowercase.
## SQL query to find duplicate values in a Column – Solution
Here are three ways to solve this problem in SQL query, first by using group by clause, second by using self-join and then third by using subquery with exists clause. While I agree that this problem can be solved in a different way, but it is also a perfect example of how you can use the
SQL GROUP BY and HAVING clause.
### 1. Finding Duplicate elements By using GROUP BY
The simplest solution to this problem is by using GROUP BY and HAVING Clause. Use GROUP BY to group the result set on email, this will bring all duplicate email in one group, now if the count for a particular email is greater than 1 it means it is a duplicate email. Here is the SQL query to find duplicate emails :
# Write your MySQL query statement below
1 `SELECT Email FROM Person GROUP BY Email HAVING COUNT(Email) > ``1`
This is also my accepted answer on LeetCode.
### 2. Finding Duplicate values in a column By using Self Join
By the way, there are a couple of more ways to solve this problem, one is by using Self Join. If you remember, In Self Join we join two instances of the same table to compare one record to another. Now if email from one record in the first instance of the table is equal to the email of another record in the second table it means the email is duplicate. Here is the SQL query using Self Join
# Write your MySQL query statement below
1 `SELECT DISTINCT a.Email FROM Person a JOIN Person b ON a.Email = b. Email WHERE a.Id != b.Id`
Remember to use the keyword distinct here because it will print the duplicate email as many times it appears in the table. This is also an accepted solution in Leetcode.
### 3. Finding duplicate emails By using Sub-query with EXISTS:
You can even solve this problem using a correlated subquery. In a correlated subquery, the inner query is executed for each record in the outer query. So one email is compared to the rest of the email in the same table using a correlated subquery. Here is the solution query :
12345678 `SELECT DISTINCT p1.Email``FROM Person p1``WHERE EXISTS(`` ``SELECT *`` ``FROM Person p2`` ``WHERE p2.Email = p1.Email`` ``AND p2.Id != p1.Id``)`
If you still need more guidance then this Julia Evans zine is also a great way to understand how to find duplicate elements in a table using GROUP BY with HAVING clause
That’s all about how to find duplicate emails in SQL using the GROUP BY and HAVING clause. I have also shown you how you can solve this problem using Self-join and a subquery with the EXISTS clause as well. Once you get familiar with the pattern you can solve many such problems. If you want to learn more check out the following resources.
Thanks for reading this article, if you like this SQL article, then please share with your friends and colleagues. If you have any questions or feedback, then please drop a note.
Published on Java Code Geeks with permission by Javin Paul, partner at our JCG program. See the original article here: SQL GROUP BY and HAVING Example – Write SQL Query to find Duplicate Emails – LeetCode SolutionOpinions expressed by Java Code Geeks contributors are their own.
### Javin Paul
I have been working in Java, FIX Tutorial and Tibco RV messaging technology from past 7 years. I am interested in writing and meeting people, reading and learning about new subjects.
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1 Comment
Inline Feedbacks | 915 | 3,998 | {"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} | 2.71875 | 3 | CC-MAIN-2024-26 | latest | en | 0.832757 |
https://www.physicsforums.com/threads/frames-in-gr.683662/ | 1,529,743,700,000,000,000 | text/html | crawl-data/CC-MAIN-2018-26/segments/1529267864953.36/warc/CC-MAIN-20180623074142-20180623094142-00188.warc.gz | 870,194,247 | 15,987 | # Frames in GR
1. Apr 6, 2013
### HomogenousCow
Hello I am having problems in GR because I do not understand how observations frames work in this theory. I have a few more specific questions.
-In the schwarzschild metric, which frame is the metric written in? (The one with swarzschild coordinates)
-In the non-vaccum solutions, how do we actually prescribe the components for the stress energy tensor, more specifically from which frame do we take the observed values from?
All other tips and pieces of wisdom are welcome
Last edited: Apr 6, 2013
2. Apr 6, 2013
### bcrowell
Staff Emeritus
This is SR, not GR. The metric has the same form in any inertial frame.
At a given point in spacetime, they can be expressed in any frame you like. Their values in different frames are related by the tensor transformation law. You don't even need to have a frame of reference; you can just have some coordinate system that covers some open subset of spacetime.
3. Apr 6, 2013
4. Apr 6, 2013
### pervect
Staff Emeritus
I would suggest that you not worry about frames at all, and simply understand GR as a theory about generalized coordinates. You can worry about frame-fields later.
The metric is written in coordinates, not in a frame.
Lets assume you have some coordinates for some point A = (t,p,q,r), t being your time coordinate.
Then the metric gives you the Lorentz interval from A to B, where B is a nearby point with coordinates t+dt, p+dp, q+dq, r+dr
Because the Lorentz interval is independent of frame, you don't need to specify what "frame" you use, as long as A and B are sufficiently close.
So you forget about the "frames" entirely, and concentrate on the Lorentz Interval, as something that you can measure that's *INDEPENENT* of your choice of frame.
Again, you don't need a frame. It's easier to visualize with one less dimension, so imagine one time coordinate t and two space coordinates p and q.
Then planes of constant t will slice your box up into slabs. Adding in planes of constant p and q dice your box into small cubes.
The stress-energy tensor gives the energy and momentum density in one of these small cubes.
5. Apr 6, 2013
### bcrowell
Staff Emeritus
OK. The Schwarzschild coordinates do not form a frame of reference. In GR, frames of reference are local, not global.
6. Apr 7, 2013
### HomogenousCow
But then what does the coordinate time in the metric actually mean?
7. Apr 7, 2013
### pervect
Staff Emeritus
It's a fairly "natural" choice of coordinate for a stationary observer. But like most coordinates, it doesn't have any great physical significance. If you realize this early on, you can save yourself a lot of misunderstandings.
If you want to leverage your intuition about flat space-time as much as possible, the isotropic form of Schwarschild coordinates is useful:
http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/IsotropicCoordinates.htm
Isotropic coordinates share the same time coordinates as the non-isotropic Schwarschild coordinates, but have a different definition for the radial coordinate.
Trying to assign coordinates to a curved space-time is rather similar to the task of assigning coordiantes to a curved Earth. Any representation of the surface of the Earth on a flat sheet of paper will be distorted. Similarly, any representation of curved space-time via global coordinates will be distorted.
The metric coefficients when properly interpreted tell you in what manner your choice of coordinates is distorted.
If your metric coordiates are all nearly a unit diagonal, the distortions due to curvature are minimal, and you can use your intuition with good results.
If you want to use your intuition in some region where the metric is not nearly a unit diagonal, it can be handy to choose a different set of coordinates that make the metric nearly a unit diagonal. Unfortunately, this can only be done locally, you can't make the metric a unit diagional everywhere, though you can make it a unit diagional at any particular point of interest.
8. Apr 7, 2013
### Agerhell
The Schwarzschild radius becomes different using isotropic coordinates (according to wikipedia).
What is the orbital velocity of a body in circular orbit using isotropic coordinates, is it still $v=\sqrt{GM/r}$?
If you drop something from standstill using isotropic coordinates (all velocities equal to zero) what is the initial acceleration, $\frac{d^2r}{dt^2}$?
The proper time will be given by the same expression even if isotropic coordinates are used?
9. Apr 7, 2013
### Agerhell
In the Schwarzschild solution in Schwarzschild coordinates the coordinate time can be said to be the time as measured by a clock located infinitely far away from the center of the gravitational field and with no veloctiy relative to it. In the solar system we could think of a clock infinitely far away from the center of mass of the solarsystem as the "coordinate time".
In general, clocks will be effected by nearby masses and there motion, they always measure "proper time".
Coordinate time is useful as a concept if you want to compare, for instance a clock somewhere on earth with a clock in a spacecraft encirculing one of the moons of saturn. You try to calculate how fast each clock ticks in coordinate time and then you compare them with each other... | 1,200 | 5,316 | {"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} | 3.359375 | 3 | CC-MAIN-2018-26 | latest | en | 0.920176 |
http://mathhelpforum.com/calculus/12596-derivative-natural-log-print.html | 1,500,585,873,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549423486.26/warc/CC-MAIN-20170720201925-20170720221925-00115.warc.gz | 201,019,835 | 5,137 | # Derivative of Natural Log
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• Mar 15th 2007, 01:17 PM
zachb
Derivative of Natural Log
Can someone please tell me what I'm doing wrong here, and show me how to do it correctly?
I need to differentiate g(x) = LN a-x/a+x
So,
g'(x) = 1/a-x/a+x *d/dx (a-x/a+x)
g'(x) = a+x/a-x *[(a+x)*d/dx(a-x) - (a-x)*d/dx(a+x)/(a+x)^2]
g'(x) = a+x/a-x *[(a+x)*(1-1) - (a-x)(1+1)/(a+x)^2]
g'(x) = a+x/a-x *[(a+x)*(0) - (a-x)(2)/(a+x)^2]
g'(x) = a+x/a-x *[0 - 2(a-x)/(a+x)^2]
g'(x) = a+x/a-x * -2a+2x/a^2+2ax+x^2
g'(x) = a+x*(-2a+2x)/a-x*(a^2+2ax+x^2)
g'(x) = -2a^2+2ax-2ax+2x^2/a^3+2a^2x+ax^2-a^2x-2ax^2-x^3)
g'(x) = -2a^2+2x^2/(a^3+a^2x-ax^2-x^3)
the answer is supposed to be g'(x) = -2a/a^2-x^2, but that's not what I get when I simplify . . .
• Mar 15th 2007, 01:29 PM
qbkr21
Re:
Here is the answer. I know it is right. The answer you have at the bottom is incorrect.
http://item.slide.com/r/1/38/i/L6mVO...x7ZkeiNuwW62H/
• Mar 15th 2007, 01:35 PM
zachb
My book says that the answer is g'(x) = -2a/a^2-x^2, but I'll look over your work.
The problem is g(x) = the natural log of a-x/a+x, not the natural log of a-x/ a+x, so I think you may ahve done it wrong . . .
• Mar 15th 2007, 01:48 PM
qbkr21
Quote:
Originally Posted by zachb
Can someone please tell me what I'm doing wrong here, and show me how to do it correctly?
I need to differentiate g(x) = LN a-x/a+x
So,
g'(x) = 1/a-x/a+x *d/dx (a-x/a+x)
g'(x) = a+x/a-x *[(a+x)*d/dx(a-x) - (a-x)*d/dx(a+x)/(a+x)^2]
g'(x) = a+x/a-x *[(a+x)*(1-1) - (a-x)(1+1)/(a+x)^2]
g'(x) = a+x/a-x *[(a+x)*(0) - (a-x)(2)/(a+x)^2]
g'(x) = a+x/a-x *[0 - 2(a-x)/(a+x)^2]
g'(x) = a+x/a-x * -2a+2x/a^2+2ax+x^2
g'(x) = a+x*(-2a+2x)/a-x*(a^2+2ax+x^2)
g'(x) = -2a^2+2ax-2ax+2x^2/a^3+2a^2x+ax^2-a^2x-2ax^2-x^3)
g'(x) = -2a^2+2x^2/(a^3+a^2x-ax^2-x^3)
the answer is supposed to be g'(x) = -2a/a^2-x^2, but that's not what I get when I simplify . . .
With your notation this is what you told me. I just went by what you said. Try using parenthesis. This is what you told me:
http://item.slide.com/r/1/102/i/kt1F...2GeQXISr0ogCN/
• Mar 15th 2007, 02:11 PM
frenzy
ln[(a-x)/(a+x)]=ln(a-x)-ln(a+x)
take the derivative
-1/(a-x)-1/(a+x)=-2a/(a^2-x^2)
• Mar 15th 2007, 02:27 PM
qbkr21
Quote:
Originally Posted by frenzy
ln[(a-x)/(a+x)]=ln(a-x)-ln(a+x)
take the derivative
-1/(a-x)-1/(a+x)=-2a/(a^2-x^2)
When you take the derivative of ln(a-x)-ln(a+x) you do not get 2a/(a^2-x^2)
you get:
http://item.slide.com/r/1/191/i/EZJp...x_b3nqpiz4mrp/
The way that first problem was posted was terrible absolutely terrible notation.
• Mar 15th 2007, 02:56 PM
frenzy
Quote:
Originally Posted by qbkr21
When you take the derivative of ln(a-x)-ln(a+x) you do not get 2a/(a^2-x^2)
you get:
http://item.slide.com/r/1/191/i/EZJp...x_b3nqpiz4mrp/
The way that first problem was posted was terrible absolutely terrible notation.
You might want to retry that derivative again.
What I wrote is correct.
• Mar 15th 2007, 03:38 PM
qbkr21
Re:
I am not going to argue with you but the derivative of ln(f(x))= (1/(f(x))* f'(x):cool:
• Mar 15th 2007, 03:41 PM
zachb
Frenzy, can you please show your work so that I can see how you got that answer? I already know what the answer should be, what I'm interested in is how you got there. :)
• Mar 15th 2007, 03:55 PM
frenzy
Quote:
Originally Posted by qbkr21
I am not going to argue with you but the derivative of ln(f(x))= (1/(f(x))* f'(x):cool:
This is correct...BUT, how you took the derivative is NOT correct.
Again...look at what you have written. It is wrong.
Quote:
Originally Posted by zachb
Frenzy, can you please show your work so that I can see how you got that answer? I already know what the answer should be, what I'm interested in is how you got there. :)
d/dx ln(a-x)= [1/(a-x)]*d/dx[(a-x)]=1/(a-x)]*(-1)=-1/(a-x)
d/dx -ln(a+x)= -[1/(a+x)]*d/dx[(a+x)]=-1/(a+x)]*(+1)=-1/(a+x)
-1/(a-x)-1/(a+x)=(-1(a+x)-1(a-x))/[(a-x)*(a+x)]
=(-a-x-a+x))/[a^2+ax-ax-x^2]=(-a-a)/(a^2-x^2)=-2a/(a^2-x^2)
• Mar 15th 2007, 04:04 PM
qbkr21
Re:
If you go to my second post where I show the problem aligned in the calculator. Then if you take the derivative of whats on the screen and put it on paper you get my exact answerr. Now maybe this whole scenario revolves around the fact that I didn't understand the way:
g(x)= ln x-a/x+a
There again this is terrible notation because it could me several different things:
g(x)= ln(x-a)/x+a ???
g(x)= ln(x)-a/x+a ???
or g(x)= ln(x-a/x+a) ???
I mean if my professor handed me this in class I would hand it right back to him simply because this in its entirety is "mathematically incorrect". A professor who does this obviously should be teaching grade school or not at all.
• Mar 15th 2007, 04:22 PM
frenzy
Quote:
Originally Posted by qbkr21
If you go to my second post where I show the problem aligned in the calculator. Then if you take the derivative of whats on the screen and put it on paper you get my exact answerr. Now maybe this whole scenario revolves around the fact that I didn't understand the way:
g(x)= ln x-a/x+a
There again this is terrible notation because it could me several different things:
g(x)= ln(x-a)/x+a ???
g(x)= ln(x)-a/x+a ???
or g(x)= ln(x-a/x+a) ???
I mean if my professor handed me this in class I would hand it right back to him simply because this in its entirety is "mathematically incorrect". A professor who does this obviously should be teaching grade school or not at all.
no doubt that the problem was missing some prentices
Going back to your 2nd post you have..
d/dx(a-x)=1-1=0
and
d/dx(a+x)=1+1=2
These are not correct.
you should have
d/dx(a-x)=0-1=-1
and
d/dx(a+x)=0+1=1
• Mar 15th 2007, 05:59 PM
zachb
Quote:
Originally Posted by frenzy
no doubt that the problem was missing some prentices
Going back to your 2nd post you have..
d/dx(a-x)=1-1=0
and
d/dx(a+x)=1+1=2
These are not correct.
you should have
d/dx(a-x)=0-1=-1
and
d/dx(a+x)=0+1=1
I don't understand why d/dx(a-x)=0-1=-1
and
d/dx(a+x)=0+1=1
Is that because "a" is a constant term?
• Mar 15th 2007, 06:05 PM
frenzy
Quote:
Originally Posted by zachb
I don't understand why d/dx(a-x)=0-1=-1
and
d/dx(a+x)=0+1=1
Is that because "a" is NOT a constant term?
a is a constant therefore
d/dx(a)=0
The derivative of any constant is zero.
• Mar 15th 2007, 06:12 PM
zachb
Okay, but I don't understand how we're supposed to know that "a" IS a constant term, and x is not. :( Is there some kind of rule for determining which term is constant when you have two or more varaibles? I thought that constant terms were just integers. For example, in f(x)=x^2-7x+5 the constant term is 5 and when you find f'(x) it's f'(x)=2x-7+0.:confused:
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last | 2,527 | 6,775 | {"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-2017-30 | longest | en | 0.807497 |
http://physics.stackexchange.com/questions/44304/calculational-method-for-determining-surface-tensions-from-photograph-of-menisci/56372 | 1,469,612,349,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257826759.85/warc/CC-MAIN-20160723071026-00110-ip-10-185-27-174.ec2.internal.warc.gz | 206,665,218 | 18,884 | # Calculational method for determining surface tensions from photograph of menisci?
How can I get from a photograph of a liquid surface to a value for the surface tension.
-
There is a possibility that this is a good question, but you need to tell us more: do you have something in the image or not to tell you what the absolute scales are (something like a ruler or knowing the diameter of the tube), or at least a guarantee that all the images have the same scale? Do you understand the geometry of the liquid column? What part of this physics do you understand and where are you stuck? Give us something to go on here. – dmckee Nov 15 '12 at 21:29
Have a look at this simple link : ehow.com/how_6019173_calculate-surface-tension.html . There is a formula for getting newtons/meter . You cannot do it just from a photograph if you do not know the temperature. Yes, the slit is covered with water because of surface tension. You could make wider and wider slits until this no longer happens and then use approximations to calculate the tension, given temperature. Complicated method. (this refers to your related slit question) – anna v Nov 16 '12 at 5:07
The word you are looking for is Contact Angle and your photograph would need to be good enough to measure this. – Mark Rovetta Nov 19 '12 at 18:40
The Sessile Drop Technique is a method used for the characterization of solid surface energies, and in some cases, aspects of liquid surface energies. – Mark Rovetta Nov 19 '12 at 20:47
The sessile drop technique that @MarkRovetta mentions is also an option to determine the surface tension but it is less commonly used because you introduce an extra `experimental difficulty'. Namely you need an (almost) perfectly smooth surface to make sure that the droplet is axisymmetric, which is almost guaranteed for the pendant drop technique. The sessile drop principle is almost the same: you put a droplet of known volume on a surface and determine contact angle and circumference from a sideview. This image you can then fit to a mathematical model which balances the Laplace pressure with the hydrostatic pressure: $$\frac{2}{R_0}+\rho g z = \gamma \left(\frac{1}{R_1}+\frac{1}{R_2}\right), \;R_1=\frac{\partial s}{\partial \theta},\; R_2=\frac{x}{\sin \theta}$$ where the parameters are indicated in the image below. With simple geometrics this system can be converted to a system of ODEs dependent on $s$. By integrating over $s$ to the desired contact angle $\theta$ and adjusting $R_0$ to get the correct volume $\left(\frac{\partial V}{\partial s}=\pi x^2 \sin \theta\right)$ the surface tension can be found by fitting the experimentally obtained droplet profile. | 638 | 2,674 | {"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} | 3.59375 | 4 | CC-MAIN-2016-30 | latest | en | 0.923882 |
https://commerceschool.in/cbse-q-40-dk-goel-fundamentals-of-partnership-2024-25/ | 1,718,939,926,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862036.35/warc/CC-MAIN-20240621031127-20240621061127-00287.warc.gz | 147,931,815 | 29,589 | # [CBSE] Q. 40 DK Goel Fundamentals of Partnership [2024-25]
Solution of Question Number 40 of the Fundamentals of partnership firm DK Goel CBSE Board (2024-25)
On 1st April, 2023 X, Y and Z started a business in partnership. X contributes ₹ 90,000 at first but withdraws ₹ 30,000 at the end of six months. Y introduces ₹ 75,000 at first and increases it to ₹ 90,000 at the end of four months, but withdraws ₹ 30,000 at the end of eight months. Z brings in ₹ 75,000 at first but increases it by ₹ 60,000 at the end of seven months.
During the year ended 31st March, 2024, they make a net profit of ₹ 42,000. Show how the partners should divided this amount on the basis of effective capital employed by each partner.
[Ans. Profit sharing ratio of X, Y and Z = 3 : 3 : 4.]
Solution:-
Here is the complete index of solutions | 234 | 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} | 2.765625 | 3 | CC-MAIN-2024-26 | latest | en | 0.937488 |
https://www.indiabix.com/aptitude/races-and-games/discussion-618-1-3 | 1,713,174,238,000,000,000 | text/html | crawl-data/CC-MAIN-2024-18/segments/1712296816954.20/warc/CC-MAIN-20240415080257-20240415110257-00803.warc.gz | 735,699,447 | 8,262 | # Aptitude - Races and Games - Discussion
Discussion Forum : Races and Games - General Questions (Q.No. 13)
13.
In a 100 m race, A can beat B by 25 m and B can beat C by 4 m. In the same race, A can beat C by:
21 m
26 m
28 m
29 m
Explanation:
A : B = 100 : 75
B : C = 100 : 96.
A : C = A x B = 100 x 100 = 100 = 100 : 72. B C 75 96 72
A beats C by (100 - 72) m = 28 m.
Discussion:
11 comments Page 1 of 2.
Praveen said: 9 years ago
When A Travels 100 m, B travels 75 m => A:B = 100:75.
When B Travels 100 m, C travels 96 m.
When B Travels 75 m, C travels (96 x 75)/100 = 72 m => B:C = 75:72.
Therefore, A:B:C = 100:75:72.
So, when A Travels 100 m, C travels 72 m.
Therefore, A beat C by 28 m.
(2)
Priya said: 4 years ago
When A covers 100m.
B covers 100-25=75m.
When B covers 100m.
C covers 96m.
Hence, when A cover 100m.
C cover (4+25)=29-1(we are considering it twice), hence the answer is 28m.
(2)
Syed Tajudeen.S said: 1 decade ago
When A runs 100m then B runs 75m and C runs 71m and A beats C by 28m. Then how come 29m? please do clear my doubts
(1)
@Syed,
Here it states that "In a 100 m race, A can beat B by 25 m and B can beat C by 4 m." It mesns that B beat C by 4 m in 100m race. So B beat C by 3 m in 75m race.
Hence A beats C in 100 m race by 28m (25+3).
(1)
Khushboo said: 9 years ago
Here it is B can beat C by 4m which means C is far from B by 4m, so it has traveled 21m
(1)
Vivek said: 7 years ago
How did you get 72? @Praveen.
(1)
Khandaker Sajib Ahmed said: 7 years ago
It means, When A covers 100m, B covers=100-25=75m. And again, When B covers=100m, C covers=100-4=96m.
ATQ, When B covers 100m, C covers 96m.
when B covers 75m, C covers 96*75/100=72m.
So A can beat C by=100-72=28m.
(1)
Kajal said: 8 years ago
Guys, just read the question, the comma after 100 m indicates 100 m is applicable for the statement before 'and' and after 'and'.
Leo said: 6 years ago
Thanks for the informations.
Prince Raj Kumar said: 5 years ago
Please give me or short tricks in this number. Thanks. | 757 | 2,049 | {"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-2024-18 | latest | en | 0.829143 |
http://blogs.mathworks.com/pick/2008/03/31/puzzler-coin-game/ | 1,502,933,899,000,000,000 | text/html | crawl-data/CC-MAIN-2017-34/segments/1502886102819.58/warc/CC-MAIN-20170817013033-20170817033033-00218.warc.gz | 58,266,386 | 20,154 | # Puzzler: Coin game11
Posted by Doug Hull,
This is a simplified version of the first problem I ever attempted to solve in MATLAB. For my original problem, I was trying to figure out the probability of being on any square in the board game Monopoly. That is a difficult problem from a book-keeping sense, but the essence of the problem is in today’s Puzzler.
With the rules shown in the diagram, what is the probability of being on each of the four squares after each of the first twenty turns?
I can think of two good ways of solving today’s Puzzler. I look forward to seeing how everyone else does it.
Take a little bit and work on the Puzzler. I don’t want to give away strategies before you get a chance to work on the puzzle, so I am just showing the results of each method in the preview below, but not saying what the method was.
[There is a typo in the comments on this second video. It should say the probability of moving from Square 4 to Square 1, not the other way around. Sorry!]
### Note
DanK replied on : 1 of 11
Here’s a brute force, and perhaps less than elegant solution to the problem.
gencoins = @(x)rand(2,x)>0.5; % anonymous function to generate the coins probs(1,:) = [1,0,0,0]; % First step probability is always square 1 nTrials = 1e6; % How many trials are we running to determine probabilities curPos = zeros(1,nTrials); for n=2:20 coins = gencoins(nTrials); twoTails = ~any(coins); curPos = mod(curPos+sum(coins),4); curPos(twoTails)=0; probs(n,:) = sum(bsxfun(@eq,curPos.',[0,1,2,3]))/nTrials; end
Thanks for running these puzzles, Doug!
Dan
Doug replied on : 2 of 11
Dan,
I like this solution. It follows exactly the idea I put in the last wrap-up where you make your code as readable as possible, and then modify it for speed when needed.
Your implementation is very similar to my solution (video one above), but yours is a lot faster. The reason yours is faster is the use of BSXFUN. In this case, a little less clarity is worth the performance increase. I have to admit, I needed to read the doc for this function, as I have never used it!
I would have placed the definition of twoTails immediately before the line where it was used, so it is easier to follow, but this is pretty clear code with some amount of commenting where needed.
While this code is fast and pretty accurate, I am curious if anyone will come up with other methods that are exactly accurate and even faster like my second video.
Thanks!
Doug
Francois replied on : 3 of 11
Hello Doug,
Here is a function working for a slightly more general situation:
function probs = game(steps)
N = 4; % number of game states
% Basic transition matrices
% move(k) corresponds to "move k squares"
move = @(k)circshift(eye(N),[0 k]);
% go(k) corresponds to "go to square k"
go = @(k)ones(N,1)*((1:N)==k);
% Build the actual transition matrix
M = 1/2 * move(1) + 1/4 * move(2) + 1/4 * go(1);
% Compute probabilities
if steps < inf
% Finite number of steps: intermediate distributions are computed
% using successive matrix-vector products
state = (1:N)==1;
for i = 1:steps
state = state*M;
probs(i, 1:N) = state;
end
else
% Infinite number of steps: stationary distribution is computed
% (assuming it exists) by solving a linear system
probs = [zeros(1,N) 1] /[M-eye(N) ones(N,1)];
end
Richard Brown replied on : 4 of 11
Alternatively, as a finite time-homogeneous Markov chain
p20 = (0.25 * [1 2 1 0; 1 0 2 1; 2 0 0 2; 3 1 0 0])^20;
disp(p20(1, :))
cheers,
Richard
Daniel Armyr replied on : 5 of 11
Yeah, the markov chain is of couse the obvious method here, assuming you are familiar with statistics. Richard here beat me to it though. My solution is identical to his, although quite a bit more verbose:
startPos = [1; 0; 0; 0];
numberOfMoves = 20;
probabilityMatrix = …
[ 1/4 1/2 1/4 0; …
1/4 0 1/2 1/4; …
1/2 0 0 1/2; …
3/4 1/4 0 0]’;
probabilityAfter20Turns = probabilityMatrix^numberOfMoves*startPos
Daniel Armyr replied on : 6 of 11
Hi.
One general comment about how you write your blog. I noticed that Loren allways puts a link to the comments section in the text of her entries. I read these blogs with a feed reader so I don’t get the automatic footer with links to the comments section as I get when I visit the homepage. If there is a link in the text, I can go directly to the comments section which is nice.
Sincerely
Daniel Armyr
Doug replied on : 7 of 11
Francois,
A beautiful solution. I like how you derived the M matrix rather than it being a “Magic Number Matrix” (which is what I did).
When I was designing this challenge, I was going to have an add-on where the number of coins and number of squares was variable. I thought that simpler was better. I see that your code would have been able to handle such a complication nicely.
Thanks for a great solution that works in a different way than the earlier one.
Doug
Doug replied on : 8 of 11
Daniel, Richard
I am surprised that so many people jumped no the Markov chain method, I expected that the exhaustive search or monte carlo methods would dominate. Maybe that is because I worked with distributed computing clusters for MATLAB for a while but never really studied Markov Chains.
Thanks for playing!
Doug
Doug replied on : 9 of 11
Daniel,
About the comments. Great idea. I will make it a best practice.
Thanks,
Doug
Urs (us) Schwarz replied on : 10 of 11
as with last weeks puzzler, when i had to second guess the task from a few examples (with consecutive bad proposals, of course), i’m again NOT able to run the videos from behind our two firewalls (the flash player [9,0,115,0] works fine, though, for other stuff): it shows about one 3rd of the first and kills the browser for the second…
it’s just a bit frustrating…
best from zurich
urs
ps: maybe i’m not meant to contribute… or it’s just the date…
craig replied on : 11 of 11
Output: x
p = [.25 .25 .5 .75 ; .5 0 0 .25 ; .25 .5 0 0 ; 0 .25 .5 0 ];
[P D] = eig(p);
D=(abs(D)>.99).*D;
x = real(P * D * P^(-1) * [1 ; 0 ; 0 ; 0]);
The idea here is to use the eigenvalue decomposition, noting that the eigenvalues with norm less than 1 will vanish. Remove those eigenvalues and reassemble the matrix. Multiply by the starting vector and you have the result. This is accurate to machine precision. The real is to remove the roundoff imaginary components. | 1,709 | 6,328 | {"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": 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} | 3.84375 | 4 | CC-MAIN-2017-34 | latest | en | 0.93128 |
https://www.cfd-online.com/Forums/main/8943-adaptive-mesh.html | 1,511,531,955,000,000,000 | text/html | crawl-data/CC-MAIN-2017-47/segments/1510934808133.70/warc/CC-MAIN-20171124123222-20171124143222-00115.warc.gz | 774,339,845 | 15,413 | Register Blogs Members List Search Today's Posts Mark Forums Read
March 31, 2005, 14:50 adaptive mesh #1 Millena Guest Posts: n/a I am applying the adaptive mesh in an elliptic equation, and when I use Neumann conditions for one linear or quadratic equation, the convergence is reached, but when the trigonometric equations is used the maximum residual doesn't reach the tolerance that I want (e10-8), or be, it stall. I am thinking that the problem can be in the order of the polynomials that I am using to do the interpolations between the coarse and fine mesh for the ghost cells or in the restriction and in the prolongation of the multigrid method. Will I have a good behavior for this case.?
April 1, 2005, 04:36 Re: adaptive mesh #2 versi Guest Posts: n/a My experience with multigrid for a equation with Neumann boundary condions, care is needed for the corresponding Neumann conditions on the coarse grid. The incremental form of the Neumann conditions is desirable, rather than the original form.
April 1, 2005, 13:21 Re: adaptive mesh #3 Millena Guest Posts: n/a but when you said "The incremental form of the Neumann" are you talking about the third order in the discretization of the neumann condition??? because I am using the second order.
April 3, 2005, 09:18 Re: adaptive mesh #4 versi Guest Posts: n/a The initial boundary values Q_b on a coarse grid is restricted from the next finer grid, now you are going to iterate Q as well as Q_b. It is suggested to solve LHS * Delta Q = RHS on the coarse grid, where Delta Q is itearted incremental form. Once Delta Q is obtained, Q_i=Q_i^n+ Delta Q_i+1 (first order accuracy), if i=1 is the boundary point.
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Similar Threads Thread Thread Starter Forum Replies Last Post Niklas Wikstrom (Wikstrom) OpenFOAM Running, Solving & CFD 122 June 15, 2014 06:20 maalan FLUENT 0 June 2, 2011 02:12 sc298 OpenFOAM Native Meshers: snappyHexMesh and Others 2 March 27, 2011 21:11 whitecat CFX 0 July 6, 2009 09:43 Joe CFX 2 March 26, 2007 18:10
All times are GMT -4. The time now is 09:59. | 615 | 2,348 | {"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} | 2.828125 | 3 | CC-MAIN-2017-47 | latest | en | 0.898505 |
https://www.techgig.com/practice/question/VW5oOVlqTlh4NS9CS0NlOHk2TGZQUT09 | 1,579,469,510,000,000,000 | text/html | crawl-data/CC-MAIN-2020-05/segments/1579250595282.35/warc/CC-MAIN-20200119205448-20200119233448-00512.warc.gz | 1,112,481,631 | 23,019 | ZS the Coder has recently found an interesting concept called the Birthday Paradox. It states that given a random set of 23 people, there is around 50% chance that some two of them share the same birthday. ZS the Coder finds this very interesting, and decides to test this with the inhabitants of Udayland.
In Udayland, there are 2^n days in a year. ZS the Coder wants to interview k people from Udayland, each of them has birthday in one of 2^n days (each day with equal probability). He is interested in the probability of at least two of them have the birthday at the same day.
ZS the Coder knows that the answer can be written as an irreducible fraction A/B . He wants to find the values of A and B (he does not like to deal with floating point numbers). Can you help him?
Input Format
The first and only line of the input contains two integers n and k, meaning that there are 2^n days in a year and that ZS the Coder wants to interview exactly k people.
1 <= n <=10^15
2 <= k <= 10^15
Output Format
If the probability of at least two k people having the same birthday in 2^n days long year equals A/B gcd(A,B) = 1, print the A and B in a single line.
Since these numbers may be too large, print them modulo 10^6+ 3. Note that A and B must be coprime before their remainders modulo 10^6 + 3 are taken.
Sample TestCase 1
`3 2`
`1 8`
Explanation
In the sample case, there are 2^3 = 8 days in Udayland. The probability that 2 people have the same birthday among 2 people is clearly 1/8, so A = 1, B = 8.
`Normal``Line: 0 Col: 0`
This site uses cookies so that we can remember you and understand how you interact with our website. This allows us to improve and customize your browsing experience. | 447 | 1,702 | {"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} | 3.109375 | 3 | CC-MAIN-2020-05 | longest | en | 0.948645 |
https://www.numbersaplenty.com/1013312343211 | 1,632,376,263,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057417.10/warc/CC-MAIN-20210923044248-20210923074248-00335.warc.gz | 929,667,977 | 3,537 | Search a number
1013312343211 = 7282726779341
BaseRepresentation
bin11101011111011100001…
…11110100110010101011
310120212112110221112021121
432233232013310302223
5113100230424440321
62053302000120111
7133131552542200
oct16575607646253
93525473845247
101013312343211
1136081a445433
12144478306037
137472a29a5bc
143708a479ca7
151b55a314441
hexebee1f4cab
1013312343211 has 24 divisors (see below), whose sum is σ = 1180743675696. Its totient is φ = 867086297280.
The previous prime is 1013312343209. The next prime is 1013312343247. The reversal of 1013312343211 is 1123432133101.
It is not a de Polignac number, because 1013312343211 - 21 = 1013312343209 is a prime.
It is a Duffinian number.
It is not an unprimeable number, because it can be changed into a prime (1013312343271) by changing a digit.
It is a polite number, since it can be written in 23 ways as a sum of consecutive naturals, for example, 108475401 + ... + 108484741.
It is an arithmetic number, because the mean of its divisors is an integer number (49197653154).
Almost surely, 21013312343211 is an apocalyptic number.
1013312343211 is a deficient number, since it is larger than the sum of its proper divisors (167431332485).
1013312343211 is an equidigital number, since it uses as much as digits as its factorization.
1013312343211 is an odious number, because the sum of its binary digits is odd.
The sum of its prime factors is 12859 (or 12852 counting only the distinct ones).
The product of its (nonzero) digits is 1296, while the sum is 25.
Adding to 1013312343211 its reverse (1123432133101), we get a palindrome (2136744476312).
The spelling of 1013312343211 in words is "one trillion, thirteen billion, three hundred twelve million, three hundred forty-three thousand, two hundred eleven". | 540 | 1,783 | {"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} | 3.328125 | 3 | CC-MAIN-2021-39 | latest | en | 0.816656 |
https://jrh794.wordpress.com/2011/09/ | 1,500,885,179,000,000,000 | text/html | crawl-data/CC-MAIN-2017-30/segments/1500549424770.15/warc/CC-MAIN-20170724082331-20170724102331-00644.warc.gz | 670,312,202 | 30,104 | # Monthly Archives: September 2011
## Playing with the Form of an Answer – The Shad-Fack Transom problem
A recent article in The College Mathematics Journal entitled The Shad-Fack Transom by Annalisa Crannell explores several methods of finding the radius of the small circle tucked up in the corner of a square circumscribed about another larger circle. In … Continue reading
## Completing the Square – Fast and Formal
Here is a strictly formal way to complete the square. We want to express in the form . We may want to do this for many reasons, for example, to graph a quadratic function (parabola), to solve a quadratic equation, … Continue reading
## Exaggerate to Teach
My wife was watching a Clinton Anderson video on training horses. I was fascinated. The man was training a horse at the same time he was teaching us how to train a horse and the horse was not necessarily his … Continue reading
## Real Numbers in Calculus
In my last post I alluded to the hand-waving we do discussing limits in calculus because students have a imprecise conception of real numbers. The clever article in the current The College Mathematics Journal, The Intermediate Value Theorem is NOT … Continue reading | 264 | 1,209 | {"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} | 2.6875 | 3 | CC-MAIN-2017-30 | longest | en | 0.902975 |
https://stclairdrake.net/calculate-the-wavelength-of-a-photon-having-3-26/ | 1,638,539,914,000,000,000 | text/html | crawl-data/CC-MAIN-2021-49/segments/1637964362879.45/warc/CC-MAIN-20211203121459-20211203151459-00384.warc.gz | 603,203,831 | 4,632 | A warm metal emits photons of irradiate with energy 3.0 xx 10^(-19) J . Calculate the frequency and wavel...
You are watching: Calculate the wavelength of a photon having 3.26
A hot metal emits photons of light with power 3.0 xx 10^(-19) J . Calculation the frequency and wavel...
If the energy of a photon matching to a wavelength that 6000 Å is 3.32 xx 10^(-19) J , the photon...
If the energy of a photon corresponding to a wavelength the 6000 Å is 3.2 xx 10^(-19) J , the photon ...
If the power of photons equivalent to wavelength the 6000 Å is 3.2 xx 10^(-19) J . The photon ene...
The minimum power required for the photoemission that electron indigenous the surface ar of a steel is 4.95xx1...
If 6000 Å Wavelength power of photon 3 . 2 xx 10^(-19) Jul is climate 4000 Å will be the power of pho...
DisclaimerThe questions posted top top the website are specifically user generated, stclairdrake.net has no ownership or manage over the nature and content the those questions. stclairdrake.net is no responsible for any type of discrepancies worrying the duplicity of content over those questions.
which would have actually the longer wavelength, a photon with energy of 4.59 x 10^-19 J or a photon v en...
If Plank"s constant is 6.6 x 10(-34) J/Hz, climate what is the approximate frequency that a photon havi...
calculation the wavelength the a photon having power of 1.257 X 10-24 joules. (Plancks constant is 6....
one X-ray tube emits X-rays v a wavelength the 1.00 x 10-11 m. Calculate the photon energy, in jou...
calculate the adjust in energy of one atom the emits a photon the wavelength 2.21 meters. (Planck"s ...
A photon has actually a wavelength the 6.2 meters. Calculation the power of the photon in joules. (Planck"s co...
calculate the energy of a photon having actually a wavelength in thefollowing ranges.(a) microwave, with ? =...
A photon with a wavelength that 2.29 10^7 meter strikes a mercury atom in the soil state. Calculat...
See more: Wolf Girl And Black Prince Anime English Dub, Wolf Girl And Black Prince Season 1 Episode 1
Light v a wavelength the 5.0 10-7 m strikes a surface ar that needs 2.0 ev come eject one electron.... | 569 | 2,169 | {"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} | 2.75 | 3 | CC-MAIN-2021-49 | latest | en | 0.829549 |
https://simplewebtool.com/converters/time/monthstoweeks/monthstoweeks.html | 1,669,975,614,000,000,000 | text/html | crawl-data/CC-MAIN-2022-49/segments/1669446710900.9/warc/CC-MAIN-20221202082526-20221202112526-00750.warc.gz | 548,587,140 | 5,665 | # Months to Weeks conversion
## Enter the time in months and press the Convert button:
How to convert Months to Weeks: 1 month = 4.348125 weeks or 1 week = 0.229984 month Months to Weeks formula The time t in weeks is equal to the time t in months multiplied by 4.348125: t(w) = t(mos) * 4.348125 or t(mos) = t(w) * 0.229984
Example: Convert 120 months to weeks: t(w) = 118(mos) * 4.348125 = 521.775(mos)
Months to Weeks conversion table
Months (mos) Weeks (w)
1 mos 4.3481 w
2 mos 8.6963 w
3 mos 13.04 w
4 mos 17.39 w
5 mos 21.74 w
6 mos 26.09 w
7 mos 30.44 w
8 mos 34.79 w
9 mos 39.13 w
10 mos 43.48 w
11 mos 47.83 w
12 mos 52.18 w
13 mos 56.53 w
14 mos 60.87 w
15 mos 65.22 w
16 mos 69.57 w
17 mos 73.92 w
18 mos 78.27 w
19 mos 82.61 w
20 mos 86.96 w
21 mos 91.31 w
22 mos 95.66 w
23 mos 100.01 w
24 mos 104.36 w
25 mos 108.7 w
26 mos 113.05 w
27 mos 117.4 w
28 mos 121.75 w
29 mos 126.1 w
30 mos 130.44 w
31 mos 134.79 w
32 mos 139.14 w
33 mos 143.49 w
34 mos 147.84 w
35 mos 152.18 w
36 mos 156.53 w
37 mos 160.88 w
38 mos 165.23 w
39 mos 169.58 w
40 mos 173.93 w | 469 | 1,075 | {"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} | 3.265625 | 3 | CC-MAIN-2022-49 | latest | en | 0.341636 |
http://www.chembuddy.com/?left=balancing-stoichiometry&right=balancing-reactions | 1,516,593,397,000,000,000 | application/xhtml+xml | crawl-data/CC-MAIN-2018-05/segments/1516084890991.69/warc/CC-MAIN-20180122034327-20180122054327-00246.warc.gz | 416,593,784 | 4,596 | # Reaction to balance & stoichiometry questions
Balancer and stoichiometry calculator
operating systems:
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single user license price:
€24.95 - approximately \$33
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## Equation balancing & stoichiometry lectures » balancing reactions
All chemical reactions occurring within the test tubes, industrial reactors, or nature can be described by reaction equations.
For example reaction of water synthesis can be written as
H2 + O2 -> H2O
This reaction contains the correct reactants - hydrogen and oxygen in its diatomic forms, and correct product - water molecule. We call such a reaction equation (unbalanced, but correctly listing all reactants and products) skeletal.
Knowing the skeletal reaction equation we know what the reactants are and what are the products, but for quantitative predictions we need to balance the reaction equation.
Reaction balancing is based on mass preservation. We know that atoms don't appear nor disappear. If the atom is present in a reactant (compound entering the reaction), it must be present in one of reaction products. The same happens with a charge - charge is also preserved, just like mass is.
We also know that compounds always have the same composition. A water molecule will always consist of one atom of oxygen and two atoms of hydrogen. It can be made when hydrogen reacts with oxygen - but these gases are usually present in the diatomic form of H2 and O2 molecules. To describe the reaction as it occurs we have to combine both mass preservation and molecules composition.
Let's take a look again at the above reaction of water synthesis - there are two oxygen atoms on the left side (in the form of O2) but only one atom of oxygen on the right side - one atom in one water molecule. Reaction equation is not balanced.
Now let's take a look at the same reaction with added coefficients:
2H2 + O2 -> 2H2O
We read it: two molecules of diatomic hydrogen react with one molecule of diatomic oxygen, producing two molecules of water. (Or, alternatively - two moles of diatomic hydrogen react with one mole of diatomic oxygen to produce two moles of water). Is this equation balanced? The ultimate test which allows you to check whether the reaction is correctly balanced or not, is to count all types of atoms on both sides of the equation - they must be identical, and to check whether the charge on both sides of the equation is identical.
Let's check oxygen - there is one molecule on the left side, containing two atoms - so there are two atoms of oxygen on the left side. On the right side there are two molecules, each containing one atom of oxygen - so there are two oxygen atoms on the rights side as well. As number of atoms of oxygen is identical on both sides reaction is balanced with respect to oxygen. Please check for yourself, that the same reaction is also balanced with respect to hydrogen, with four hydrogen atoms on both sides.
So when is the reaction balanced? Firstly, it must have the same number of atoms on both sides. Secondly, all coefficients must be integer. Finally, by convention they should have the smallest possible denominator.
Thus
H2 + 1/2O2 -> H2O
and
4H2 + 2O2 -> 4H2O
are incorrect - even if they are balanced in terms of number of atoms. Note, that such incorrect equations can appear during balancing - and they are perfectly valid as intermediate forms, they just have to be cleaned up before becoming the final version. In this case multiplying the first reaction equation by 2 and dividing the second one by 2 leads us to correctly balanced equations.
Please remember, that when balancing equations you should never touch subscripts, since that will change the composition and therefore the substance itself. All you can modify are the coefficients telling us how many molecules of the reagent entered the reaction, or have left it.
If there are any charged species, you should also check if the charge is balanced, just as atoms are. But there is one, important difference - charge may be negative and positive and sum of these charges can be zero, or negative, while number of atoms is always a positive number. So, we can easily tell that the neutralization reaction
H+ + OH- -> H2O
is balanced - there are identical number of atoms on both sides, and charges on the left side sum up to 0, which is also a total charge on the right side. At the same time equation
H2 -> 2H+
is not balanced - while there are identical numbers of atoms on both sides, charge appeared on the right side from nowhere.
Finally note that not all reaction equations can be balanced - for example
H2O2 -> H2O
will be never balanced, no matter what coefficients you will use. More on that in the when balancing fails section.
Once you will know how to balance equations on paper, you may check our equation balancing and stoichiometry calculator EBAS - it makes most calculations immediately.
Questions? Comments? Feel free to discuss this page content at the chemistry quizzes forum (requires free registration). | 1,074 | 5,095 | {"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} | 2.875 | 3 | CC-MAIN-2018-05 | longest | en | 0.937213 |
http://mathforum.org/library/drmath/view/58659.html | 1,498,574,425,000,000,000 | text/html | crawl-data/CC-MAIN-2017-26/segments/1498128321426.45/warc/CC-MAIN-20170627134151-20170627154151-00424.warc.gz | 246,671,016 | 2,719 | Associated Topics || Dr. Math Home || Search Dr. Math
### How Many Stickers...?
```
Date: 8/25/96 at 4:41:26
From: Raymond
Subject: How Many Stickers...?
James has twice as many stickers as John. John has twice as many
stickers as Pete. If the three boys have 840 stickers altogether,
how may stickers does John have?
```
```
Date: 8/25/96 at 18:36:35
From: Doctor Paul
Subject: Re: How Many Stickers...?
James has x stickers. So then John was 2*x stickers and Pete has
2*(2*x) stickers. All together they have 840, so:
7*x = 840
Divide both sides by seven and get:
x = 120
so James has 120 stickers..
Regards,
-Doctor Paul, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
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Math Forum Home || Math Library || Quick Reference || Math Forum Search | 299 | 1,137 | {"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} | 3.25 | 3 | CC-MAIN-2017-26 | longest | en | 0.901713 |
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Match the length of a right triangle's legs with the exact length of its hypotenuse. 1. square root of thirty seven and square root of one hundred eleven 2. eight and nine 3. five and square root of one hundred nine 4. square root of twenty one and eleven a. square root of one hundred forty-two b. square root of one hundred thirty-four c. square root of one hundred forty-eight d. square root of one hundred and forty-five
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
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Not the answer you are looking for? Search for more explanations. | 346 | 1,366 | {"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 | 3 | CC-MAIN-2017-47 | latest | en | 0.438561 |
http://mathhelpforum.com/pre-calculus/196971-finding-period-tan-function-x-2-exponent.html | 1,481,294,309,000,000,000 | text/html | crawl-data/CC-MAIN-2016-50/segments/1480698542712.12/warc/CC-MAIN-20161202170902-00439-ip-10-31-129-80.ec2.internal.warc.gz | 171,039,230 | 10,388 | # Thread: Finding period of a tan function with x^2 as an exponent
1. ## Finding period of a tan function with x^2 as an exponent
Find four solutions of tan(2x^2 + x − 1) = 5
I work this out to:
2x^2 + x -2.373 = 0
Use the quadratic formula to get x= .8676, -1.3676
I don't know how to find two other answers since I don't the period.
If it didn't have x^2 I could put it in the form: A tan((2pi/B)(x-C)+D where B is the period.
How does one find the other solutions?
2. ## Re: Finding period of a tan function with x^2 as an exponent
As the tangent is positive (+5) there are solutions in the first and third quadrants.
So inverse tan of 5=1.373 or pi+1.373 or 2pi+1.373 or 3pi+1.373 and so on.
3. ## Re: Finding period of a tan function with x^2 as an exponent
Originally Posted by GorFree
Find four solutions of tan(2x^2 + x − 1) = 5
I work this out to:
2x^2 + x -2.373 = 0
Use the quadratic formula to get x= .8676, -1.3676
I don't know how to find two other answers since I don't the period.
If it didn't have x^2 I could put it in the form: A tan((2pi/B)(x-C)+D where B is the period.
How does one find the other solutions?
It's not periodic. But to solve this equation...
\displaystyle \begin{align*} \tan{ \left( 2x^2 + x - 1 \right) } &= 5 \\ 2x^2 + x - 1 &= \arctan{(5)} + \pi n, n \in \mathbf{Z} \\ 2x^2 + x - 1 - \arctan{(5)} - \pi n &= 0 \\ x &= \frac{-1 \pm \sqrt{1^2 - 4(2)[-1 - \arctan{(5)} - \pi n]}}{2(2)} \\ x &= \frac{-1 \pm \sqrt{1 + 8 + 8\arctan{(5)} + 8\pi n }}{4} \\ x &= \frac{-1 \pm \sqrt{9 + 8\arctan{(5)} + 8\pi n}}{4} \end{align*} | 584 | 1,576 | {"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": 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} | 4.375 | 4 | CC-MAIN-2016-50 | longest | en | 0.855758 |
http://web2.0calc.com/questions/hollya_1 | 1,508,457,187,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187823482.25/warc/CC-MAIN-20171019231858-20171020011858-00681.warc.gz | 362,903,286 | 5,777 | +0
# HollyA
+3
145
1
+12
Which best describes how forces must interact for a kite to sail up into the air?
a.The force of gravity must be equal to the force of wind
b.The force of gravity must be greater than the force of the wind
c.The force of the wind must be greater than the force of gravity
d.The force of the person flying the kite must be equal to the force of the wind
HollyA May 16, 2017
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https://codedump.io/share/UArFeJfIuilf/1/how-can-i-plot-2-peaks-in-a-3d-graph-with-different-colors-in-r-using-the-x-axis-not-the-z-axis | 1,632,507,618,000,000,000 | text/html | crawl-data/CC-MAIN-2021-39/segments/1631780057564.48/warc/CC-MAIN-20210924171348-20210924201348-00073.warc.gz | 223,852,121 | 9,376 | M. Beausoleil -4 years ago 78
R Question
How can I plot 2 peaks in a 3D graph with different colors in R (using the x-axis, not the Z-axis)?
I want to color two peaks, with different colors, in a 3d plots but it's not working. Most of the help I find is to color by the z-axis, but I want to color just the peaks.
Here is what I got so far:
``````x <- seq(-10, 10, length = 100)
y <- x
norm <- function(x,y, mean1 = 3, var1 =2,mean2 = -3, var2 =2) {
1/sqrt(2*var1^2*1)*exp(-((y-mean2)^2/(2*var2^2))) *
1/sqrt(2*var2^2*1)*exp(-((x-mean2)^2/(2*var1^2)))+
1/sqrt(2*var2^2*1)*exp(-((y-mean1)^2/(2*var2^2))) *
1/sqrt(2*var1^2*1)*exp(-((x-mean1)^2/(2*var1^2))) }
z <- outer(x, y, norm)
z[is.na(z)] <- 1
rbPal <- colorRampPalette(c('red','blue'))
Col <- rbPal(10)[as.numeric(cut(x = x,breaks = 2))]
persp(x, y, z, col = "blue",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
border = border,
zlim=c(0,.2))
``````
Other try, this is closer to what I want, but as you can see, not optimal since the "floor" is overlapping the other peak!
``````norm1 <- function(x,y, mean1 = 3, var1 =2,mean2 = -3, var2 =2) {
1/sqrt(2*var1^2*1)*exp(-((y-mean2)^2/(2*var2^2))) *
1/sqrt(2*var2^2*1)*exp(-((x-mean2)^2/(2*var1^2)))
}
norm2 <- function(x,y, mean1 = 3, var1 =2,mean2 = -3, var2 =2) {
1/sqrt(2*var2^2*1)*exp(-((y-mean1)^2/(2*var2^2))) *
1/sqrt(2*var1^2*1)*exp(-((x-mean1)^2/(2*var1^2))) }
z1 <- outer(x, y, norm1)
z1[is.na(z1)] <- 1
z2 <- outer(x, y, norm2)
z2[is.na(z1)] <- 1
persp(x, y, z1, col ="red",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
border = border,
zlim=c(0,.2))
par(new=TRUE)
red.a = adjustcolor( "blue", alpha.f = .70)
persp(x, y, z2, col ="blue",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
border = border,
zlim=c(0,.2))
``````
One approach is to make the full plot as a collage of different pieces. In the code below, I split the plot along the diagonal (in the x-y plane) to give the two peaks different colors. But I think there's nothing stopping you from getting fancier here with different colors for multiple regions of the x-y plane.
``````x <- seq(-10, 10, length = 100)
y <- x
norm <- function(x,y, mean1 = 3, var1 =2,mean2 = -3, var2 =2) {
1/sqrt(2*var1^2*1)*exp(-((y-mean2)^2/(2*var2^2))) *
1/sqrt(2*var2^2*1)*exp(-((x-mean2)^2/(2*var1^2)))+
1/sqrt(2*var2^2*1)*exp(-((y-mean1)^2/(2*var2^2))) *
1/sqrt(2*var1^2*1)*exp(-((x-mean1)^2/(2*var1^2))) }
z <- outer(x, y, norm)
z[is.na(z)] <- 1
rbPal <- colorRampPalette(c('red','blue'))
Col <- rbPal(10)[as.numeric(cut(x = x,breaks = 2))]
persp(x, y, z, col = "blue",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
# border = border,
zlim=c(0,.2))
z2 <- z
for(i in 1:100){
for(j in 1:(100-i)){
z2[i,j] <- NA
}
}
par(new=T)
graphics::persp(x, y, z2, col = "red",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
# border = border,
zlim=c(0,.2))
``````
Or if you just want the peaks colored (and everything else gray, for example), you could do
``````z3 <- z
z3[z3<.03] <- NA
z4 <- z2
z4[z4<.03] <- NA
persp(x, y, z, col = "gray90",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
# border = border,
zlim=c(0,.2))
par(new=T)
persp(x, y, z3, col = "skyblue1",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
# border = border,
zlim=c(0,.2))
par(new=T)
persp(x, y, z4, col = "indianred1",
xlab = "",
ylab = "", zlab = "",
theta = 60,
phi = 15,d = 10,
# border = border, | 1,437 | 3,478 | {"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} | 3.40625 | 3 | CC-MAIN-2021-39 | latest | en | 0.744494 |
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Bank Clerk :: CR Quiz 134
Home > Bank Clerk > CR Quiz 134 > General Questions
1 .
Direction (Q. 1 - 5): In each question below, there are three statements followed by four conclusions numbered I, II, III and IV. You have to take the given statements to be true even if they seem to be at variance with commonly known facts and then decide which of the given conclusions logically follow (s) from the given statements.
Statements: All kites are nights. No night is a flight.
Some kites are bites.
Conclusions: I. No kite is a flight.
II. Some kites are flights.
III. Some bites are nights.
IV. Some bites are not nights.
Only I follows Only IV follows I and III follow Either III or IV and I follow
2 .
Some stereos are monos.
Conclusions: I. Some monos are not radios.
II. Some FMs are not stereos.
IV. All monos are FMs
Only I follows Only II follows I and III follow None of these
3 .
Statements: All floppies are discs.
All discs are ROMs.
All ROMs are RAMs.
Conclusions: I. All discs are RAMs.
II. All floppies are RAMs.
III. All ROMs are discs.
IV. Some RAMs are floppies.
All follows I, II and III follow I and II follow None of these
4 .
Statements: All guitars are sitars.
Some sitars are harmoniums.
No harmoniums are sarods.
Conclusions: I. Some harmoniums are guitars.
II. Some sitars are sarods.
III. Some sitars are not sarods.
IV. Some guitars are not harmoniums.
Only I follows Only III follows Either I or IV and III follow None of these
5 .
Statements: Some teas are coffee.
No tea is milk. All coffees are cream.
Conclusions: I. No milk is cream.
II. Some teas are cream.
III. Some coffees are milk.
IV. Some coffees are not milk.
Only I follows Only II follows Either III or IV follows Only II and IV follow
6 .
Direction (Q. 6 - 10): Each of the questions below consists of a question and two statements numbered I and II given below it. You have to decide whether the data provided in the statements are sufficient to answer the question. Read both the statements.
How many times has Ramesh topped in school ?
I. We had three examinations every year in our school.
II. Ramesh was beaten only twice by Navin in his entire school career.
if the data in statement I alone are sufficient to answer the question while the data in statement II alone are not sufficient to answer the question. if the data in statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question. if the data either in statement I alone or in statement II alone are sufficient to answer the question. if the data even in both the statements I and II together are not sufficient to answer the question.
7 .
Where are the National Games being held this year ?
I. The National Games are being held at a city 80 km from Delhi.
II. Only Gurgaon is a city 80 km from Delhi.
if the data in statement I alone are sufficient to answer the question while the data in statement II alone are not sufficient to answer the question. if the data in statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question. if the data either in statement I alone or in statement II alone are sufficient to answer the question. if the data in both statements I and II together are necessary to answer the question.
8 .
I. Madhav was born 10 days after Sanjit was born
II Sanj it will be 22 years of age on the 13 th ofJanuary 2010.
if the data in statement I alone are sufficient to answer the question while the data in statement II alone are not sufficient to answer the question. if the data in statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question. if the data either in statement I alone or in statement II alone are sufficient to answer the question. if the data in both statements I and II together are necessary to answer the question.
9 .
How many children does I have ?
I. E has only two sons.
II. E, wife of L, has only one daughter.
if the data in statement I alone are sufficient to answer the question while the data in statement II alone are not sufficient to answer the question. if the data in statement II alone are sufficient to answer the question while the data in statement I alone are not sufficient to answer the question. if the data either in statement I alone or in statement II alone are sufficient to answer the question. if the data in both statements I and II together are necessary to answer the question. | 1,102 | 4,750 | {"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} | 2.53125 | 3 | CC-MAIN-2017-26 | longest | en | 0.905281 |
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# Am I eligible to apply for Accounting Phd?
Author Message
Intern
Joined: 04 Jan 2012
Posts: 4
Followers: 0
Kudos [?]: 0 [0], given: 0
Am I eligible to apply for Accounting Phd? [#permalink]
### Show Tags
04 Jan 2012, 02:47
I am a full-time student in Master of Accounting program right now, about to graduate in a year. I am thinking about applying for Accounting Phd, but really not sure if I got any chance. It's getting more competitive these years. Can you guys help to have a look at my profile? If I am not eligible at all, I will drop this idea soon enough. I appreciate!
My profile:
Bachelor
-One of top 10 schools in China
-Accounting major/GPA 3.6
Master
-University of Nebraska at Omaha (a small regional school)
-Accounting/GPA 3.9
-Research assistant for Accounting 1 year+
-Accounting Lab tutor 1 semester
-Taking classes research method & econometrics
Work
- Two and a half year auditor in Big 4 in China
-CPA in China & will pass CPA exam of US soon
GMAT
Last edited by huiminchen on 04 Jan 2012, 14:15, edited 1 time in total.
Current Student
Joined: 12 Sep 2011
Posts: 900
Concentration: Finance, Finance
GMAT 1: 710 Q48 V40
Followers: 139
Kudos [?]: 905 [0], given: 114
Re: Am I eligible to apply for Accounting Phd? [#permalink]
### Show Tags
04 Jan 2012, 09:41
I don't see why you wouldn't be a candidate! You sound like you have a strong background, but it may be recommended that you speak with a professional on the topic. Potentially a school couseler at your current school, a current teacher at your school, or maybe review the website and class profile of your desired school(s). If you need help with your GMAT, this is the place to be!
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Re: Am I eligible to apply for Accounting Phd? [#permalink]
### Show Tags
04 Jan 2012, 14:14
Thank you, GMATLA!
I guess I will give it a shot...Doing research is fun for me. I think I will figure out what schools to apply for in August, after taking GMAT exam.
Re: Am I eligible to apply for Accounting Phd? [#permalink] 04 Jan 2012, 14:14
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Strange (??) eligibility norms for PhD in US 0 12 Nov 2013, 19:49
Am I able to get into a Phd Program in US? 4 29 Jan 2012, 00:46
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1 Am I good enough for a top PhD program? 16 20 Dec 2010, 06:25
Am I eligible to apply 1 26 Oct 2010, 05:35
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# Am I eligible to apply for Accounting Phd?
Moderator: carcass
Powered by phpBB © phpBB Group and phpBB SEO Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 992 | 3,616 | {"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} | 2.625 | 3 | CC-MAIN-2017-13 | longest | en | 0.886595 |
http://mathhelpforum.com/advanced-applied-math/4819-help-me-print.html | 1,527,041,773,000,000,000 | text/html | crawl-data/CC-MAIN-2018-22/segments/1526794865023.41/warc/CC-MAIN-20180523004548-20180523024548-00469.warc.gz | 186,161,025 | 2,747 | # Help me!
• Aug 8th 2006, 12:42 PM
babygirl
Help me!
The size of your eardrum (the tympanum) partially determines the upper frequency limit of your audible region, usually between 16,000 Hz and 20,000 Hz. If the wavelength is about twice the diameter of the eardrum and the air temperature is 20oC, how wide is your eardrum? Is your answer reasonable?
• Aug 8th 2006, 01:16 PM
CaptainBlack
Quote:
Originally Posted by babygirl
The size of your eardrum (the tympanum) partially determines the upper frequency limit of your audible region, usually between 16,000 Hz and 20,000 Hz. If the wavelength is about twice the diameter of the eardrum and the air temperature is 20oC, how wide is your eardrum? Is your answer reasonable?
The speed of sound at 20C is 343.4 m/s, and as before:
$\displaystyle \lambda=c/f$,
Lets work with a cut-off frequency of 18000 Hz. Then the wavelength is:
$\displaystyle \lambda=\frac{34304}{18000}\approx 0.019 \mbox{ m}$,
or about 2cm. So the eardrum dia would be half this or about 1cm. This seems
about the right sort of size (perhaps a little bigger than I would have expected).
RonL | 317 | 1,123 | {"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} | 3.53125 | 4 | CC-MAIN-2018-22 | latest | en | 0.930403 |
https://hyperleap.com/topic/Sampling_(signal_processing) | 1,604,138,190,000,000,000 | text/html | crawl-data/CC-MAIN-2020-45/segments/1603107917390.91/warc/CC-MAIN-20201031092246-20201031122246-00199.warc.gz | 361,341,795 | 43,245 | # Sampling (signal processing)
sampling ratesamplingsample ratesamplesampledsampling frequencysamplessamplersSampling (information theory)sound samplers
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.wikipedia
487 Related Articles
### Aliasing
aliasaliasedtemporal aliasing
That fidelity is reduced when s(t) contains frequency components whose periodicity is smaller than two samples; or equivalently the ratio of cycles to samples exceeds ½ (see Aliasing).
In signal processing and related disciplines, aliasing is an effect that causes different signals to become indistinguishable (or aliases of one another) when sampled.
### Discrete time and continuous time
discrete timediscrete-timecontinuous-time
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.
Unlike a continuous-time signal, a discrete-time signal is not a function of a continuous argument; however, it may have been obtained by sampling from a continuous-time signal.
### Oversampling
oversampledoverachievingoversample
Although the use of oversampling can completely eliminate aperture error and aliasing by shifting them out of the pass band, this technique cannot be practically used above a few GHz, and may be prohibitively expensive at much lower frequencies.
In signal processing, oversampling is the process of sampling a signal at a sampling frequency significantly higher than the Nyquist rate.
### Analog-to-digital converter
In practice, the continuous signal is sampled using an analog-to-digital converter (ADC), a device with various physical limitations. One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
Furthermore, instead of continuously performing the conversion, an ADC does the conversion periodically, sampling the input, limiting the allowable bandwidth of the input signal.
### Nyquist frequency
Nyquist limitNyquistN/2 different frequencies
The original signal is retrievable from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of low pass filter called a reconstruction filter.
The Nyquist frequency, named after electronic engineer Harry Nyquist, is half of the sampling rate of a discrete signal processing system.
### Nyquist–Shannon sampling theorem
sampling theoremNyquist-Shannon sampling theoremNyquist theorem
The approximately double-rate requirement is a consequence of the Nyquist theorem.
It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.
### Whittaker–Shannon interpolation formula
interpolation/sampling theoryreconstructingsinc interpolation
The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
When the x[n] sequence represents time samples, at interval T, of a continuous function, the quantity f s = 1/T is known as the sample rate, and f s /2 is the corresponding Nyquist frequency.
### Quantization (signal processing)
quantizationquantization errorquantized
An analog-to-digital converter (ADC) can be modeled as two processes: sampling and quantization.
### Digital-to-analog converter
DACDACsD/A
One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
There are several DAC architectures; the suitability of a DAC for a particular application is determined by figures of merit including: resolution, maximum sampling frequency and others.
### Low-pass filter
low-passlow pass filterlowpass filter
The original signal is retrievable from a sequence of samples, up to the Nyquist limit, by passing the sequence of samples through a type of low pass filter called a reconstruction filter. The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
A low-pass filter is used as an anti-aliasing filter prior to sampling and for reconstruction in digital-to-analog conversion.
### Digital audio
digital musicdigitalaudio
Digital audio uses pulse-code modulation and digital signals for sound reproduction.
The ADC runs at a specified sampling rate and converts at a known bit resolution.
### Anti-aliasing filter
anti-aliasinganti-aliasing (AA) filteroptical low-pass filter
The Audio Engineering Society recommends 48 kHz sampling rate for most applications but gives recognition to 44.1 kHz for Compact Disc (CD) and other consumer uses, 32 kHz for transmission-related applications, and 96 kHz for higher bandwidth or relaxed anti-aliasing filtering.
Since the theorem states that unambiguous reconstruction of the signal from its samples is possible when the power of frequencies above the Nyquist frequency is zero, a real anti-aliasing filter trades off between bandwidth and aliasing.
### Dirac comb
Sampling functionimpulse traininfinite impulse train
When the time interval between adjacent samples is a constant (T), the sequence of delta functions is called a Dirac comb.
Owing to the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling by multiplication with it, but it also allows modelling periodization by convolution with it.
### Digital Audio Tape
DATDATsDAT recorder
In appearance it is similar to a Compact Cassette, using 3.81 mm / 0.15" (commonly referred to as 4 mm) magnetic tape enclosed in a protective shell, but is roughly half the size at 73 mm × 54 mm × 10.5 mm. The recording is digital rather than analog. DAT can record at sampling rates equal to, as well as higher and lower than a CD (44.1, 48 or 32 kHz sampling rate respectively) at 16 bits quantization. If a comparable digital source is copied without returning to the analogue domain, then the DAT will produce an exact clone, unlike other digital media such as Digital Compact Cassette or non-Hi-MD MiniDisc, both of which use a lossy data reduction system.
### 44,100 Hz
44.1 kHz44.1kHz44,100 samples per second
In digital audio, 44,100 Hz (alternately represented as 44.1 kHz) is a common sampling frequency.
### Signal processing
signal analysissignalsignal processor
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.
### DV
MiniDVDVCAMDVCPRO
Audio can be stored in either of two forms: 16-bit Linear PCM stereo at 48 kHz sampling rate (768 kbit/s per channel, 1.5 Mbit/s stereo), or four nonlinear 12-bit PCM channels at 32 kHz sampling rate (384 kbit/s per channel, 1.5 MBit/s for four channels).
### Pulse-code modulation
PCMLPCMLinear PCM
Digital audio uses pulse-code modulation and digital signals for sound reproduction.
In a PCM stream, the amplitude of the analog signal is sampled regularly at uniform intervals, and each sample is quantized to the nearest value within a range of digital steps.
### Dirac delta function
The Whittaker–Shannon interpolation formula is mathematically equivalent to an ideal lowpass filter whose input is a sequence of Dirac delta functions that are modulated (multiplied) by the sample values.
A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Shah distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis.
### Direct Stream Digital
DSDDSD-CD(DSD)
The signal is stored as delta-sigma modulated digital audio, a sequence of single-bit values at a sampling rate of 2.8224 MHz (64 times the CD audio sampling rate of 44.1 kHz, but only at 1⁄32768 of its 16-bit resolution).
### Digital signal processing
DSPsignal processingdigital
However, digital signal processing operations can have very high dynamic range, consequently it is common to perform mixing and mastering operations at 32-bit precision and then convert to 16- or 24-bit for distribution.
The signals processed in this manner are a sequence of numbers that represent samples of a continuous variable in a domain such as time, space, or frequency.
### Delta-sigma modulation
delta-sigmasigma-deltaSigma-delta modulation
One advantage of higher sampling rates is that they can relax the low-pass filter design requirements for ADCs and DACs, but with modern oversampling sigma-delta converters this advantage is less important.
In a conventional ADC, an analog signal is sampled with a sampling frequency and subsequently quantized in a multi-level quantizer into a digital signal. | 1,858 | 8,938 | {"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} | 2.515625 | 3 | CC-MAIN-2020-45 | latest | en | 0.896406 |
https://www.geeksforgeeks.org/gate-gate-cs-2015-set-2-question-44/?type=article&id=145183 | 1,686,240,220,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224655027.51/warc/CC-MAIN-20230608135911-20230608165911-00481.warc.gz | 832,092,183 | 30,807 | GeeksforGeeks App
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# GATE | GATE-CS-2015 (Set 2) | Question 65
Assume that the bandwidth for a TCP connection is 1048560 bits/sec. Let α be the value of RTT in milliseconds (rounded off to the nearest integer) after which the TCP window scale option is needed. Let β be the maximum possible window size with window scale option. Then the values of α and β are.
(A) 63 milliseconds 65535 × 214
(B) 63 milliseconds 65535 × 216
(C) 500 milliseconds 65535 × 214
(D) 500 milliseconds 65535 × 216 | 149 | 517 | {"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} | 2.703125 | 3 | CC-MAIN-2023-23 | latest | en | 0.686652 |
https://www.programming-idioms.org/idiom/257/traverse-list-backwards/5127/csharp | 1,695,444,383,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233506479.32/warc/CC-MAIN-20230923030601-20230923060601-00229.warc.gz | 1,051,152,343 | 7,021 | # Idiom #257 Traverse list backwards
Print each index i and value x from the list items, from the last down to the first.
``````for(int i = items.Count - 1; i >= 0; i--)
{
Console.WriteLine(\$"Index = {i}, Item = {items[i]}");
}``````
``````var i = items.length ;
for (var e in items.reversed) {
i--;
print("\$i, \$e");
}``````
``````do i=size(items),1,-1
print *,i,items(i)
end do``````
``import "fmt"``
``````for i := len(items) - 1; i >= 0; i-- {
x := items[i]
fmt.Printf("Item %d = %v \n", i, x)
}``````
``````[...items].reverse().forEach((item, index) =>
console.log(Math.abs(index -= items.length), item));``````
``````for(int i = items.length-1;i>=0;i--){
System.out.println("Index:"+ i +", Value:" + items[i]);
}``````
``````import java.util.ArrayList;
import java.util.Collections;``````
``````Collections.reverse(items);
int i = 0;
for(T x : items) {
System.out.printf("%d %s%n", i, x);
i++;
}``````
``for i := items.count-1 downto 0 do writeln(i,' ',items[i]);``
``````for (\$i = \$#items; \$i >= 0; \$i--) {
print "\$i \$items[\$i]\n";
}``````
``````for i in range(len(items)-1, -1, -1):
print(i, items[i])``````
``````for i, x in enumerate(reversed(items)):
print(f'{i} {x}')
``````
``````items.each_with_index.reverse_each{|x, i| puts "#{i} #{x}" }
``````
``````for (i, item) in items.iter().enumerate().rev() {
println!("{} = {}", i, *item);
}``````
`````` for (i, x) in items.iter().rev().enumerate() {
println!("{i} = {x}");
}``````
``````(define (reversed-inspect items)
(foldl (lambda (x i)
(display (format "~a ~a\n" i x))
(sub1 i))
(sub1 (length items))
(reverse items)))``````
programming-idioms.org | 553 | 1,625 | {"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} | 2.515625 | 3 | CC-MAIN-2023-40 | latest | en | 0.350902 |
http://www.javaguides.net/2018/12/java-arithmetic-operators-example.html | 1,547,746,512,000,000,000 | text/html | crawl-data/CC-MAIN-2019-04/segments/1547583659056.44/warc/CC-MAIN-20190117163938-20190117185938-00328.warc.gz | 334,951,959 | 29,587 | ### Java Arithmetic Operators Example
Arithmetic operators are used in mathematical expressions in the same way that they are used in algebra.
The following table lists the arithmetic operators:
The operands of the arithmetic operators must be of a numeric type. You cannot use them on boolean types, but you can use them on char types, since the char type in Java is, essentially, a subset of an int.
### The Basic Arithmetic Operators Example
The basic arithmetic operations — additionsubtractionmultiplication, and division — all behave as you would expect for all numeric types. The unary minus operator negates its single operand.
The following simple example program demonstrates the arithmetic operators. It also illustrates the difference between floating-point division and integer division.
```package net.javaguides.corejava.operators.arithmetic;
public class BasicMath {
public static void main(String args[]) {
// arithmetic using integers
System.out.println("Integer Arithmetic");
int a = 1 + 1;
int b = a * 3;
int c = b / 4;
int d = c - a;
int e = -d;
System.out.println("a = " + a);
System.out.println("b = " + b);
System.out.println("c = " + c);
System.out.println("d = " + d);
System.out.println("e = " + e);
// arithmetic using doubles
System.out.println("\nFloating Point Arithmetic");
double da = 1 + 1;
double db = da * 3;
double dc = db / 4;
double dd = dc - a;
double de = -dd;
System.out.println("da = " + da);
System.out.println("db = " + db);
System.out.println("dc = " + dc);
System.out.println("dd = " + dd);
System.out.println("de = " + de);
}
}```
Output:
``````Integer Arithmetic
a = 2
b = 6
c = 1
d = -1
e = 1
Floating Point Arithmetic
da = 2.0
db = 6.0
dc = 1.5
dd = -0.5
de = 0.5``````
### The Modulus Operator Example
The modulus operator, %, returns the remainder of a division operation. It can be applied to floating-point types as well as integer types. The following example program demonstrates the %:
```package net.javaguides.corejava.operators.arithmetic;
public class Modulus {
public static void main(String args[]) {
int x = 42;
double y = 42.25;
System.out.println("x mod 10 = " + x % 10);
System.out.println("y mod 10 = " + y % 10);
}
}```
Output:
``````x mod 10 = 2
y mod 10 = 2.25``````
### Arithmetic Compound Assignment Operator Example
Java provides special operators that can be used to combine an arithmetic operation with an assignment. As we know that, statements like the following are quite common in programming:
``````a = a + 4;
``````
In Java, you can rewrite this statement as shown here:
``````a += 4;
``````
This version uses the += compound assignment operator.
Here is a sample program that shows several op = assignments in action:
```package net.javaguides.corejava.operators.arithmetic;
public class OpEquals {
public static void main(String args[]) {
int a = 1;
int b = 2;
int c = 3;
a += 5;
b *= 4;
c += a * b;
c %= 6;
System.out.println("a = " + a);
System.out.println("b = " + b);
System.out.println("c = " + c);
}
}```
Output:
``````a = 6
b = 8
c = 3``````
### Increment and Decrement Examples
The ++ and – – are Java’s increment and decrement operators.
The increment operator increases its operand by one. For example:
``````x = x + 1; or x++;
``````
The decrement operator decreases its operand by one. For example:
``````x = x - 1; or x--;
``````
The following program demonstrates the increment operator:
```package net.javaguides.corejava.operators.arithmetic;
public class IncrementDecrement {
public static void main(String args[]) {
int a = 1;
int b = 2;
int c;
int d;
c = ++b;
d = a++;
c++;
System.out.println("a = " + a);
System.out.println("b = " + b);
System.out.println("c = " + c);
System.out.println("d = " + d);
}
}```
Output:
``````a = 2
b = 3
c = 4
d = 1`````` | 1,050 | 3,770 | {"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} | 2.9375 | 3 | CC-MAIN-2019-04 | latest | en | 0.680348 |
http://bookshadow.com/weblog/2017/08/?page=2 | 1,571,825,833,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570987833089.90/warc/CC-MAIN-20191023094558-20191023122058-00522.warc.gz | 31,722,199 | 19,191 | # 归档 2017年8月
## LeetCode Weekly Contest 46解题报告 作者是 在线疯狂 发布于 2017年8月21日 在 LeetCode.
LeetCode Weekly Contest 46是LeetCode举办的第五十一场正式周赛，共4道题目，比赛时长1.5小时。
LeetCode 661. Image Smoother
LeetCode 662. Maximum Width of Binary Tree
LeetCode 663. Equal Tree Partition
LeetCode 664. Strange Printer
## 题目描述：
LeetCode 664. Strange Printer
There is a strange printer with the following two special requirements:
1. The printer can only print a sequence of the same character each time.
2. At each turn, the printer can print new characters starting from ...
## 题目描述：
LeetCode 663. Equal Tree Partition
Given a binary tree with `n` nodes, your task is to check if it's possible to partition the tree to two trees which have the equal sum of values after removing exactly one edge ...
## 题目描述：
LeetCode 662. Maximum Width of Binary Tree
Given a binary tree, write a function to get the maximum width of the given tree. The width of a tree is the maximum width among all levels. The binary tree has ...
## 题目描述：
LeetCode 661. Image Smoother
Given a 2D integer matrix M representing the gray scale of an image, you need to design a smoother to make the gray scale of each cell becomes the average gray scale (rounding down) of ... | 338 | 1,229 | {"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} | 2.953125 | 3 | CC-MAIN-2019-43 | latest | en | 0.629965 |
https://www.studypool.com/discuss/467822/use-a-calculator-to-simplify-or-solve-the-following?free | 1,508,550,992,000,000,000 | text/html | crawl-data/CC-MAIN-2017-43/segments/1508187824537.24/warc/CC-MAIN-20171021005202-20171021025202-00522.warc.gz | 982,258,243 | 14,491 | Time remaining:
##### use a calculator to simplify or solve the following
label Algebra
account_circle Unassigned
schedule 0 Hours
account_balance_wallet \$5
Joel killmer has a pumpkin patch with 13 rows of pumpkin plants. There are 3 rows with 150 pumpkins each and 10 rows with 99 pumpkins each.How many pumpkins does joel have ?
Oct 20th, 2017
3 rows of 150 pumpkins = 3*150 = 450
10 rows of 99 pumpkins = 10 * 99 = 990
Now total pumpkins = 990 + 450 = 1440
Apr 9th, 2015
...
Oct 20th, 2017
...
Oct 20th, 2017
Oct 21st, 2017
check_circle | 178 | 548 | {"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} | 3.359375 | 3 | CC-MAIN-2017-43 | latest | en | 0.856889 |
https://docs.teradata.com/reader/qs3BwCv08lKpQd7ePrYOvA/HLYccD105APPk4FWNpUyHQ | 1,571,213,854,000,000,000 | text/html | crawl-data/CC-MAIN-2019-43/segments/1570986666467.20/warc/CC-MAIN-20191016063833-20191016091333-00017.warc.gz | 456,482,762 | 51,472 | # Descriptive Statistics - Teradata Warehouse Miner - 5.4.4
## Teradata Warehouse Miner User Guide - Volume 1Introduction and Profiling
prodname
vrm_release
5.4.4
category
User Guide
featnum
B035-2300-077K
The descriptive statistics available with Teradata Warehouse Miner provide a variety of functions to statistically analyze and explore a Teradata database. Descriptive statistical analysis is valuable for the following reasons.
• It can provide business insight in its own right.
• It uncovers data quality issues which, if not corrected or compensated for, can jeopardize the accuracy of any analytic models that are based on the data.
• It isolates the data that should be used in building analytic models. For example, outlying values should sometimes be excluded from a model; in other cases, these values might be required to solve a particular business problem. Further, some statistical processes used in analytic modeling require a certain type of distribution of data.
Descriptive statistical analysis can determine the suitability of various data elements for model input and can suggest transformations that may be required for these data elements.
In the case of the Descriptive Statistics, NULL values are handled through the generated SQL’s aggregate functions. In this case, SQL ignores the NULL value and adjusts the number of observations in its calculation. This effectively provides a listwise deletion of NULL values.
The following are the descriptive statistical functions currently available in Teradata Warehouse Miner:
• Adaptive Histogram — Determine the distribution of a numeric column(s) giving counts, sub-binning column(s) with higher counts and determining data spikes.
• Correlation Matrix — Build and view a correlation matrix.
• Data Explorer — Automated exploration of any number of tables or views within an entire database.
• Frequency — Compute frequency of column values or multi-column combined values. Optionally, compute frequency of values for pairs of columns in a single column list or two column lists, and generate simple statistics for any other column within a table.
• Histogram — Determine the distribution of a numeric column(s) giving counts with optional overlay counts and statistics.
• Overlap — Count overlapping column values in combinations of tables (i.e., find “key” values in common between tables).
• Scatter Plot — Plot sampled values of two to three variables in 2-D or 3-D.
• Statistical Analysis — Determine any of the following descriptive statistics for numeric column(s):
1. Minimum Value
2. Maximum Value
3. Mean Value
4. Standard Deviation
5. Skewness
6. Kurtosis
7. Standard Mean Error
8. Coefficient of Variance
9. Variance
10. Sum
11. Uncorrected Sums of squares
12. Corrected Sums of squares
13. Values Count
14. Modal Value
15. Percentiles
16. Top 5/Bottom 5 Rank and Values
• Text Field Analyzer — Analyze character data and help distinguish whether the field is a numeric type, a date, a time, a timestamp, or character data.
• Values Analysis — Count the number of values of various kinds for a given column or columns, including:
1. Number of Rows
2. Rows with Non-NULL Values
3. Rows with NULL Values
4. Unique Values
5. Rows with Value ‘0’
6. Rows with a Positive Value
7. Rows with a Negative Value
8. Rows Containing Blank Values
In order to add a Descriptive Statistical analysis to a Teradata Warehouse Miner Data Mining Project, create a new analysis with any of the mechanisms described in Using Teradata Warehouse Miner. This will produce the following dialog. | 747 | 3,561 | {"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} | 2.78125 | 3 | CC-MAIN-2019-43 | latest | en | 0.802078 |
https://community.cesium.com/t/question-about-drawing-an-ellipse-and-about-cartesian3-angles/881 | 1,716,692,130,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058861.60/warc/CC-MAIN-20240526013241-20240526043241-00391.warc.gz | 144,847,270 | 4,616 | Question 1:
In Cesium.Shapes there is a static function named "computeEllipseBoundary". One of the params it takes is "rotation". I understand what this value is, but I'm not sure how to get it without hard-coding.
I want the rotation value to change depending on where my mouse is at a given time. For example, if my mouse is at a 45 degree angle on the 3D globe, I want to know that. I'm trying to constantly re-draw an ellipse until I release the left click on the mouse, so I need my params to constantly update until I'm finished drawing the ellipse.
Question 2:
In Cesium.Cartesian3 there is a static function named "angleBetween". The documentation says it returns "the angle between the Cartesians". What does that mean? What angle? I was hoping it would return a value I could use as the "rotation" in my first question, but it doesn't look like that is what it's trying to do.
Any help on either question is much appreciated!
In Cesium an ellipse is drawn along a line of longitude that passes through the center point.
Noting that positive x points East and positive y points North you can do something like this which is a modification of the Ellipse example in SandCastle.
``````// Green ellipse with height
// Default orientation of ellipse is along north/south
// Remember that positive x is East and positive y is North
var north = Cesium.Cartesian3.fromArray([0.0, 1.0, 0.0]);
var mouse = Cesium.Cartesian3.fromArray([1.0, 1.0, 0.0]);
var rotationAngle = Cesium.Cartesian3.angleBetween(north, mouse);
ellipseGeometry = new Cesium.EllipseGeometry({
center : ellipsoid.cartographicToCartesian(Cesium.Cartographic.fromDegrees(-95.0, 35.0)),
semiMinorAxis : 200000.0,
semiMajorAxis : 400000.0,
rotation : rotationAngle,
height: 200000.0,
vertexFormat : Cesium.PerInstanceColorAppearance.VERTEX_FORMAT
});`````` | 453 | 1,831 | {"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} | 3.015625 | 3 | CC-MAIN-2024-22 | latest | en | 0.879153 |
https://www.physicsforums.com/threads/first-order-diffy-q-problem-with-bernoulli-integrating-factors.995696/ | 1,701,637,112,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100508.53/warc/CC-MAIN-20231203193127-20231203223127-00003.warc.gz | 1,072,659,259 | 16,639 | # First Order Diffy Q Problem with Bernoulli/Integrating Factors
It looks like the second term is just the antiderivative of the first term, but it's not clear what it should be.
Homework Statement
If dy/dy + y = (1-1/x)(1/(2y)), solve when x=1, y=1/sqrt(2). Hint: Use a substitution to get into the form of a first order differential equation
Relevant Equations
integrating factor = e^integral(P(x))
v=y^(1-n)
dy/dx +P(x)y=Q(x)y^n
I seem to be getting an unsolvable integral here (integral calculator says it's an Ei function, which I've never seen). My thought was to use Bernoulli to make it linear and then integrating factors. Is that wrong? The basic idea is below:
P(x) 1, Q(x) = 1/2(1-1/x), n=-1, so use v=y^1- -1)=y^2
so y=sqrt(v), dy/dx=1/(2sqrt(v))dv/dx
Thus the equation becomes 1/(2sqrt(v))dv/dx + sqrt(v) = 1/sqrt(v) * 1/2(1-1/x))
Multiplying by 2* sqrt(v) gives dv/dx +2 v = (1-1/x)
Use the integrating factor e^2x gives: d/dx(v*e^(2x)) =e^(2x)(1-1/x) = e^(2x) - e^(2x)*1/x
ve^(2x) then just equals the antiderivative of e^(2x) - e^(2x)*1/x, but that second term seems to give a problem. Is my method incorrect? Not sure how to proceed.
I can't spot any mistakes in your solution (except that you got to learn using ##\LaTeX## )so it probably is correct.
I tried wolfram and it also gives the solution in terms of the Ei(2x) function
https://www.wolframalpha.com/input/?i=y'+y=(1-1/x)0.5(1/y)
Maybe check again the exact statement of the problem. There might be some typo after all.
Assuming you mean ##\frac{dy}{dx}+y=(1-\frac 1x)(\frac 1{2y})## (you wrote dy/dy), it already is a first order ODE.
This does suggest a typo,
Delta2
haruspex said:
Assuming you mean ##\frac{dy}{dx}+y=(1-\frac 1x)(\frac 1{2y})## (you wrote dy/dy), it already is a first order ODE.
This does suggest a typo,
I think there he wanted to write first order linear
Delta2 said:
I think there he wanted to write first order linear
Yes, that fits. I agree it leads to a nasty integral; I had it in the form of a double exponential. | 657 | 2,031 | {"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-2023-50 | latest | en | 0.892192 |
http://mathhelpforum.com/advanced-algebra/188517-complex-polar-forms-sin-cos-angles-print.html | 1,524,575,884,000,000,000 | text/html | crawl-data/CC-MAIN-2018-17/segments/1524125946688.88/warc/CC-MAIN-20180424115900-20180424135900-00110.warc.gz | 202,909,407 | 3,155 | # Complex Polar Forms, Sin and Cos Angles
• Sep 21st 2011, 01:55 PM
tangibleLime
Complex Polar Forms, Sin and Cos Angles
Problem: Find all solutions of $\displaystyle z^{3} = -8$.
I can do the entire problem except for one part. After putting it into the correct form,
$\displaystyle |z|^{3}(cos3\theta+isin3\theta)$,
I do not know how to find the values of $\displaystyle cos3\theta$ or $\displaystyle sin3\theta$. I know that $\displaystyle |z|^3$ is 8, but I can't figure out the values of cos and sin.
Any help is appreciated.
• Sep 21st 2011, 01:59 PM
alexmahone
Re: Complex Polar Forms, Sin and Cos Angles
Quote:
Originally Posted by tangibleLime
Problem: Find all solutions of $\displaystyle z^{3} = -8$.
I can do the entire problem except for one part. After putting it into the correct form,
$\displaystyle |z|^{3}(cos3\theta+isin3\theta)$,
I do not know how to find the values of $\displaystyle cos3\theta$ or $\displaystyle sin3\theta$. I know that $\displaystyle |z|^3$ is 8, but I can't figure out the values of cos and sin.
Any help is appreciated.
$\displaystyle |z| = 2$.
So, $\displaystyle cos\ 3\theta+isin\ 3\theta=-1$.
Equate real and imaginary terms to find $\displaystyle \theta$.
• Sep 21st 2011, 02:04 PM
tangibleLime
Re: Complex Polar Forms, Sin and Cos Angles
Thanks,
So since $\displaystyle |z|^3 = 8$, I need $\displaystyle cos3\theta+isin3\theta$ to equal -1 to satisfy the initial equation where $\displaystyle z^3 = -8$. The only way to get -1 from $\displaystyle cos3\theta+isin3\theta$ is to have $\displaystyle cos3\theta = -1$ and $\displaystyle sin3\theta = 0$ to get rid of the imaginary number and return (-1 + i0).
Correct?
• Sep 21st 2011, 02:08 PM
alexmahone
Re: Complex Polar Forms, Sin and Cos Angles
Quote:
Originally Posted by tangibleLime
Thanks,
So since $\displaystyle |z|^3 = 8$, I need $\displaystyle cos3\theta+isin3\theta$ to equal -1 to satisfy the initial equation where $\displaystyle z^3 = -8$. The only way to get -1 from $\displaystyle cos3\theta+isin3\theta$ is to have $\displaystyle cos3\theta = -1$ and $\displaystyle sin3\theta = 0$ to get rid of the imaginary number and return (-1 + i0).
Correct?
Yes. You should get three values of $\displaystyle \theta$ in the interval $\displaystyle [0,\ 2\pi]$.
• Sep 21st 2011, 02:11 PM
Plato
Re: Complex Polar Forms, Sin and Cos Angles
Quote:
Originally Posted by tangibleLime
Problem: Find all solutions of $\displaystyle z^{3} = -8$.
Here is some notation: $\displaystyle \exp(i\theta)=\cos{\theta)+i\sin(\theta)$
So we can write $\displaystyle -8=8\exp(\pi i)$
The cube roots of that is then $\displaystyle 2\exp \left( {\frac{{i\pi }}{3} + \frac{{2\pi ik}}{3}} \right),~~k=0,1,2$ | 847 | 2,708 | {"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-2018-17 | latest | en | 0.753553 |
http://philippinekabu.com/best/iphone/low/15850930f1aef780d66f4ef2e-failing-abstract-algebra | 1,674,769,753,000,000,000 | text/html | crawl-data/CC-MAIN-2023-06/segments/1674764494826.88/warc/CC-MAIN-20230126210844-20230127000844-00006.warc.gz | 34,860,606 | 23,233 | failing abstract algebra
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The notation will be the same as that which is used in Gallian's Contemporary Abstract Algebra[1]. He majored in pre-law, following in his father's footsteps, but also took some computer science courses. standard algebra only when x = 0 or x = 1. MATH 379. Many Title I at-risk students fail algebra during the first semester in a large, urban comprehensive high school. The more you figure out on your own, the better your understanding of the material, and the better you'll do both on the exams and in your future endeavours that might require abstract algebra. Only 100 showed up, even though administrators called each student's home. The data, composed of records from a cohort of students ( N = 10,100) who entered ninth grade in 2001, were analyzed via logistic regression. 5 + 4 is 9 and 4 + 5 is 9. Online Abstract Algebra Problems Assistance. Some reasons could have led to your not doing well at Abstract Math problems. This would be fine if "abstract algebra" would be a well established subarea. is called the cyclotomic polynomial. One overview of research on this topic was . Many mathematics experts also consider algebra knowledge and skills It happened that my double major in Physics kept me away from the lecture time for the course. Abstract Algebra, 3rd Edition. Abstract: It is shown that any -finite, wild AW*-algebra of type II 1 fails to have non-trivial separable representations.
Wanna send me your hard exams? Failing to comply with the Code may result in charges and sanctions, as outlined in the Code. -4 learners passed the test without studying. Pass/Fail only. 0000-0002-1385-8351 Metadata Show full item record Abstract The purpose of this study is to examine the differences between Title I at-risk students who do pass algebra and those who do not. Global Seminar. 1. To address such limits of neural solvers, we introduce the concept of a . This is the idea behind abstract algebra. Professors often say that students do not fail calculus but fail the algebra in their calculus classes. Available at . I am of an age to insist, stubbornly, that Lang's Algebra is really the benchmark in the present context, even though I . -60 learners failed the test without having studied.
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これから伸び行くアジアの雄「フィリピン」で株の売買をはじめましょう！ | 5,001 | 23,014 | {"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} | 2.734375 | 3 | CC-MAIN-2023-06 | latest | en | 0.952535 |
https://oeis.org/A002514 | 1,585,732,017,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585370505550.17/warc/CC-MAIN-20200401065031-20200401095031-00254.warc.gz | 599,028,897 | 3,951 | The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.
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A002514 Coefficients in the asymptotic expansions of modified Hankel functions h_1(z) and h_2(z), rounded to nearest integer. (Formerly M2992 N1212) 2
0, 0, 0, 0, 1, 3, 15, 79, 474, 3207, 24087, 198923, 1791902, 17484377, 183707380, 2067904033, 24827519376, 316694549817, 4277112686513, 60971132411393, 914869422343564, 14413525170009350, 237888443951757586, 4104608160094692304 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,6 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). The Staff of the Computational Laboratory, Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives. Annals of the Computation Laboratory of Harvard University, Vol. 2, Harvard Univ. Press, Cambridge, Massachusetts, 1945. see p. XXXV. LINKS Robert Israel, Table of n, a(n) for n = 1..459 The Staff of the Computational Laboratory, Tables of the Modified Hankel Functions of Order One-Third and of Their Derivatives. [Annotated scans of two pages] FORMULA a(n) = round((Product_{k=1..n} (9 * (2*k-1)^2 - 4)) / (2^(4*n) * 3^n * n!)). - Sean A. Irvine, Oct 18 2015 a(n) = round(Gamma(n+5/6)*Gamma(n+1/6)*3^n/(2^(2*n+1)*Pi*n!)). - Robert Israel, Oct 19 2015 MAPLE seq(round(simplify(GAMMA(n+5/6)*GAMMA(n+1/6)*3^n/(2^(2*n+1)*Pi*n!))), n=1..50); # Robert Israel, Oct 19 2015 CROSSREFS Sequence in context: A292097 A052755 A137957 * A319216 A093889 A020044 Adjacent sequences: A002511 A002512 A002513 * A002515 A002516 A002517 KEYWORD nonn AUTHOR EXTENSIONS More terms from Sean A. Irvine, Oct 18 2015 STATUS approved
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Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.) | 714 | 2,234 | {"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} | 2.875 | 3 | CC-MAIN-2020-16 | latest | en | 0.627213 |
https://www.esaral.com/q/which-of-the-following-pairs-of-linear-equations-has-unique-solution-no-solution-or-infi-nitely-many-solutions/ | 1,642,609,033,000,000,000 | text/html | crawl-data/CC-MAIN-2022-05/segments/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00702.warc.gz | 797,727,782 | 23,284 | Which of the following pairs of linear equations has unique solution, no solution, or infi nitely many solutions.
Question.
Which of the following pairs of linear equations has unique solution, no solution, or infi nitely many solutions. In case there is a unique solution, find it by using cross multiplication method.
(i) x – 3y – 3 = 0
3x – 9y – 2 = 0
(ii) 2x + y = 5
3x + 2y = 8
(iii) 3x – 5y = 20
6x – 10y = 40
(iv) x – 3 y – 7 = 0
3x – 3y – 15 = 0
Solution:
(i) x – 3y – 3 = 0, 3x – 9y – 2 = 0
$\frac{a_{1}}{a_{2}}=\frac{1}{3}, \frac{h_{1}}{b_{2}}=\frac{-3}{-9}=\frac{1}{3}, \frac{c_{1}}{c_{2}}=\frac{3}{2}$
$\Rightarrow \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$
Hence, no solution.
(ii) 2x + y = 5 …(i) and 3x + 2y = 8 …(ii)
$\frac{a_{1}}{a_{2}} \neq \frac{h_{1}}{h_{2}}\left(\frac{a_{1}}{a_{2}}=\frac{2}{3}, \frac{h_{1}}{b_{2}}=\frac{1}{2}\right)$
Here, we have a unique solution. By cross multiplication, we have
$\Rightarrow \frac{x}{\{(1)(-8)-(2)(-5)\}}=\frac{y}{\{(-5)(3)-(-8)(2)\}}$
$=\frac{1}{\{(Q)(2)-(3)(1)\}}$
$\Rightarrow \frac{x}{(-8+10)}=\frac{y}{(-15+16)}=\frac{1}{(4-3)}$
$\Rightarrow \frac{x}{2}=\frac{y}{1}=\frac{1}{1} \Rightarrow \frac{x}{2}=\frac{1}{1}$ and $\frac{y}{1}=\frac{1}{1}$
$\Rightarrow x=2$ and $y=1$
(iii) 3x – 5y = 20 ……..(i)
6x – 10y = 40 …….(ii)
$\frac{a_{1}}{a_{2}}=\frac{3}{6}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{-5}{-10}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{20}{40}=\frac{1}{2}$
$\therefore \quad \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Hence, infinite solutions
(iv) x – 3y – 7 = 0 ……..(i)
3x – 3y – 15 = 0 ……..(ii)
$\frac{a_{1}}{a_{2}}=\frac{1}{3}, \frac{b_{1}}{b_{2}}=1, \frac{c_{1}}{c_{2}}=\frac{7}{15}$
$\therefore \quad \frac{\mathbf{a}_{1}}{\mathbf{a}_{2}} \neq \frac{\mathbf{h}_{1}}{\mathbf{h}_{2}}$
Hence, unique solution
$\frac{x}{(-3)(-15)-(-3)(-7)}=\frac{y}{3 \times(-7)-1 \times(-15)}$
$=\frac{1}{1 \times(-3)-3(-3)}$
$\Rightarrow \frac{x}{45-21}=\frac{y}{-21+15}=\frac{1}{-3+9}$
$\Rightarrow \frac{x}{24}=\frac{y}{-6}=\frac{1}{6}$
$x=\frac{\mathbf{2 4}}{\mathbf{6}}=4, \quad y=\frac{-\mathbf{6}}{\mathbf{6}}=-1$
Editor | 1,023 | 2,163 | {"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} | 4.5625 | 5 | CC-MAIN-2022-05 | latest | en | 0.445444 |
https://republicofsouthossetia.org/question/12v-2y-8-28v-4-9-what-is-the-answer-14780461-11/ | 1,652,676,020,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662509990.19/warc/CC-MAIN-20220516041337-20220516071337-00685.warc.gz | 571,802,951 | 14,852 | ## 12vx^2y^8–28v^4x^9 What is the answer
Question
12vx^2y^8–28v^4x^9
What is the answer
in progress 0
5 months 2021-12-27T12:11:22+00:00 2 Answers 0 views 0
## Answers ( )
1. Answer:
6vx2 • (3v4x6 – 2y6)
Step-by-step explanation:
18v5x8-12vx2y6
Final result :
6vx2 • (3v4x6 – 2y6)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((18•(v5))•(x8))-((22•3vx2)•y6)
Step 2 :
Equation at the end of step 2 :
((2•32v5) • x8) – (22•3vx2y6)
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
18v5x8 – 12vx2y6 = 6vx2 • (3v4x6 – 2y6)
Trying to factor as a Difference of Squares :
4.2 Factoring: 3v4x6 – 2y6
Theory : A difference of two perfect squares, A2 – B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 – AB + BA – B2 =
A2 – AB + AB – B2 =
A2 – B2
Note : AB = BA is the commutative property of multiplication.
Note : – AB + AB equals zero and is therefore eliminated from the expression.
Check : 3 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor as a Difference of Cubes:
4.3 Factoring: 3v4x6 – 2y6
Theory : A difference of two perfect cubes, a3 – b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
6vx2 • (3v4x6 – 2y6)18v5x8-12vx2y6
Final result :
6vx2 • (3v4x6 – 2y6)
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((18•(v5))•(x8))-((22•3vx2)•y6)
Step 2 :
Equation at the end of step 2 :
((2•32v5) • x8) – (22•3vx2y6)
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
18v5x8 – 12vx2y6 = 6vx2 • (3v4x6 – 2y6)
Trying to factor as a Difference of Squares :
4.2 Factoring: 3v4x6 – 2y6
Theory : A difference of two perfect squares, A2 – B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 – AB + BA – B2 =
A2 – AB + AB – B2 =
A2 – B2
Note : AB = BA is the commutative property of multiplication.
Note : – AB + AB equals zero and is therefore eliminated from the expression.
Check : 3 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Trying to factor as a Difference of Cubes:
4.3 Factoring: 3v4x6 – 2y6
Theory : A difference of two perfect cubes, a3 – b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Final result :
6vx2 • (3v4x6 – 2y6)
2. 48vx^x(16y^2-21v) a lot of steps but you got this | 1,206 | 2,882 | {"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} | 4.46875 | 4 | CC-MAIN-2022-21 | latest | en | 0.531741 |
https://jeelabs.org/2010/08/27/flippin%E2%80%99-bits/index.html | 1,702,052,575,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100762.64/warc/CC-MAIN-20231208144732-20231208174732-00389.warc.gz | 371,227,149 | 10,196 | # Computing stuff tied to the physical world
## Flippin’ bits
In AVR, Hardware, Software on Aug 27, 2010 at 00:01
After yesterday’s post about setting and clearing bits, let’s explore reversing bits, i.e. changing them from 0 to 1 and back. And let’s do it by blinking an LED attached to DIO of port 1 – i.e. Arduino digital pin 4:
The “if (onOff = 0)” etc is the logic that toggles onOff between 0 and 1 on each pass through the loop:
`````` if (onOff == 0) onOff = 1; else onOff = 0;
``````
But there are lots of ways to do the same thing, coded differently:
`````` if (onOff == 0) onOff = 1; else onOff = 0;
if (onOff == 0) onOff = bit(0); else onOff = 0;
if (onOff == 0) bitSet(onOff, 0); else bitClear(onOff, 0);
onOff = onOff ? 0 : 1;
onOff = (~ onOff) & 1;
onOff = (onOff + 1) & 1;
onOff = ! onOff;
onOff = 1 - onOff;
onOff = onOff ^ 1;
onOff ^= 1;
``````
See if you can figure out all of these.
Take your pick. Those last two use C’s XOR operator. I tend to prefer shorter source code, so I’d use that last notation (note that the resulting compiled code is not necessarily shorter than the other examples).
Now suppose you have a byte value “X” and you want to flip the 4th bit in it, while not changing anything else. That’s a bit more work. It could be done like this, for example:
`````` if (bitRead(X, 4) == 0) bitSet(X, 4); else bitClear(X, 4);
``````
Or like either of these:
`````` X = X ^ bit(4);
X ^= bit(4);
``````
This shows clearly that the “^” XOR operator does exactly what we need: flip bits.
Back to blinking an actual LED, as done with the above sketch. Here’s a little mind bender – another sketch, doing the same using raw ports and the XOR operator:
The first example was doing things “the Arduino way”, using pinMode() and digitalWrite(). It compiles to 890 bytes of code. This second example goes straight to the hardware and uses 554 bytes of code:
• Arduino digital pin 4 is bit 4 on the “D port” of an ATmega
• “DDRD” is the “Data Direction Register”, where we set up pin 4 as an output
• “PORTD” is the out “Port Register”, which controls the actual output signal
You can see the XOR in action in that last example. It takes all the output bits of port D (Arduino pins 0 .. 7), and flips just a single bit, i.e. bit 4.
Just for kicks, I’ll show you one more way to blink the LED:
This uses a relatively little-known feature of the hardware, which actually has “bit flipping” built-in. The “PIND” register is normally used for input, i.e. for reading the state of a pin as an input signal. But you can also write to that register. When you do, it will be used to flip output pins, but only for the bits which were set to 1. It’s essentially a built-in XOR.
That last example uses 550 bytes of code, most of which is overhead from the Arduino run-time library (setting up the milliseconds timer, etc). So what’s in a measly 4 bytes, right? Wrong. There is a minute, but sometimes important difference: the other approaches all had to read the register value first, flip the bit, and then write the value back. This last version only writes a (constant) value to a register. With interrupts, that can be very important: this last version can’t ever go wrong, it will always flip the requested bit. The other version could have an interrupt occur between the read and the write. It’s a known issue for the Arduino Mega. It can lead to code which runs for a week, and then fails mysteriously. Bugs like these are fiendishly hard to properly diagnose.
Bit-flipping can be quite useful for physical computing. Not only does it let you easily toggle specific bits, and change the state of some output pins, it can also be a way to clear a bit. Let’s say you need to generate a (very) quick pulse. Here are four ways to accomplish the same thing:
`````` bitSet(PORTD, 4); bitClear(PORTD, 4);
PORTD |= bit(4); PORTD ^= bit(4);
PORTD |= bit(4); PIND = bit(4);
PIND = bit(4); PIND = bit(4);
``````
That second one based on XOR works, because bit 4 is known to be one, so setting it to zero is always the same as flipping it. That’s also why the third PORTD/PIND example works, with PIND doing the XOR in hardware. Lastly, the fourth approach will only work if bit 4 was initially zero. It’s the fastest one, and does not suffer from the interrupt race condition mentioned above.
Ok, that’s enough flippin’ for one day!
Tomorrow, I’m going to go into, ehm… “fractional bits” (haha!) ;)
Update – see comment below on why “bitSet(PORTD, 4); bitClear(PORTD, 4);” are also interrupt-safe (mostly – but not on every pin of an Arduino Mega!).
1. The ‘bitSet(PORTD, 4); bitClear(PORTD, 4);’ example does a read/modify/write but in one instruction and two cycles (so it should work OK with interrupts).
Funny that the datasheet mentions this XORing feature of PINx but still lists all the bits in PINx as read-only. ;] And remember that this feature works only on the newer devices (ATmega88 and co., not on the old ATtiny2313 or ATmega8).
2. Ah, good catch. | 1,367 | 5,017 | {"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} | 2.953125 | 3 | CC-MAIN-2023-50 | latest | en | 0.846337 |
https://classroom.thenational.academy/lessons/solve-problems-involving-division-with-remainders-part-1-6hjp6t | 1,618,320,159,000,000,000 | text/html | crawl-data/CC-MAIN-2021-17/segments/1618038072366.31/warc/CC-MAIN-20210413122252-20210413152252-00389.warc.gz | 276,152,237 | 39,534 | # Solve problems involving division with remainders (Part 1)
In this lesson, we will solve division equations that involve remainders.
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
1/5
2/5
3/5
4/5
5/5
Quiz:
# Intro quiz - Recap from previous lesson
Before we start this lesson, let’s see what you can remember from this topic. Here’s a quick quiz!
1/5
2/5
3/5
4/5
5/5
# Video
Click on the play button to start the video. If your teacher asks you to pause the video and look at the worksheet you should:
• Click "Close Video"
• Click "Next" to view the activity
Your video will re-appear on the next page, and will stay paused in the right place.
# Worksheet
These slides will take you through some tasks for the lesson. If you need to re-play the video, click the ‘Resume Video’ icon. If you are asked to add answers to the slides, first download or print out the worksheet. Once you have finished all the tasks, click ‘Next’ below.
Quiz:
# Division with remainders
Don’t worry if you get a question wrong! Forgetting is an important step in learning. We will recap next lesson.
1/6
2/6
3/6
4/6
5/6
6/6
Quiz:
# Division with remainders
Don’t worry if you get a question wrong! Forgetting is an important step in learning. We will recap next lesson.
1/6
2/6
3/6
4/6
5/6 | 376 | 1,409 | {"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} | 2.734375 | 3 | CC-MAIN-2021-17 | longest | en | 0.869786 |
https://blog.ronrecord.com/index.php/tag/videos/ | 1,719,072,341,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862404.32/warc/CC-MAIN-20240622144011-20240622174011-00229.warc.gz | 118,751,562 | 29,778 | # videos
## My YouTube Playlists
I’m adding many of my YouTube playlists here, both in the sidebar and as pages reflected in menus elsewhere. Here is the current list of pages containing YouTube playlists: In…
## Videos
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## Fractals
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Here however, we attempt to provide a visual introduction to fractals that does not require any mathematics.
Fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractional dimension, of which many examples have been formulated and studied in great depth. Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, architecture and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.
Blog posts by Doctorwhen in the Fractals category:
## The Buddhabrot Fractal, Mandelbrot Set, and The Logistic Map
The Buddhabrot is a fractal rendering technique related to the Mandelbrot set. Its name reflects its pareidolic resemblance to classical depictions of Gautama Buddha, seated in a meditation pose with a forehead mark (tikka) and traditional topknot (ushnisha).…
Posted in Art, Fractals, Math, Video | Tagged , , , |
## Big Lyapunov Exponent Bug
Fractal visually depicting the Lyapunov exponents of an iterated map of the plane. Generated with a fractal software laboratory written by Dr. Ronald Joe Record. I find it interesting that,…
Posted in Fractals, Interesting, Math |
## Mandelbrot Fractal Zoom (8k 60fps)
A nice straight forward deep zoom into the Mandelbrot Set with some nice shapes and structures appearing at magnification 1e124. The frames for this video were rendered with Ultra Fractal…
Posted in Fractals, Math, Video | Tagged , , |
## Electric Sheep HD Fractal Animation
Electric Sheep is a distributed computing project for animating and evolving fractal flames, which are in turn distributed to the networked computers, which display them as a screensaver. It’s crowd sourced evolving art. Below is…
Posted in Art, Fractals, Math, Video | Tagged , , , |
## Frax Fractal Compilation
A short video I made combining several fractals generated with Frax HD on my iPad. These illustrate some of the features of Frax including zoom, lighting, textures, panning, and spinning…
Posted in Art, Fractals, Math, Video | Tagged , , , , , , |
## My YouTube Playlists
I’m adding many of my YouTube playlists here, both in the sidebar and as pages reflected in menus elsewhere. Here is the current list of pages containing YouTube playlists: In…
Posted in Fractals, Music, Playlist, Video | Tagged , , , , |
## Mandelbox Trip – 3D Mandelbrot Fly Through
Key-frame animation test in Mandelbulber. Animation was rendered using Mandelbulber 0.80 https://sourceforge.net/projects/mandelbulber/h Mandelbox Trip video © Copyright by Krzysztof Marczak
Posted in Fractals, Interesting, Math, Video | Tagged , , , |
## Mandelbulb Flight
A super cool zoom into a 3D Mandelbrot Set (z^8+c Mandelbulb)
Posted in Fractals, Interesting, Math, Video | Tagged , , , , |
## Mandelbox Zoom
An amazing zoom on the Mandelbox made with Mandelbulb 3D v1.53 and VirtualDub. In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in…
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By popular demand, I will write a few posts about some games that Katie has enjoyed playing over the past couple of years. I would also love to hear about other people’s favorite kids games.
We started playing ‘board’ games with Katie when she was three, and that seemed like a good age. You can probably start a bit earlier with some kids, but my guess is not by much. Here are some of Katie’s earliest games:
Zingo – Katie got this game for her third birthday and it was the first one we played with her. Similar to bingo, but requires the kid to pay a bit more attention. When a picture comes up, only the first person to notice that they have it gets to put the tile on their card. It also comes with a fun tile dispenser which Katie absolutely loved and insisted on controlling every game. We don’t play is so much any more, but this was definitely a favorite for quite a while. There are also variations where the tiles have words or numbers on them.
Sum Swamp – On one hand, this game is nothing special; the board consists of a snaky path and the players take turns throwing dice to determine how many steps they get to advance. The one thing that sets this apart from games like chutes-and-ladders is the non-standard dice, which there are three of. Two of the dice have the numbers 1-6 on them and the third one has + and – signs. In order to find out how far to advance, the child has to solve a small addition or subtraction problem. This game is mostly responsible for making Katie comfortable with single-digit arithmetic.
Dominoes – This one needs no description. However, we have a set where instead of dots, the tiles have pictures of characters from one of Katie’s favorite Russian cartoons. I think this really helped make her more enthusiastic about the game. So if your kid likes princesses or cars, or anything else that you can find on dominoes, I’d recommend starting with those rather than the traditional ones.
Match It! Memory – There are several variations of these games where you have to match words to pictures, numbers to addition/subtraction expressions, first letter of a word to the rest of it, etc. Katie has a few of the variations, although by far not all. You are supposed to play memory with the cards, but Katie just likes to find the matching pairs in the open. For some reason, this is one of Katie’s most frequent go-to games when she plays by herself (ok, it can’t quite compete with dolls, but might actually be a second). When she would sometimes wake up way too early in the morning and we’d send her off to her room to play, we would often find her playing with one of these sets.
Ok, that’s it for now. In future posts I hope to write about games that we have been playing more recently with Katie.
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About aofradkin
I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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3 Responses to Starting Games
1. Daniel says:
Have you tried with SET?
Like
• aofradkin says:
I have played SET with her, but not much. However, I am a big fan of the game so I’m sure that we’ll be playing it more in the future.
Like
2. Daniel says:
Thank you. I tried with my daughter when she was just 5 but she found it too difficult. So we’ll try next year!
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https://newpathworksheets.com/math/grade-6/kansas-standards | 1,719,158,052,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198862474.84/warc/CC-MAIN-20240623131446-20240623161446-00338.warc.gz | 370,195,722 | 21,373 | ## Kansas Academic Standards for Sixth Grade Math
EstimationEstimation is the process of rounding a number either up or down to the nearest place value requested. Estimation makes it easier to perform mathematical operations quickly. Read more...iWorksheets: 6Study Guides: 1
GraphsFreeA graph is a diagram that shows information in an organized way. Read more...iWorksheets: 16Study Guides: 1
ProbabilityProbability word problems worksheets. Probability is the possibility that a certain event will occur. An event that is certain to occur has a probability of 1. An event that cannot occur has a probability of 0. Therefore, the probability of an event occurring is always between 0 and 1. The closer a probability is to 1, the more certain that an event will occur. Read more...iWorksheets: 3Study Guides: 1
Commutative/Associative PropertiesThe commutative property allows us to change the order of the numbers without changing the outcome of the problem. The associative property allows us to change the grouping of the numbers. Read more...iWorksheets: 4Study Guides: 1
Area and Circumference of CirclesFreeThe circumference of a circle is the distance around the outside. The area of a circle is the space contained within the circumference. It is measured in square units. Read more...iWorksheets: 6Study Guides: 1
Diameter of CircleThe diameter of a circle is a line segment that passes through the center of a circle connecting one side of the circle to the other. Read more...iWorksheets: 3Study Guides: 1
PerimeterA perimeter is the measurement of the distance around a figure. It is measured in units and can be measured by inches, feet, blocks, meters, centimeters or millimeters. Read more...iWorksheets: 3Study Guides: 1
Add/Subtract FractionsAdding or substracting fractions means to add or subtract the numerators and write the sum over the common denominator. Read more...iWorksheets: 9Study Guides: 1
Adding FractionsAdding fractions is the operation of adding two or more different fractions. Read more...iWorksheets: 3Study Guides: 1
MultiplicationMultiplication is a mathematical operation in which numbers, called factors, are multiplied together to get a result, called a product. Multiplication can be used with numbers or decimals of any size. Read more...iWorksheets: 3Study Guides: 1
Multiply FractionsMultiplying fractions is the operation of multiplying two or more fractions together to find a product. Read more...iWorksheets: 3Study Guides: 1
Number PatternsA number pattern is a group of numbers that are related to one another in some sort of pattern. Finding a pattern is a simpler way to solve a problem. Read more...iWorksheets: 3Study Guides: 1
Simplify FractionsSimplifying fractions is the process of reducing fractions and putting them into their lowest terms. Read more...iWorksheets: 3Study Guides: 1
TablesTables refer to the different types of diagram used to display data. <br>There are many types of tables such as data table, frequency table, line chart and stern-and-leaf plot. Read more...iWorksheets: 3Study Guides: 1
Mixed NumbersA mixed number has both a whole number and a fraction. Read more...iWorksheets: 4Study Guides: 1
Whole Numbers to TrillionsThe number system we use is based on a place value system. Although there are only 10 different digits in this system, it is possible to order them in so many variations that the numbers represented are infinite. Read more...iWorksheets: 4Study Guides: 1
### KS.6.RP. Ratios and Proportional Relationships
#### Understand ratio concepts and use ratio reasoning to solve problems.
##### 6.RP.1. Use ratio language to describe a relationship between two quantities. Distinguish between part-to-part and part-to-whole relationships. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Proportions/Equivalent FractionsEquivalent fractions represent the same ratio between two values. Read more...iWorksheets :3Study Guides :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
RatioA ratio is a comparison of two numbers. The two numbers must have the same unit in order to be compared. Read more...iWorksheets :3Study Guides :1
Simple ProportionsA proportion is a statement that two ratios are equal. A ratio is a pair of numbers used to show a comparison. To solve a proportion, calculate equivalent fractions in order to be sure the two fractions (ratios) are equal. Read more...iWorksheets :3Study Guides :1
Numerical ProportionsNumerical proportions compare two numbers. The numbers can have the same units such as a ratio or the numbers can have different units such as rates. A proportion is usually in the form of a:b or a/b. Ratios are used to compare objects, wins and losses, sides of a figure to its area and many more. Rates are used to compare miles per hour, words per minute, and many others. A unit rate is when the denominator of a proportion is one. Read more...iWorksheets :4Study Guides :1
##### 6.RP.2. Use unit rate language (“for each one”, “for every one” and “per”) and unit rate notation to demonstrate understanding the concept of a unit rate a/b associated with a ratio ݑ:ΰݑ with ϰݑ Ϣɠ 0, For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
Numerical ProportionsNumerical proportions compare two numbers. The numbers can have the same units such as a ratio or the numbers can have different units such as rates. A proportion is usually in the form of a:b or a/b. Ratios are used to compare objects, wins and losses, sides of a figure to its area and many more. Rates are used to compare miles per hour, words per minute, and many others. A unit rate is when the denominator of a proportion is one. Read more...iWorksheets :4Study Guides :1
##### 6.RP.3. Use ratio and rate reasoning to solve real-world and mathematical problems, (e.g. by reasoning about tables of equivalent ratios, tape diagrams, double number line diagram, or using calculations.)
###### 6.RP.3a. Make tables of equivalent ratios relating quantities with whole-number measurements, find the missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Proportions/Equivalent FractionsEquivalent fractions represent the same ratio between two values. Read more...iWorksheets :3Study Guides :1
RatioRatios are used to make a comparison between two things. Read more...iWorksheets :9Study Guides :1Vocabulary :1
RatioA ratio is a comparison of two numbers. The two numbers must have the same unit in order to be compared. Read more...iWorksheets :3Study Guides :1
Simple ProportionsA proportion is a statement that two ratios are equal. A ratio is a pair of numbers used to show a comparison. To solve a proportion, calculate equivalent fractions in order to be sure the two fractions (ratios) are equal. Read more...iWorksheets :3Study Guides :1
Numerical ProportionsNumerical proportions compare two numbers. The numbers can have the same units such as a ratio or the numbers can have different units such as rates. A proportion is usually in the form of a:b or a/b. Ratios are used to compare objects, wins and losses, sides of a figure to its area and many more. Rates are used to compare miles per hour, words per minute, and many others. A unit rate is when the denominator of a proportion is one. Read more...iWorksheets :4Study Guides :1
###### 6.RP.3b. Find a percent of a quantity as a rate per 100 (e.g. 30% of a quantity means 30/100 times the quantity ); solve problems involving finding the whole, given a part and the percent.
Percent, Rate, BaseA percent is a way of comparing a number with 100. Percents are usually written with a percent sign. To solve a percent problem, multiply the value by the percent using one of the representations for the percent. Read more...iWorksheets :3Study Guides :1
PercentageThe term percent refers to a fraction in which the denominator is 100. It is a way to compare a number with 100. Read more...iWorksheets :6Study Guides :1
Multiple Representation of Rational NumbersWhat are multiple representations of rational numbers? A rational number represents a value or a part of a value. Rational numbers can be written as integers, fractions, decimals, and percents.The different representations for any given rational number are all equivalent. Read more...iWorksheets :3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Applying PercentsApplying percents is a term that refers to the different ways that percents can be used. The percent of change refers to the percent an amount either increases or decreases based on the previous amounts or numbers. Applying percents also means to calculate simple interest using the interest equation, I = P · r · t, where P is the principal; r is the rate and t is the time. Read more...iWorksheets :3Study Guides :1
###### 6.RP.3c. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
MeasurementFreeThere are many units of measurement: inches, feet, yards, miles, millimeters, meters, seconds, minutes, hours, cups, pints, quarts, gallons, ounces, pounds, etc Read more...iWorksheets :6Study Guides :1
### KS.6.NS. The Number System
#### Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
##### 6.NS.1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, requiring multiple exposures connecting various concrete and abstract models.
Multiply / Divide FractionsFreeTo multiply two fractions with unlike denominators, multiply the numerators and multiply the denominators. It is unnecessary to change the denominators for this operation. Read more...iWorksheets :6Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets :3Study Guides :1
#### Compute fluently (efficiently, accurately, and flexibly) with multi-digit numbers and find common factors and multiples.
##### 6.NS.2. Fluently (efficiently, accurately, and flexibly) divide multi-digit numbers using an efficient algorithm.
DivisionDivision is a mathematical operation is which a number, called a dividend is divided by another number, called a divisor to get a result, called a quotient. Read more...iWorksheets :3Study Guides :1
DivisionDivide three-digit numbers by one- and two-digit numbers. Read more...iWorksheets :6Study Guides :1Vocabulary :1
##### 6.NS.3. Fluently (efficiently, accurately, and flexibly) add, subtract, multiply, and divide multi-digit decimals using an efficient algorithm for each operation.
Add/Subtract/Multiply/Divide DecimalsYou add/subtract/multiply/divide decimals the same way you add/subtract/multiply/divide whole numbers BUT you also need to place the decimal in the correct spot. When multiplying decimals, the decimals may or may NOT be lined up in the multiplication problem. Read more...iWorksheets :10Study Guides :1Vocabulary :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting, multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
##### 6.NS.4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 18+48 as 6(3+8).
Distributive PropertyThe distributive property offers a choice in multiplication of two ways to treat the addends in the equation. We are multiplying a sum by a factor which results in the same product as multiplying each addend by the factor and then adding the products. Read more...iWorksheets :3Study Guides :1
Common FactorsFactors are two numbers multiplied together to get a product (an answer to a multiplication problem) Read more...iWorksheets :6Study Guides :1Vocabulary :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
#### Apply and extend previous understandings of numbers to the system of rational numbers.
##### 6.NS.5. Understand positive and negative numbers to describe quantities having opposite directions or values (e.g. temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge);
###### 6.NS.5a. Use positive and negative numbers to represent quantities in real-world contexts,
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
###### 6.NS.5b. Explaining the meaning of 0 in each situation.
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
##### 6.NS.6. Understand a rational number as a point on the number line and a coordinate pair as a location on a coordinate plane.
###### 6.NS.6b. Recognize signs of numbers in ordered pairs indicate locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Plot PointsYou use plot points to place a point on a coordinate plane by using X and Y coordinates to draw on a coordinate grid. Read more...iWorksheets :5Study Guides :1Vocabulary :1
CoordinatesFreeThe use of coordinates pertains to graphing and the quadrants that are formed by the x and y-axis. Read more...iWorksheets :14Study Guides :1
Plotting PointsIn a coordinate pair, the first number indicates the position of the point along the horizontal axis of the grid. The second number indicates the position of the point along the vertical axis. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Graphs and TablesUsing tables and graphs is a way people can interpret data. Data means information. So interpreting data just means working out what information is telling you. Information is sometimes shown in tables, charts and graphs to make the information easier to read. Read more...iWorksheets :3Study Guides :1
Area of Coordinate PolygonsCalculate the area of basic polygons drawn on a coordinate plane. Coordinate plane is a grid on which points can be plotted. The horizontal axis is labeled with positive numbers to the right of the vertical axis and negative numbers to the left of the vertical axis. Read more...iWorksheets :3Study Guides :1
###### 6.NS.6c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Plot PointsYou use plot points to place a point on a coordinate plane by using X and Y coordinates to draw on a coordinate grid. Read more...iWorksheets :5Study Guides :1Vocabulary :1
CoordinatesFreeThe use of coordinates pertains to graphing and the quadrants that are formed by the x and y-axis. Read more...iWorksheets :14Study Guides :1
Plotting PointsIn a coordinate pair, the first number indicates the position of the point along the horizontal axis of the grid. The second number indicates the position of the point along the vertical axis. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Graphs and TablesUsing tables and graphs is a way people can interpret data. Data means information. So interpreting data just means working out what information is telling you. Information is sometimes shown in tables, charts and graphs to make the information easier to read. Read more...iWorksheets :3Study Guides :1
Area of Coordinate PolygonsCalculate the area of basic polygons drawn on a coordinate plane. Coordinate plane is a grid on which points can be plotted. The horizontal axis is labeled with positive numbers to the right of the vertical axis and negative numbers to the left of the vertical axis. Read more...iWorksheets :3Study Guides :1
##### 6.NS.7. Understand ordering and absolute value of rational numbers.
###### 6.NS.7b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write −3∘ݐ > ֢Ȓ7∘ݐ to express the fact that ֢Ȓ3∘ݐ is warmer than ֢Ȓ7∘ݐ.
Fractions/DecimalsAny fraction can be changed into a decimal and any decimal can be changed into a fraction. Read more...iWorksheets :3Study Guides :1
Ordering DecimalsWhen putting decimals in order from least to greatest, we must look at the highest place value first. Read more...iWorksheets :7Study Guides :1Vocabulary :1
Compare and Order FractionsWhen comparing two fractions that have a common denominator, you can looks at the numerators to decide which fraction is greater Read more...iWorksheets :4Study Guides :1Vocabulary :1
Ordering FractionsThe order of rational numbers depends on their relationship to each other and to zero. Rational numbers can be dispersed along a number line in both directions from zero. Read more...iWorksheets :6Study Guides :1
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
Ordering FractionsA fraction consists of two numbers separated by a line - numerator and denominator. To order fractions with like numerators, look at the denominators and compare them two at a time. The fraction with the smaller denominator is the larger fraction. Read more...iWorksheets :3Study Guides :1
Fractions/DecimalsHow to convert fractions to decimals: Divide the denominator (the bottom part) into the numerator (the top part). Read more...iWorksheets :3Study Guides :1
Less Than, Greater ThanCompare fractions and decimals using <, >, or =. Read more...iWorksheets :3Study Guides :1
Rational and Irrational NumbersA rational number is a number that can be made into a fraction. Decimals that repeat or terminate are rational because they can be changed into fractions. An irrational number is a number that cannot be made into a fraction. Decimals that do not repeat or end are irrational numbers. Pi is an irrational number. Read more...iWorksheets :3Study Guides :1
Exponents, Factors and FractionsFreeIn a mathematical expression where the same number is multiplied many times, it is often useful to write the number as a base with an exponent. Exponents are also used to evaluate numbers. Any number to a zero exponent is 1 and any number to a negative exponent is a number less than 1. Exponents are used in scientific notation to make very large or very small numbers easier to write. Read more...iWorksheets :8Study Guides :1
###### 6.NS.7d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
Positive & Negative IntegersPositive integers are all the whole numbers greater than zero. Negative integers are all the opposites of these whole numbers, numbers that are less than zero. Zero is considered neither positive nor negative Read more...iWorksheets :4Study Guides :1
##### 6.NS.8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Plot PointsYou use plot points to place a point on a coordinate plane by using X and Y coordinates to draw on a coordinate grid. Read more...iWorksheets :5Study Guides :1Vocabulary :1
CoordinatesFreeThe use of coordinates pertains to graphing and the quadrants that are formed by the x and y-axis. Read more...iWorksheets :14Study Guides :1
Plotting PointsIn a coordinate pair, the first number indicates the position of the point along the horizontal axis of the grid. The second number indicates the position of the point along the vertical axis. Read more...iWorksheets :4Study Guides :1Vocabulary :1
Area of Coordinate PolygonsCalculate the area of basic polygons drawn on a coordinate plane. Coordinate plane is a grid on which points can be plotted. The horizontal axis is labeled with positive numbers to the right of the vertical axis and negative numbers to the left of the vertical axis. Read more...iWorksheets :3Study Guides :1
### KS.6.EE. Expressions and Equations
#### Apply and extend previous understandings of arithmetic to algebraic expressions.
##### 6.EE.1. Write and evaluate numerical expressions involving whole-number exponents.
Exponents to Repeated MultiplicationAn exponent is a smaller-sized number which appears to the right and slightly above a number. Read more...iWorksheets :4Study Guides :1
Exponential & Scientific NotationExponential notation is shorten way of expressing a large number using exponents. Read more...iWorksheets :6Study Guides :1Vocabulary :1
Order of OperationsA numerical expression is a phrase which represents a number. Read more...iWorksheets :8Study Guides :1
Evaluate ExponentsEvaluating an expression containing a number with an exponent means to write the repeated multiplication form and perform the operation Read more...iWorksheets :3Study Guides :1
Repeated Multiplication to ExponentsThe result of raising a number to a power is the same number that would be obtained by multiplying the base number together the number of times that is equal to the exponent. Read more...iWorksheets :3Study Guides :1
ExponentsThe exponent represents the number of times to multiply the number, or base. When a number is represented in this way it is called a power. Read more...iWorksheets :4Study Guides :1
Exponents, Factors and FractionsFreeIn a mathematical expression where the same number is multiplied many times, it is often useful to write the number as a base with an exponent. Exponents are also used to evaluate numbers. Any number to a zero exponent is 1 and any number to a negative exponent is a number less than 1. Exponents are used in scientific notation to make very large or very small numbers easier to write. Read more...iWorksheets :8Study Guides :1
##### 6.EE.2. Write, read, and evaluate expressions in which letters stand for numbers.
###### 6.EE.2a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5−ݑ.
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
###### 6.EE.2b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8+7) as a product of two factors; view (8+7) as both a single entity and a sum of two terms.
Order of OperationsA numerical expression is a phrase which represents a number. Read more...iWorksheets :8Study Guides :1
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
###### 6.EE.2c. 6.EE.2c. Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas ݑ = s^3 and ɰݐ = 6s^2 to find the volume and surface area of a cube with sides of length s = 1/2.
FormulasThe formulas contain places for inputting numbers. Evaluating a formula requires inputting the correct data and performing the operations. Read more...iWorksheets :3Study Guides :1
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Algebraic EquationsFreeWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. Read more...iWorksheets :6Study Guides :1
##### 6.EE.3. Apply the properties of operations and combine like terms, with the conventions of algebraic notation, to identify and generate equivalent expressions. For example, apply the distributive property to the expression 3(2+ݑ) to produce the equivalent expression 6 + 3Űݑ; apply properties of operations to Űݑ + ưݑ + ưݑ to produce the equivalent expression 3y.
AlgebraAlgebra is the study of mathematical symbols and the rules for manipulating these symbols Read more...iWorksheets :7Study Guides :1Vocabulary :1
Distributive PropertyThe distributive property offers a choice in multiplication of two ways to treat the addends in the equation. We are multiplying a sum by a factor which results in the same product as multiplying each addend by the factor and then adding the products. Read more...iWorksheets :3Study Guides :1
Order of OperationsA numerical expression is a phrase which represents a number. Read more...iWorksheets :8Study Guides :1
FormulasThe formulas contain places for inputting numbers. Evaluating a formula requires inputting the correct data and performing the operations. Read more...iWorksheets :3Study Guides :1
One & Two Step EquationsAn algebraic equation is an expression in which a letter represents an unknown number Read more...iWorksheets :5Study Guides :1
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Algebraic EquationsFreeWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. Read more...iWorksheets :6Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to. The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
Introduction to FunctionsA function is a rule that is performed on a number, called an input, to produce a result called an output. The rule consists of one or more mathematical operations that are performed on the input. An example of a function is y = 2x + 3, where x is the input and y is the output. The operations of multiplication and addition are performed on the input, x, to produce the output, y. By substituting a number for x, an output can be determined. Read more...iWorksheets :7Study Guides :1
Nonlinear Functions and Set TheoryA function can be in the form of y = mx + b. This is an equation of a line, so it is said to be a linear function. Nonlinear functions are functions that are not straight lines. Some examples of nonlinear functions are exponential functions and parabolic functions. An exponential function, y = aˆx, is a curved line that gets closer to but does not touch the x-axis. A parabolic function, y = ax² + bx +c, is a U-shaped line that can either be facing up or facing down. Read more...iWorksheets :5Study Guides :1
#### Reason about and solve one-variable equations and inequalities.
##### 6.EE.4. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
AlgebraAlgebra is the study of mathematical symbols and the rules for manipulating these symbols Read more...iWorksheets :7Study Guides :1Vocabulary :1
One & Two Step EquationsAn algebraic equation is an expression in which a letter represents an unknown number Read more...iWorksheets :5Study Guides :1
Algebraic EquationsFreeWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. Read more...iWorksheets :6Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Using IntegersIntegers are negative numbers, zero and positive numbers. To compare integers, a number line can be used. On a number line, negative integers are on the left side of zero with the larger a negative number, the farther to the left it is. Positive integers are on the right side of zero on the number line. If a number is to the left of another number it is said to be less than that number. In the coordinate plane, the x-axis is a horizontal line with negative numbers, zero and positive numbers. Read more...iWorksheets :4Study Guides :1
Decimal OperationsDecimal operations refer to the mathematical operations that can be performed with decimals: addition, subtraction, multiplication and division. The process for adding, subtracting, multiplying and dividing decimals must be followed in order to achieve the correct answer. Read more...iWorksheets :3Study Guides :1
Fraction OperationsFraction operations are the processes of adding, subtracting, multiplying and dividing fractions and mixed numbers. A mixed number is a fraction with a whole number. Adding fractions is common in many everyday events, such as making a recipe and measuring wood. In order to add and subtract fractions, the fractions must have the same denominator. Read more...iWorksheets :3Study Guides :1
Introduction to PercentWhat Is Percent? A percent is a term that describes a decimal in terms of one hundred. Percent means per hundred. Percents, fractions and decimals all can equal each other, as in the case of 10%, 0.1 and 1/10. Percents can be greater than 100% or smaller than 1%. A markup from the cost of making an item to the actual sales price is usually greater than 100%. A salesperson's commission might be 1/2% depending on the item sold. Read more...iWorksheets :4Study Guides :1
Algebraic EquationsWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. When algebraic equations are written in words, the words must be changed into the appropriate numbers and variable in order to solve. Read more...iWorksheets :5Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to. The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
##### 6.EE.5. Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
##### 6.EE.6. Solve one-step equations involving non-negative rational numbers using addition, subtraction, multiplication and division.
AlgebraAlgebra is the study of mathematical symbols and the rules for manipulating these symbols Read more...iWorksheets :7Study Guides :1Vocabulary :1
Algebraic EquationsFreeWhat are algebraic equations? Algebraic equations are mathematical quations that contain a letter or variable, which represents a number. Read more...iWorksheets :6Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
##### 6.EE.7. Write an inequality of the form ݑ >Űݑ or аݑ <Űݑ to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form аݑ> or Űݑ<Űݑ have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Equations and InequalitiesAlgebraic equations are mathematical equations that contain a letter or variable, which represents a number. To solve an algebraic equation, inverse operations are used. The inverse operation of addition is subtraction and the inverse operation of subtraction is addition. Inequalities are mathematical equations that compare two quantities using greater than, >; greater than or equal to ≥; less than, <; and less than or equal to, ≤. Read more...iWorksheets :6Study Guides :1
Algebraic InequalitiesFreeAlgebraic inequalities are mathematical equations that compare two quantities using these criteria: greater than, less than, less than or equal to, greater than or equal to. The only rule of inequalities that must be remembered is that when a variable is multiplied or divided by a negative number the sign is reversed. Read more...iWorksheets :3Study Guides :1
#### Represent and analyze quantitative relationships between dependent and independent variables.
##### 6.EE.8. Use variables to represent two quantities in a real-world problem that change in relationship to one another.
###### 6.EE.8b. Write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Simple AlgebraSimple algebra is the term used when using expressions with letters or variables that represent numbers. Read more...iWorksheets :3Study Guides :1
Introduction to AlgebraAlgebra is the practice of using expressions with letters or variables that represent numbers. Words can be changed into a mathematical expression by using the words, plus, exceeds, diminished, less, times, the product, divided, the quotient and many more. Algebra uses variables to represent a value that is not yet known. Read more...iWorksheets :4Study Guides :1
### KS.6.G. Geometry
#### Solve real-world and mathematical problems involving area, surface area, and volume.
##### 6.G.1. Find the area of all triangles, special quadrilaterals (including parallelograms, kites and trapezoids), and polygons whose edges meet at right angles (rectilinear figure polygons) by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
AreaAn area is the amount of surface a shape covers. <br>An area is measured in inches, feet, meters or centimeters. Read more...iWorksheets :3Study Guides :1
Area of Triangles and QuadrilateralsThe area is the surface or space within an enclosed region. Area is expressed in square units. Read more...iWorksheets :7Study Guides :1Vocabulary :2
AreaArea is the number of square units needed to cover a flat surface. Read more...iWorksheets :3Study Guides :1
Exploring Area and Surface AreaArea is the amount of surface a shape covers. Area is measured in square units, whether the units are inches, feet, meters or centimeters. The area formula for a triangle is: A = 1/2 · b · h, where b is the base and h is the height. The area formula for a circle is: A = π · r², where π is usually 3.14 and r is the radius of the circle. The area formula for a parallelogram is: A = b · h, where b is the base and h is the height. Read more...iWorksheets :4Study Guides :1
##### 6.G.2. Find the volume of a right rectangular prism with fractional edge lengths by applying the formulas ݑ = ɰݑٰݑĢĎ and ݑ = ɰݐբĎ (B is the area of the base and h is the height) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (Builds on the 5th grade concept of packing unit cubes to find the volume of a rectangular prism with whole number edge lengths.)
VolumeVolume measures the amount a solid figure can hold. Read more...iWorksheets :3Study Guides :1
Finding VolumeVolume measures the amount a solid figure can hold. Volume is measured in terms of cubed units and can be measured in inches, feet, meters, centimeters, and millimeters. The formula for the volume of a rectangular prism is V = l · w · h, where l is the length, w is the width, and h is the height. Read more...iWorksheets :4Study Guides :1
##### 6.G.3. Draw polygons whose edges meet at right angles (rectilinear figure polygons) in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.
Plot PointsYou use plot points to place a point on a coordinate plane by using X and Y coordinates to draw on a coordinate grid. Read more...iWorksheets :5Study Guides :1Vocabulary :1
Area of Coordinate PolygonsCalculate the area of basic polygons drawn on a coordinate plane. Coordinate plane is a grid on which points can be plotted. The horizontal axis is labeled with positive numbers to the right of the vertical axis and negative numbers to the left of the vertical axis. Read more...iWorksheets :3Study Guides :1
##### 6.G.4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Exploring Area and Surface AreaArea is the amount of surface a shape covers. Area is measured in square units, whether the units are inches, feet, meters or centimeters. The area formula for a triangle is: A = 1/2 · b · h, where b is the base and h is the height. The area formula for a circle is: A = π · r², where π is usually 3.14 and r is the radius of the circle. The area formula for a parallelogram is: A = b · h, where b is the base and h is the height. Read more...iWorksheets :4Study Guides :1
### KS.6.SP. Statistics and Probability
#### Develop concepts of statistical measures of center and variability and an informal understanding of outlier.
##### 6.SP.3. Recognize that a measure of center (mean, median and/or mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation (range and/or interquartile range) describes how its values vary with a single number.
StatisticsA statistic is a collection of numbers related to a specific topic. Read more...iWorksheets :6Study Guides :1
StatisticsThe statistical mode is the number that occurs most frequently in a set of numbers. Read more...iWorksheets :3Study Guides :1
Data AnalysisCollecting Data. Data = information. You can collect data from other people using polls and surveys. Recording Data. You can record the numerical data you collected on a chart or graph: bar graphs, pictographs, line graphs, pie charts, column charts. Read more...iWorksheets :6Study Guides :1
Organizing DataThe data can be organized into groups, and evaluated. Mean, mode, median and range are different ways to evaluate data. The mean is the average of the data. The mode refers to the number that occurs the most often in the data. The median is the middle number when the data is arranged in order from lowest to highest. The range is the difference in numbers when the lowest number is subtracted from the highest number. Data can be organized into a table, such as a frequency table. Read more...iWorksheets :3Study Guides :1
#### Summarize and describe distributions.
##### 6.SP.5. Summarize numerical data sets in relation to their context, such as by:
###### 6.SP.5c. Giving quantitative measures of center (mean, median and/or mode) and variability (range and/or interquartile range), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
StatisticsA statistic is a collection of numbers related to a specific topic. Read more...iWorksheets :6Study Guides :1
StatisticsThe statistical mode is the number that occurs most frequently in a set of numbers. Read more...iWorksheets :3Study Guides :1
Data AnalysisCollecting Data. Data = information. You can collect data from other people using polls and surveys. Recording Data. You can record the numerical data you collected on a chart or graph: bar graphs, pictographs, line graphs, pie charts, column charts. Read more...iWorksheets :6Study Guides :1
Organizing DataThe data can be organized into groups, and evaluated. Mean, mode, median and range are different ways to evaluate data. The mean is the average of the data. The mode refers to the number that occurs the most often in the data. The median is the middle number when the data is arranged in order from lowest to highest. The range is the difference in numbers when the lowest number is subtracted from the highest number. Data can be organized into a table, such as a frequency table. Read more...iWorksheets :3Study Guides :1 | 10,838 | 49,148 | {"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-2024-26 | latest | en | 0.899565 |
http://mrnussbaum.com/wizard// | 1,472,162,266,000,000,000 | text/html | crawl-data/CC-MAIN-2016-36/segments/1471982294158.7/warc/CC-MAIN-20160823195814-00007-ip-10-153-172-175.ec2.internal.warc.gz | 167,574,042 | 11,887 | Parents and Teachers: MrNussbaum XTEND is now only \$12.00 per year (\$1.00 per month) for access to ALL PROGRAMS for the 2016-2017 school year if you purchase over the summer (until September 7). Click HERE to learn more. Also, check out my JOBS for Teachers and Parents section. More jobs will be posted soon.
# Garage Sale Wizard – Let’s Make a Deal!
### Watch a How to Play this Game VIDEO
#### Activities for This Game | Worksheets | Age Levels | Lesson
You play the role of a cranky old man trying to make a few bucks by selling his stuff at a garage sale. For each item, there will be three people vying for its purchase. Each person will offer a different amount of money. Count up the coins and click on the person who offers the best deal (the most money). Your skills will be put to the test and you will earn ten points if you choose the best deal one point if you choose the second best deal, and you will lose ten points for choosing the worst deal. Have fun and sell lots of junk!
Do you find this game useful? If so, please link to it from your website or blog, or, like or share on Facebook:
### Picture-based Math
* Working with All Coins Garage Sale Wizard
* Equivalent Sums of Coins
* Word Problems Using Money
### Printable, Fun Worksheets for This Game
* Garage Sale Wizard Assessment
* Coin Search
### Age Appropriateness
You play the role of a cranky old man trying to make a few bucks by selling his stuff at a garage sale. For each item, there will be three people vying for its purchase. Each person will offer a different amount of money. Count up the coins and click on the person who offers the best deal (the most money). Your skills will be put to the test and you will earn ten points if you choose the best deal one point if you choose the second best deal, and you will lose ten points for choosing the worst deal. Have fun and sell lots of junk!
Garage Sale Wizard is perfect for kids ages 7 – 10, Grade Levels: 2.9 – 4.1 (apx.)
### Scoring Breakdown and Grade Level Equivalency
Score Grade Level Equivalency 21 – 30 points 2.9 31 – 40 points 3.2 41 – 50 points 3.5 51 – 60 3.8 61 + 4.1 (mastery)
### Skills Reinforced:
Counting change Counting change over \$1.00 Comparing different sums of change | 555 | 2,252 | {"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} | 2.84375 | 3 | CC-MAIN-2016-36 | longest | en | 0.880386 |
https://oeis.org/A349877 | 1,702,107,149,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100873.6/warc/CC-MAIN-20231209071722-20231209101722-00380.warc.gz | 463,648,861 | 4,873 | The OEIS is supported by the many generous donors to the OEIS Foundation.
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A349877 a(n) is the number of times the map x -> A353314(x) needs to be applied to n to reach a multiple of 3, or -1 if the trajectory never reaches a multiple of 3. 5
0, 2, 14, 0, 1, 13, 0, 4, 1, 0, 12, 3, 0, 1, 3, 0, 4, 1, 0, 11, 2, 0, 1, 2, 0, 2, 1, 0, 2, 3, 0, 1, 3, 0, 10, 1, 0, 4, 5, 0, 1, 7, 0, 3, 1, 0, 3, 2, 0, 1, 2, 0, 2, 1, 0, 2, 4, 0, 1, 9, 0, 3, 1, 0, 3, 4, 0, 1, 5, 0, 6, 1, 0, 4, 2, 0, 1, 2, 0, 2, 1, 0, 2, 7, 0, 1, 4, 0, 6, 1, 0, 6, 3, 0, 1, 3, 0, 5, 1, 0, 8, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET 0,2 COMMENTS Equally, number of iterations of A353313 needed to reach a multiple of 3, or -1 if no multiple of 3 is ever reached. - Antti Karttunen, Apr 14 2022 LINKS Antti Karttunen, Table of n, a(n) for n = 0..19683 FORMULA From Antti Karttunen, Apr 14 2022: (Start) If A010872(n) = 0 then a(n) = 0, otherwise a(n) = 1 + a(A353314(n)). a(n) < A353311(n) for all n. (End) EXAMPLE a(1) = 2 : 1 -> 4 -> 9 (as it takes two applications of A353314 to reach a multiple of three), a(2) = 14 : 2 -> 5 -> 10 -> 19 -> 34 -> 59 -> 100 -> 169 -> 284 -> 475 -> 794 -> 1325 -> 2210 -> 3685 -> 6144 a(3) = 0 : 3 (as the starting point 3 is already a multiple of 3). a(4) = 1 : 4 -> 9 a(7) = 4 : 7 -> 14 -> 25 -> 44 -> 75. PROG (Python) import itertools def f(n): for i in itertools.count(): quot, rem = divmod(n, 3) if rem == 0: return i n = (5 * quot) + rem + 3 (PARI) A353314(n) = { my(r=(n%3)); if(!r, n, ((5*((n-r)/3)) + r + 3)); }; A349877(n) = { my(k=0); while(n%3, k++; n = A353314(n)); (k); }; \\ Antti Karttunen, Apr 14 2022 CROSSREFS Cf. A010872, A349876, A353311, A353313. Sequence in context: A219221 A249510 A324219 * A196815 A260120 A221234 Adjacent sequences: A349874 A349875 A349876 * A349878 A349879 A349880 KEYWORD nonn,easy AUTHOR Nicholas Drozd, Dec 03 2021 EXTENSIONS Definition corrected and more terms from Antti Karttunen, Apr 14 2022 STATUS approved
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Last modified December 9 02:18 EST 2023. Contains 367681 sequences. (Running on oeis4.) | 1,099 | 2,682 | {"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} | 3.359375 | 3 | CC-MAIN-2023-50 | latest | en | 0.694214 |
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2010-08-15, 14:17 #34
R.D. Silverman
"Bob Silverman"
Nov 2003
North of Boston
22·5·373 Posts
Quote:
Originally Posted by xilman AFAIK, the general consensus is that it unlikely that P and NP are not equal. However, the general consensus has been wrong on numerous occasions in the past. FWIW, my guess is that P !=NP but that there is a fair chance that integer factorization is in P. Paul
Huh? Read what you wrote: "unlikely that P and NP are not equal"
I think you need to delete the 'not'..........
2010-08-15, 14:40 #35
CRGreathouse
Aug 2006
3×1,993 Posts
Quote:
Originally Posted by xilman FWIW, my guess is that P !=NP but that there is a fair chance that integer factorization is in P.
Interesting... Peter Shor said the same thing recently:
Quote:
[Q: Is integer factorization outside of P?] I have to say that this isn't clear either way.
Is this the general consensus?
2010-08-15, 15:19 #36
R.D. Silverman
"Bob Silverman"
Nov 2003
North of Boston
22·5·373 Posts
Quote:
Originally Posted by CRGreathouse Interesting... Peter Shor said the same thing recently: Is this the general consensus?
A minor nitpick.
The definition/wording in Wikipaedia is a bit vague:
"given an integer N and an integer M with 1 โค M โค N, does N have a factor d with 1 < d < M?"
If I give you M = [sqrt(N)] + 1, the answer is trivially yes. (or N is prime;
which is in P)
If M ~ (log(N))^k for any k, then the question can be answered
in polynomial time.
The meaning of "an integer M", is unclear, because it is clear that
for some M depending on N the question is in P.
2010-08-15, 16:19 #37
CRGreathouse
Aug 2006
3·1,993 Posts
Quote:
Originally Posted by R.D. Silverman The meaning of "an integer M", is unclear, because it is clear that for some M depending on N the question is in P.
I don't really think so. The question is, in terms of N only, what is the worst-case complexity of determining whether there is a factor between 1 and M?
Sure, there are easy cases, but there are easy cases of SAT, too.
So what's your view? How likely is it that FACTORIZATION is in P? (xilman, you're welcome to answer as well!) And if you're willing to go out on a limb, how likely to you think it is that:
• integer factorization can be performed in time $N^\varepsilon$ for any $\varepsilon>0$?
• integer factorization can be performed in half-exponential time, $f(f(x))=\exp(x)$?
• the appropriate decision version of integer factorization* is in BPP (or even RP)?
I would have called all of these extremely unlikely, but now that I'm seeing people suggest that FACTORIZATION might be in P it seems like these should be reconsidered (since they are 'at least as likely').
* Either the one you gave above, or, say, "What is the m-th bit of the largest prime factor of N?" for 1 < m < lg N.
2010-08-15, 17:23 #38
xilman
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2C7A16 Posts
Quote:
Originally Posted by R.D. Silverman Huh? Read what you wrote: "unlikely that P and NP are not equal" I think you need to delete the 'not'..........
Thanks for picking up this error. You are quite correct and I somehow managed to get lost among the repeated negatives.
To be clear, I believe that P != NP.
Paul
2010-08-15, 20:38 #39
R.D. Silverman
"Bob Silverman"
Nov 2003
North of Boston
22·5·373 Posts
Quote:
Originally Posted by CRGreathouse I don't really think so. The question is, in terms of N only, what is the worst-case complexity of determining whether there is a factor between 1 and M? Sure, there are easy cases, but there are easy cases of SAT, too. So what's your view? How likely is it that FACTORIZATION is in P? (xilman, you're welcome to answer as well!) And if you're willing to go out on a limb, how likely to you think it is that: integer factorization can be performed in time $N^\varepsilon$ for any $\varepsilon>0$? integer factorization can be performed in half-exponential time, $f(f(x))=\exp(x)$? the appropriate decision version of integer factorization* is in BPP (or even RP)? I would have called all of these extremely unlikely, but now that I'm seeing people suggest that FACTORIZATION might be in P it seems like these should be reconsidered (since they are 'at least as likely'). * Either the one you gave above, or, say, "What is the m-th bit of the largest prime factor of N?" for 1 < m < lg N.
I have no idea if factors is in P. Insufficient information.
2010-08-16, 07:12 #40
xilman
Bamboozled!
"๐บ๐๐ท๐ท๐ญ"
May 2003
Down not across
2×5,693 Posts
Quote:
Originally Posted by CRGreathouse So what's your view? How likely is it that FACTORIZATION is in P? (xilman, you're welcome to answer as well!)
First off, let me say that assigning a probability in the mathematical sense is not well defined. The proposition is either true or false.
However, we could define it in another way, that used by gamblers who are not mathematicians. The probability is then the answer to the question: if you wished to place a bet, what odds would you negotiate before committing your money? My gut reaction, and it is only that, is that I'd settle for odds somewhere between 100:1 and 1000:1. That is a "fair chance" in my estimation.
The reasoning behind my estimation that FACTORIZATION has a fair chance of being in P is entirely the result of analogy and wishful thinking. I don't know the answer --- which is pretty clear because you would have heard about it if I did. Anyway, history shows us that until recently factorization was as hard as primality proving and only exponential time algorithms were known for each. About forty years ago, the two problems were still of equal complexity but a subexponential algorithm was found for factoring. Dixon's algorithm, for instance, takes time which is in a loose sense half-way between exponential and polynomial. The looseness can be removed but I won't bother in this post. A decade or so later, GNFS factors in (heuristic) 1/3 exponential -- 2/3 polynomial time, again in the same loose sense. By that measure, we're already 2/3 of the way to finding a P-time algorithm. I emphasise that this is not at all rigorous and is almost entirely vigorous hand-waving.
Turning to PRIMES, the factoring algorithms can still be used but asymptotically better ones have been discovered. ECPP and APRT-CL both run in (heuristically for ECPP) very nearly polynomial time. A few years ago, the polynomial time AKS algorithm was discovered.
Paul
Last fiddled with by xilman on 2010-08-16 at 07:57 Reason: Fix a couple of small bugs
2010-08-16, 07:32 #41 CRGreathouse Aug 2006 10111010110112 Posts Paul, thank you for your carefully-considered answer. (And yes, you have understood me correctly in terms of 'probability' -- think of it in terms of Bayesian knowledge sets if you must. )
2010-08-17, 06:31 #42
davieddy
"Lucan"
Dec 2006
England
2·3·13·83 Posts
Quote:
Originally Posted by Batalov ...well at least now his life is still pointful.
But nobody (except Xilman miscounting negatives) was trying
to suggest P = NP.
I quoted Paul Seymour because I thought others here might
find his criteria for plausibility (long and unintelligible) hilarious,
especially coming from the chief editor of a major mathematical journal.
(More or less the same as Bob's initial reaction).
OTOH, the statement that P=NP would make much of his
mathematical life pointless conveys the importance of the
problem. Can someone explain simply why he said that?
David
2010-08-17, 12:54 #43
CRGreathouse
Aug 2006
3·1,993 Posts
Quote:
Originally Posted by davieddy OTOH, the statement that P=NP would make much of his mathematical life pointless conveys the importance of the problem. Can someone explain simply why he said that?
Not that I agree with the sentiment, but I think the theory is that if P = NP, proving theorem is easy, so who needs a mathematician? Just hook up the polynomial-time SAT solver computer.
2010-08-17, 21:37 #44
davieddy
"Lucan"
Dec 2006
England
2·3·13·83 Posts
Quote:
Originally Posted by CRGreathouse Not that I agree with the sentiment, but I think the theory is that if P = NP, proving theorem is easy, so who needs a mathematician? Just hook up the polynomial-time SAT solver computer.
Hmm. Sometimes my eyes just glaze over.
David
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247
15:3*1=5sm width (15+5) *2=40sm perimeter 15*5=75sm2 ploshchadshirina = 15:3=5 cm Perimeter = (15+5) *2=40 cm Area = 15*5 = 75 cm of quarter.
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So, here is my solution to the problem of the last post. Let us call f(t) the expected win at the optimal strategy, if we get the value t. Then this function must have the following property.
$$f(t) = \max \{ t, \int_t^1 f(s) \, ds \}$$
The reason for this is that we can either keep our value t or continue. If we continue, we get the average win value of all larger numbers.
It is possible to prove that
$$f(t)=t$$
for all
$$t \ge a$$,
and for $$t < a$$
$$f(t) = \int_t^1 f(s) \, ds$$
The point a must be chosen so that
$$a = \int_a^1 s \,ds$$
One gets
$$a=\sqrt{2}-1$$
Differentiating the above equation for t < a, we see that
$$f(t) = \lambda e^{-t} , \qquad t<a.$$
with
$$\lambda = a / e^{-a}$$.
So the optimal strategy is to keep anything larger than a, and the expected win is lambda.
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# Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University
## Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University
Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University:- we will provide complete details of Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University.
### Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University
Explain the difference between variable and absorption costing. How unit product cost is computed under two methods?
Variable and absorption are two different costing methods. Almost all successful companies in the world use both the methods. Variable costing and absorption costing cannot be substituted for one another because both the systems have their own benefits and limitations.
These costing approaches are known by various names. For example, variable costing is also known as direct costing or marginal costing and absorption costing is also known as full costing or traditional costing.
The information provided by variable costing method is mostly used by internal management for decision making purposes. Absorption costing provides information that is used by internal management as well as by external parties like creditors, government agencies and auditors etc.
## Computation of unit product cost under two methods:
Under absorption costing system, the product cost consists of all variable as well as all fixed manufacturing costs i.e., direct materials, direct labor and factory overhead (FOH). But when variable costing system is used, the fixed cost (both manufacturing and non-manufacturing) is treated as a period or capacity cost and is, therefore, not included in the product cost.
Following exhibition summarizes the difference between variable costing and absorption costing:
Variable versus absorption costing
For further clarification of the concept, consider the following examples:
Example 1
A company manufactures and sells 5000 units of product X per year . Suppose one unit of product X requires the following costs:
Direct materials: \$5 per unit
Direct labor: \$4 per unit
Variable manufacturing overhead: \$1 per unit
Fixed manufacturing overhead: \$20,000 per year
The unit product cost of the company is computed as follows:
Absorption Costing: \$5 + \$4 + \$1 + \$4* = \$14
Variable Costing: \$5 + \$4 + \$1 = \$10
* \$20,000 / 5,000
Notice that the fixed manufacturing overhead cost has not been included in the unit cost under variable costing system but it has been included in the unit cost under absorption costing system. This is the primary difference between variable and absorption costing.
Example 2
Sunshine company produces and sells only washing machines. The company uses variable costing for internal reporting and absorption costing for external reporting. The data for the year 2016 is given below:
Direct materials: \$150/unit
Direct labor: \$45/unit
Fixed manufacturing overhead: \$160,000 per year
Fixed marketing and administrative expenses: \$110,000 per year
Variable marketing and administrative expenses: \$15/unit sold
Company produced and sold 8,000 machines during the year 2016.
Required: Compute the unite product cost under variable costing and absorption costing.
### Solution:
*\$160,000 / 8,000 Units = \$20
Note: Marketing and administrative expenses are period costs and are not relevant in the computation of unit product cost.
### Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University
Variable and absorption costing generate different levels of cost and net income in cost accounting, so it’s important to understand the differences so you can select a costing method to use internally for decision-making.
Say your business manufactures handsaws. Here is a summary of production, sales, and costs in Year 1.
Production (units) 3,000 Sales (units) 2,500 Sales (at \$25 per unit) \$62,500 Fixed manufacturing costs \$21,000 All other product costs \$33,000
Because you didn’t sell all of your production, you created ending inventory:
Ending inventory = units produced – units sold
Ending inventory = 3,000 – 2,500
Ending inventory = 500
Your fixed manufacturing costs are \$7 per unit produced (\$21,000 ÷ 3,000 units). Absorption costing requires you to assign \$3,500 of fixed manufacturing costs to ending inventory (\$7 x 500 units). The next table outlines the profit in Year 1, comparing variable and absorption costing.
Variable Costing Absorption Costing Sales (at \$25 per unit) \$62,500 \$62,500 Fixed manufacturing costs \$21,000 \$17,500 All other costs \$27,500 \$27,500 Total costs \$48,500 \$45,000 Profit \$14,000 \$17,500
Absorption costing deferred \$3,500 of fixed manufacturing costs. The fixed manufacturing costs are only \$17,500. You see that absorption costing has a \$3,500 higher profit (\$17,500 versus \$14,000).
In Year 2, assume that your sales and sales price are the same. You also sell all your production, plus the 500 units that were in ending inventory. Your sales (2,500 units) are 500 units more than your production (2,000 units). Because you produced less in Year 2, the all-other-cost number declines to \$22,500. Less production means less cost. Check out this next table.
Production (units) 2,000 Sales (units) 2,500 Sales (at \$25 per unit) \$62,500 Fixed manufacturing costs \$21,000 All other costs \$22,500
Variable and absorption costing are the same if you sell all of your production. You don’t produce any ending inventory, so you don’t defer any fixed manufacturing costs into inventory items. Here is the profit in Year 2.
Variable Costing Absorption Costing Sales (at \$25 per unit) \$62,500 \$62,500 Fixed manufacturing costs \$21,000 \$24,500 All other costs \$27,500 \$27,500 Total costs \$48,500 \$52,000 Profit \$14,000 \$10,500
Five hundred units from Year 1 ending inventory are sold in Year 2. In the third table, production of 2,000 is 500 units less than sales of 2,500. You had 500 units available for sale at the beginning of Year 2.
Fixed manufacturing costs for Year 2 are the same for both methods (\$21,000). However, absorption costing added the \$3,500 fixed manufacturing cost that was deferred in Year 1. The fixed manufacturing cost is \$24,500 (\$21,000 + \$3,500).
The variable costing profit in Year 2 is \$3,500 higher than the absorption costing profit (\$14,000 versus \$10,500). In Year 1, variable costing profit was \$3,500 lower than the absorption costing. When Year 1 ending inventory is sold in Year 2, absorption picks up the fixed manufacturing cost that was deferred.
Over two years, all the production is sold. The total profit over two years is the same for both costing methods.
You’re probably wondering about which method to use. Your profit eventually is the same under either method. In the long run, there is no advantage to using one method over another.
You should select a method and stick with it. By doing so, you’re applying the principle of consistency. For a financial statement reader to compare your results year by year, you need to use the same method. It’s the old idea of an apples-to-apples comparison.
### Unit III Variable and Absorption Costing for Managerial Accounting Mcom Delhi University
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This is a translation of the absolute value graph ULJKW XQLWVDQGGRZQ XQLWV 6LQFHWKHLQHTXDOLW\V\PEROLV WKHERXQGDU\OLQHLV solid. Find 25 photos of the 68 Boundary Line Rd home on Zillow. When using the slope-intercept form to graph linear inequalities, how do you know which side of the line to shade on? Medium’s site status, or find something interesting to read. CDN$19.99 CDN$ 19. ft. home is a 2 bed, 1.0 bath property. This home was built in 1900 and last sold on for. The Integral Calculator lets you calculate integrals and antiderivatives of functions online — for free! This is an awesome piece of real estate for your new home. One of the most common reasons for a boundary line agreement is when a neighbor has encroached on your property by … The equation can be written . We use these equations along with the boundary conditions and loads for our beams to derive closed-form solutions to the beam configurations shown on this page (simply supported and cantilver beams). Example. The 1,792 sq. Code to add this calci to your website . 62/87,21 The boundary of the graph is the graph of . For second order differential equations, which will be looking at pretty much exclusively here, any of the following can, and will, be used for boundary conditions. The International Date Line is located halfway around the world from the prime meridian (0° longitude) or about 180° east (or west) of Greenwich, London, UK, the reference point of time zones.It is also known as the line … All hosts on a subnetwork have the same network prefix, unlike the host identifier which is a unique local identification. visboundaries(BW) draws boundaries of regions in the binary image BW on the current axes.BW is a 2-D binary image where pixels that are logical true belong to the foreground region and pixels that are logical false constitute the background.visboundaries uses bwboundaries to find the boundary pixel locations in … 4. 4.5 out of 5 stars 735. Second calculator finds the line equation in parametric form, that is, . The boundary line is also listed as mapping coordinates on the property owner’s deed. Shade the appropriate area. Description. Calculators. Texas Instruments TI-30X IIS 2-Line Scientific Calculator, Black with Blue Accents. You can tell which region to shade by testing some points in the inequality. Tell us some details about your needs and get connected to pre-screened companies in your area. However, there are certain rules that come into play if a fence is built right at the boundary between two properties. the flat plate, the bed of a river, or the wall of a pipe, the fluid touching the surface is brought to rest by the shear stress to at the wall.The region in which flow adjusts from zero velocity at the wall to a maximum in the main stream of the flow is termed the boundary layer. It also outputs direction vector and displays line and direction vector on a graph. Now, we know that slope m = 5 and y-intercept b = -2. 68 Boundary Line Rd , Fort Fairfield, ME 04742 is a single-family home listed for-sale at $139,900. As the floor is usually the closest boundary to a speaker floor modes should dominate though. If the resulting sentence is true, then shade the region where that test point is located, indicating that all the points on that side of the boundary line will make the original sentence true. It is drawn as a dashed line if the points on the line do not satisfy the inequality, as in the cases of < and >. Select a point not on the boundary line and substitute its x and y values into the original inequality. In simpler speak, a linear inequality is just everything on ONE side of a line … Notional Boundary - there may be a risk of fire spread between buildings even when on land in the same occupation. Boundary line agreements are specifically used when there is a dispute over land and its use. 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http://math.stackexchange.com/questions/88505/infinitely-many-primes-of-the-form-6-cdot-k1-where-k-is-an-odd-number | 1,469,463,984,000,000,000 | text/html | crawl-data/CC-MAIN-2016-30/segments/1469257824319.59/warc/CC-MAIN-20160723071024-00280-ip-10-185-27-174.ec2.internal.warc.gz | 148,674,990 | 19,699 | # Infinitely many primes of the form $6\cdot k+1$ , where $k$ is an odd number?
How to prove that there are infinitely many primes of the form $6k+1$ , where $k$ is an odd number ?
Here is a proof that there are infinitely many primes of the form $6k+1$ :
We will assume that there are a finite number of primes of the form $6k+1$ .
Let $p_1 = 6k_1+1, p_2 = 6k_2+1$, ... Then,
$p_1p_2 = (6k_1+1)(6k_2+1) = 36k_1k_2 + 6k_1 + 6k_2 + 1$
$p_1p_2 = 6\cdot f(k_1,k_2) + 1$, where $f(k_1,k_2) = 6k_1k_2 + k_1 + k_2$
i.e. the product of two numbers of the form $6k+1$ is also a number of the form $6k+1$.
Hence $p_1p_2...p_n$ will be a number of the form $6k+1$
i.e. $p_1\cdot p_2...p_n = 6f(k_1,k-2,...,k-n) + 1$
$(2p_1p_2...p_n)^2 = 4(36f^2 +12f + 1) = 6g + 4$, where $g = 24f^2 + 8f$,
$(2p_1p_2...p_n)^2 + 3 = 6g + 4 + 3$
$(2p_1p_2...p_n)^2 + 3 = 6g + 6 + 1$
$(2p_1p_2...p_n)^2 + 3 = 6g' + 1$
i.e. $N = 6g' + 1$, a number of the form $6k+1$
Now, by construction, $N = (2p_1p_2...p_n)^2 + 3$, and is not divisible by any of the $p_i$.
Hence it is either prime itself, or divisible by another prime greater than $p_n$, contradicting the assumption.
I.e. There are an infinite number of primes of the form $6k+1$
Can I use the method of this proof to prove that there are infinitely many primes of the form $6k+1$ , where $k$ is an odd number or to choose some other proof strategy ?
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Are you looking for a proof, similar to Euclid's argument, which will work exclusively for this case? Else, in case you are not aware, en.wikipedia.org/wiki/… – user17762 Dec 5 '11 at 8:17
I don't think your proof works. If $N$ is not prime, how do you know that its prime divisors aren't all of the form $6k+5$? – TonyK Dec 5 '11 at 8:20
@SivaramAmbikasaran,I think that I cannot conclude that there are infinitely many primes of the form $6k+1$ , where $k$ is an odd number if I have proved that there are infinitely many primes of the form $6k+1$... – pedja Dec 5 '11 at 8:29
@TonyK There is a big step missing at the end of the proof. The number N constructed is a number of the form x^2+3y^2 with (x,y)=1, and all the prime divisors of any such number are of the form 6n+1. – Ted Dec 5 '11 at 8:34
@pedja: Proving infinitely many primes of the form $6k+1$ where $k$ is odd is equivalent to proving infinitely many primes of the form $6(2m+1)+1 = 12m+7$. – user17762 Dec 5 '11 at 8:39
edit, Monday, December 5: Pedja's argument can me made to work with the addition of a quadratic reciprocity step...
original: Pedja, you use Dirichlet on the arithmetic progression $12 n + 7,$ which is what you are describing by saying $6k+1$ with $k$ odd. Write $k = 2n+1,$ then $6k+1 = 6 (2n+1) + 1 = 12n + 6 + 1 = 12 n + 7.$
Meanwhile, if you are willing to accept the Cebotarev Density Theorem, asymptotically the two positive quadratic forms $x^2 + 12 y^2$ and $3 x^2 + 4 y^2$ represent the same proportion of primes. The first form represents primes $p \equiv 1 \pmod {12},$ while the second form represents 3 and the primes $q \equiv 7 \pmod {12}.$ I am quoting Theorem 9.12 on page 188 of Primes of the Form $x^2 + n y^2$ by David A. Cox.
EDIT: Actually, your attempt is almost correct, there is an elementary proof of this one. As you write above, take the collection of primes $q_j \equiv 7 \pmod{12}$ that is assumed to be complete, up to some largest we will call $q_r.$ Then, just as you wrote, define $$w = 3 + 4 \left( q_1 q_2 \ldots q_r \right)^2$$ We know that $w \equiv 7 \pmod {12}.$ We also know that $w$ is primitively represented by the quadratic form $3 x^2 + 4 y^2.$ That means that $\gcd(x,y)=1,$ where here $x=1,$ and we are writing $w = 3 x^2 + 4 y^2.$
Now, $w$ is not divisible by 2 or 3. The fact that $\gcd(x,y)=1$ means precisely that $w$ is not divisible by any prime $s \equiv 2 \pmod 3.$ This fact comes under the general heading of quadratic reciprocity. All prime factors of $w$ are, in fact , $1 \pmod 3.$ That is, all prime factors of $w$ are either $1 \pmod {12}$ or $7 \pmod {12}.$ If all prime factors of $w$ were $1 \pmod {12},$ then $w$ itself would also be $1 \pmod {12},$ which it is not.
So, in fact, either $w$ is prime, or $w$ has at least one prime factor, call it $q_{r+1},$ such that $q_{r+1} \equiv 7 \pmod {12}.$ By construction, for any $j \leq r,$ we have $\gcd(q_j,q_{r+1} ) = 1.$ That is, either $w$ itself or $q_{r+1}$ is not in the assumed finite list of primes $7 \pmod {12},$ contradicting the assumption that there is a finite list. $\bigcirc \bigcirc \bigcirc \bigcirc \bigcirc \bigcirc$
EDIT TOO: Maybe not everyone will know this, so I will give the short proof. Suppose we have some prime $s \equiv 2 \pmod 3,$ with $s \neq 2,$ that divides some $z = 3 u^2 + 4 v^2.$ Then suppose that $s | z,$ which is also written $z \equiv 0 \pmod s.$ I claim that, in fact, both $s | u$ and $s | v,$ with the result that $\gcd(u,v)$ is larger than one. PROOF: We have $3 u^2 + 4 v^2 \equiv 0 \pmod s.$ Assume that $v \neq 0 \pmod s.$ Then $v$ has a multiplicative inverse $\pmod s.$ So we get $$3 u^2 + 4 v^2 \equiv 0 \pmod s,$$ $$3 u^2 \equiv - 4 v^2 \pmod s,$$ $$\frac{3 u^2}{v^2} \equiv - 4 \pmod s,$$ $$\frac{ u^2}{v^2} \equiv \frac{- 4}{3} \pmod s,$$ $$\left(\frac{ u}{v}\right)^2 \equiv \frac{- 4}{3} \pmod s.$$ This says that the right hand side is a quadratic residue $\pmod s.$ It also means that $-12$ is a quadratic residue $\pmod s.$ This is false, though, as we have Legendre symbol $$( -12 | s) = ( -3 | s) = ( -1 | s ) \cdot ( 3 | s).$$ Now, if $s \equiv 1 \pmod 4,$ we can ignore the $-1$ and switch the other pair. If $s \equiv 3 \pmod 4,$ then $( -1 | s ) = -1$ and $(3 | s) = - (s | 3)$. In either case, we get $$(-3 | s) = (s | 3).$$ As $s \equiv 2 \pmod 3,$ we have $$( s | 3) = ( 2 | 3) = -1.$$ So, in the end we have $$( -12 | s) = -1,$$ and this contradicts the assumption that $v$ is nonzero $\pmod s.$ Thus, $v \equiv 0 \pmod s.$ From $3 u^2 + 4 v^2 \equiv 0 \pmod s,$ we then have $u \equiv 0 \pmod s.$ So $s | \gcd(u,v)$ and $\gcd(u,v) \neq 1.$
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thanks..it is so elementary proof... – pedja Dec 5 '11 at 8:46
@pedja: No, it relies on Dirichlet's theorem, which is far from elementary! – TonyK Dec 5 '11 at 8:55
@WillJagy,I think that this fact is necessary condition that there are infinitely many Mersenne primes . Am I correct ? – pedja Dec 5 '11 at 9:03
Pedja, I am not reading anything that makes any demands about $p$ when $2^p -1$ is also prime. You might look at the tables of the $p$ at primes.utm.edu/mersenne and check for any patterns, for example $\pmod{12}.$ – Will Jagy Dec 6 '11 at 3:12 | 2,409 | 6,577 | {"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} | 4.25 | 4 | CC-MAIN-2016-30 | latest | en | 0.856099 |
https://convertoctopus.com/3-8-grams-to-kilograms | 1,623,922,116,000,000,000 | text/html | crawl-data/CC-MAIN-2021-25/segments/1623487629632.54/warc/CC-MAIN-20210617072023-20210617102023-00265.warc.gz | 167,790,630 | 7,994 | Conversion formula
The conversion factor from grams to kilograms is 0.001, which means that 1 gram is equal to 0.001 kilograms:
1 g = 0.001 kg
To convert 3.8 grams into kilograms we have to multiply 3.8 by the conversion factor in order to get the mass amount from grams to kilograms. We can also form a simple proportion to calculate the result:
1 g → 0.001 kg
3.8 g → M(kg)
Solve the above proportion to obtain the mass M in kilograms:
M(kg) = 3.8 g × 0.001 kg
M(kg) = 0.0038 kg
The final result is:
3.8 g → 0.0038 kg
We conclude that 3.8 grams is equivalent to 0.0038 kilograms:
3.8 grams = 0.0038 kilograms
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 kilogram is equal to 263.15789473684 × 3.8 grams.
Another way is saying that 3.8 grams is equal to 1 ÷ 263.15789473684 kilograms.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that three point eight grams is approximately zero point zero zero four kilograms:
3.8 g ≅ 0.004 kg
An alternative is also that one kilogram is approximately two hundred sixty-three point one five eight times three point eight grams.
Conversion table
grams to kilograms chart
For quick reference purposes, below is the conversion table you can use to convert from grams to kilograms
grams (g) kilograms (kg)
4.8 grams 0.005 kilograms
5.8 grams 0.006 kilograms
6.8 grams 0.007 kilograms
7.8 grams 0.008 kilograms
8.8 grams 0.009 kilograms
9.8 grams 0.01 kilograms
10.8 grams 0.011 kilograms
11.8 grams 0.012 kilograms
12.8 grams 0.013 kilograms
13.8 grams 0.014 kilograms | 467 | 1,667 | {"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-2021-25 | latest | en | 0.78827 |
https://ncertmcq.com/ncert-solutions-for-class-6-science-chapter-6/ | 1,701,586,647,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100489.16/warc/CC-MAIN-20231203062445-20231203092445-00083.warc.gz | 480,834,456 | 10,618 | NCERT Solutions for Class 6 Science Chapter 6 Changes Around us are part of NCERT Solutions for Class 6 Science. Here we have given NCERT Solutions for Class 6 Science Chapter 6 Changes Around us.
Board CBSE Textbook NCERT Class Class 6 Subject Science Chapter Chapter 6 Chapter Name Changes Around us Number of Questions Solved 7 Category NCERT Solutions
## NCERT Solutions for Class 6 Science Chapter 6 Changes Around us
NCERT TEXTBOOK EXERCISES
(Page 51)
Question 1.
To walk through a waterlogged area, you usually shorten the length of your dress by folding it. Can this change be reversed?
Yes, this change can be reversed.
Question 2.
You accidentally dropped your favourite toy and broke it. This is a change you did not want. Can this change be reversed?
No. This change cannot be reversed. Breaking a toy is an irreversible change.
Question 3.
Some changes are listed in the following table. For each change, write in the blank column, whether the change can be reversed or not.
S.No. Change Can be reversed (Yes/No) 1. The sawing of a piece of wood 2. The melting of ice candy 3. Dissolving sugar in water 4. The cooking of food 5. The ripening of a mango 6. Souring of milk
S.No. Change Can be reversed (Yes/No) 1. The sawing of a piece of wood No 2. The melting of ice candy Yes 3. Dissolving sugar in water Yes 4. The cooking of food No 5. The ripening of a mango No 6. Souring of milk No
Question 4.
A drawing sheet changes when you draw a picture on it. Can you reverse this change?
No. We cannot reverse this change. Because we cannot get a fresh drawing sheet once a picture is drawn on it. But if the pencil is used to draw the picture, we can reverse the change.
Question 5.
Give examples to explain the difference between changes that can or cannot be reversed.
One way we can group changes is to see if they can be reversed or not.
1. Changes that can be reversed: In such changes, we can return back to the original position if the cause of change is withdrawn; for example,
• heating of milk,
• drying of clothes,
• knitting a sweater,
• stretching a rubber band,
• melting of ice, etc.
2. Changes that cannot be reversed: The changes in which we cannot go back to the original position, even if we withdraw the cause of change; for example,
• boiling of egg,
• making curd from milk,
• making flour from grain,
• the flowering of a bud,
• producing biogas from cow dung, etc.
Question 6.
A thick coating of a paste of Plaster of Paris (POP) is applied over the bandage on a fractured bone. It becomes hard on drying to keep the fractured bone immobilized. Can the change in POP be reversed? | 633 | 2,629 | {"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} | 3.234375 | 3 | CC-MAIN-2023-50 | longest | en | 0.877044 |
https://nobisgrouplabs.com/tasmania/algorithm-design-and-applications-ed-4-goodrick.php | 1,606,371,715,000,000,000 | text/html | crawl-data/CC-MAIN-2020-50/segments/1606141186761.30/warc/CC-MAIN-20201126055652-20201126085652-00393.warc.gz | 423,387,199 | 7,489 | # Algorithm design and applications ed 4 goodrick
## How to Explain Algorithms to Kids Tynker Blog
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### How to Explain Algorithms to Kids Tynker Blog
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All Categories Cities: Bungendore Darlington Holmes Goldsborough Lochaber Walls of Jerusalem Mount Dandenong Carine St Helens Forestburg Masset Grandview Drummond Port Saunders Fort Resolution Mulgrave Kimmirut Wayside Tyne Valley Barkmere Paynton Teslin River | 1,648 | 7,893 | {"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} | 2.53125 | 3 | CC-MAIN-2020-50 | latest | en | 0.775683 |
https://cs.stackexchange.com/questions/10129/how-does-deforestation-remove-trees-from-a-program | 1,718,550,785,000,000,000 | text/html | crawl-data/CC-MAIN-2024-26/segments/1718198861665.97/warc/CC-MAIN-20240616141113-20240616171113-00876.warc.gz | 162,828,254 | 43,354 | # How does 'deforestation' remove 'trees' from a program?
I think understand how deforestation consumes and produces a list at the same time (from a fold and an unfold function -- see this good answer on CodeReview here), but when I compared that with the wikipedia entry on the technique it talked about 'removing trees' from a program.
I understand how a program can be parsed into a syntactic parse tree (is that right?), but what is the meaning of this use of deforestation for some kind of simplification (is it?) of programs? And how would I do it to my code?
Yatima2975 seems to have covered your first two questions, I'll try to cover the third. To do this I'll treat an unrealistically simple case, but I'm sure you'll be able to imagine something more realistic.
Imagine you want to compute the depth of the full binary tree of order $n$. The type of (unlabeled) binary trees is (in Haskell syntax):
type Tree = Leaf | Node Tree Tree
Now the full tree of order $n$ is:
full : Int -> Tree
full n | n == 0 = Leaf
full n = Node (full (n-1)) (full (n-1))
And the depth of a tree is computed by
depth : Tree -> Int
depth Leaf = 0
depth (Node t1 t2) = 1 + max (depth t1) (depth t2)
Now you can see that any computation of $\mathrm{depth}\ (\mathrm{full}\ n)$ will first construct the full tree of order $n$ using $\mathrm{full}$ and then deconstruct that tree using $\mathrm{depth}$. Deforestation relies on the observation that such a pattern (construction followed by deconstruction) can often be short-circuited: we can replace any calls to $\mathrm{depth}\ (\mathrm{full}\ n)$ by a single call to $\mathrm{full\_depth}$:
full_depth : Int -> Int
full_depth n | n == 0 = 0
full_depth n = 1 + max (full_depth (n-1)) (full_depth (n-1))
This avoids memory allocation of the full tree, and the need to perform pattern matching, which greatly improves performance. In addition, if you add the optimization
max t t --> t
Then you have turned an exponential time procedure into a linear time one... It would be cool if there was an additional optimization that recognized that $\mathrm{full\_depth}$ was the identity on integers, but I am not sure that any such optimization is used in practice.
The only mainstream compiler that performs automatic deforestation is GHC, and if I recall correctly, this is performed only when composing built-in functions (for technical reasons).
• Awarded because I got more out this answer from the way it was formulated than from the other answers, even though they essentially cover the same territory. Commented Mar 2, 2013 at 12:44
First, lists are a kind of trees. If we represent a list as a linked list, it's just a tree whose each node has either 1 or 0 descendants.
Parse trees are just an utilization of trees as a data structure. Trees have many many applications in computer science, including sorting, implementing maps, associative arrays, etc.
In general, list, trees etc. are recursive data structures: Each node contains some information and another instance of the same data structure. Folding is an operation over all such structures that recursively transforms nodes to values "bottom up". Unfolding is the reverse process, it converts values to nodes "top down".
For a given data structure, we can mechanically construct their folding and unfolding functions.
As an example, let's take lists. (I'll use Haskell for the examples as it's typed and its syntax is very clean.) List is either an end or a value and a "tail".
data List a = Nil | Cons a (List a)
Now let's imagine we're folding a list. At each step, we have the current node to be folded and we have already folded its recursive sub-nodes. We can represent this state as
data ListF a r = NilF | ConsF a r
where r is the intermediate value constructed by folding the sublist. This allows us to express a folding function over lists:
foldList :: (ListF a r -> r) -> List a -> r
foldList f Nil = f NilF
foldList f (Cons x xs) = f (ConsF x (foldList f xs))
We convert List into ListF by recursively folding over its sublist and then use a function defined on ListF. If you think about it, this is just another representation of standard foldr:
foldr :: (a -> r -> r) -> r -> List a -> r
foldr f z = foldList g
where
g NilF = z
g (ConsF x r) = f x r
We can construct unfoldList in the same fashion:
unfoldList :: (r -> ListF a r) -> r -> List a
unfoldList f r = case f r of
NilF -> Nil
ConsF x r' -> Cons x (unfoldList f r')
Again, it's just another representation of unfoldr:
unfoldr :: (r -> Maybe (a, r)) -> r -> [a]
(Notice that Maybe (a, r) is isomorphic to ListF a r.)
And we can construct a deforestation function too:
deforest :: (ListF a r -> r) -> (s -> ListF a s) -> s -> r
deforest f u s = f (map (deforest f u) (u s))
where
map h NilF = NilF
map h (ConsF x r) = ConsF x (h r)
It simply leaves out the intermediate List and fuses the folding and unfolding functions together.
The same procedure can be applied to any recursive data structure. For example, a tree whose nodes can have 0, 1, 2 or descendants with values on 1- or 0-branching nodes:
data Tree a = Bin (Tree a) (Tree a) | Un a (Tree a) | Leaf a
data TreeF a r = BinF r r | UnF a r | LeafF a
treeFold :: (TreeF a r -> r) -> Tree a -> r
treeFold f (Leaf x) = f (LeafF x)
treeFold f (Un x r) = f (UnF x (treeFold f r))
treeFold f (Bin r1 r2) = f (BinF (treeFold f r1) (treeFold f r2))
treeUnfold :: (r -> TreeF a r) -> r -> Tree a
treeUnfold f r = case f r of
LeafF x -> Leaf x
UnF x r -> Un x (treeUnfold f r)
BinF r1 r2 -> Bin (treeUnfold f r1) (treeUnfold f r2)
Of course, we can create deforestTree just as mechanically as before.
(Usually, we'd express treeFold more conveniently as:
treeFold' :: (r -> r -> r) -> (a -> r -> r) -> (a -> r) -> Tree a -> r
)
I'll leave out the details, I hope the pattern is obvious. | 1,562 | 5,938 | {"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": 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} | 3.703125 | 4 | CC-MAIN-2024-26 | latest | en | 0.910691 |
http://stackoverflow.com/questions/9170678/understanding-and-visualizing-recursion?answertab=active | 1,427,816,842,000,000,000 | text/html | crawl-data/CC-MAIN-2015-14/segments/1427131300735.71/warc/CC-MAIN-20150323172140-00022-ip-10-168-14-71.ec2.internal.warc.gz | 239,049,408 | 17,434 | # Understanding and visualizing recursion
I referred to several questions here about recursion but I am not able to understand how recursion works for this particular problem: Recursive program to get all combination of characters in a string in Python:
``````st= []
def combi(prefix, s):
if len(s)==0: return
else:
st.append(prefix+s[0])
''' printing values so that I can see what happens at each stage '''
print "s[0]=",s[0]
print "s[1:]=",s[1:]
print "prefix=",prefix
print "prefix+s[0]=",prefix+s[0]
print "st=",st
combi(prefix+s[0],s[1:])
combi(prefix,s[1:])
return st
print combi("",'abc')
``````
I've made it print the values so that I can see what's happening. This is the output:
``````s[0]= a
s[1:]= bc
prefix=
prefix+s[0]= a
st= ['a']
s[0]= b
s[1:]= c
prefix= a
prefix+s[0]= ab
st= ['a', 'ab']
s[0]= c
s[1:]=
prefix= ab
prefix+s[0]= abc
st= ['a', 'ab', 'abc']
s[0]= c
s[1:]=
prefix= a ----> How did prefix become 'a' here. Shouldn't it be 'abc' ?
prefix+s[0]= ac
st= ['a', 'ab', 'abc', 'ac']
.........
.........
['a', 'ab', 'abc', 'ac', 'b', 'bc', 'c'] # final output
``````
Full output: http://pastebin.com/Lg3pLGtP
As I've shown in the output, how did prefix become 'ab'?
I tried to visualize the recursive calls for the combi(prefix+s[0],s[1:]). Am I understanding it right?
-
There are two recursive calls to `combi()` in the function. Thus the path of calls is not a single line, but rather a binary tree that forks. What you are seeing is the second half of the tree.
-
I thought the 2nd recursive call i.e `combi(prefix,s[1:])` would start off as `combi('','bc')` and go though the same process forming b,bc. Here at the last step s[0] is 'c' and when recursing out prefix+s[0] becomes ''+c = c if I understand it right? Btw, I've added a pastbin link of the complete output to the question. – Bharat Feb 7 '12 at 3:48
If you're familiar with depth-first search, it's how the tree Amber mentions is being traversed (or generated, depending on how you want to look at it). – ktodisco Feb 7 '12 at 3:56
@RBK: It's the call for `combi('a', 'c')` from `combi('a','bc')` that's creating the second `prefix='a'`. – Amber Feb 7 '12 at 3:59
@ktodisco , Yes I know Depth first traversal. I think the connection with recursion in general would be that DFS uses a stack.. I will try to visualize it on paper & see... I am still not able to understand how the backtracking works exactly. – Bharat Feb 7 '12 at 4:05
@Amber, `prefix+s[0]` is passed as the 1st argument to combi, doesn't prefix now become 'ab' after the call to `combi('a','bc')` ? How does it remain 'a' ? – Bharat Feb 7 '12 at 4:11
Theres a python module for that
Generated with:
``````from rcviz import callgraph, viz
st= []
@viz
def combi(prefix, s):
if len(s)==0:
return
else:
st.append(prefix+s[0])
combi.track(st = st) #track st in rcviz
combi(prefix+s[0],s[1:])
combi(prefix,s[1:])
return st
print combi("",'abc')
callgraph.render("combi.png")
``````
-
Thanks. Looks interesting. I'll try it out. – Bharat May 25 '14 at 4:08
This library is very useful. Thanks. Up voted. – Bharat May 27 '14 at 16:21
I drew the recursion tree. By Depth First Traversal, the final output is got at the last node. This visualization helps understand what's happening.
- | 1,021 | 3,256 | {"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} | 3.3125 | 3 | CC-MAIN-2015-14 | latest | en | 0.836309 |
https://www.learncram.com/dav-solutions/dav-class-6-maths-chapter-6-brain-teasers/ | 1,716,739,401,000,000,000 | text/html | crawl-data/CC-MAIN-2024-22/segments/1715971058956.26/warc/CC-MAIN-20240526135546-20240526165546-00141.warc.gz | 754,104,994 | 16,427 | # DAV Class 6 Maths Chapter 6 Brain Teasers Solutions
The DAV Class 6 Maths Solutions and DAV Class 6 Maths Chapter 6 Brain Teasers Solutions of Introduction to Algebra offer comprehensive answers to textbook questions.
## DAV Class 6 Maths Ch 6 Brain Teasers Solutions
Question 1A.
(а) Mrs. Shalini is p years of age now, 3 years ago her age in years, was-
(i) 3 – p
(ii) 3 + p
(iii) p – 3
(iv) 3 + p
Solution:
(iii) p – 3
Present age of Mrs Shalini is = p years
Her age 3 years ago was (p – 3) years.
Hence, (iii) is the correct option.
(b) If the perimeter of a square is q metres, then length of each side in metres, is-
(i) q + 4
(ii) q ÷ 4
(iii) q – 4
(iv) 4 ÷ q
Solution:
(ii) q ÷ 4
Perimeter = 4 × side q = 4 × side
∴ side = q ÷ 4 metre
Hence, (ii) is the correct option.
(c) Expression obtained when k is multiplied by 2 and then subtracted from 5, is-
(i) 2k – 5
(ii) 2k + 5
(iii) 5 – 2k
(iv) 5k – 2
Solution:
5 – 2k
(d) Number of notebooks bought for ₹ t at the rate of ₹ 40 per notebook, is-
(i) $$\frac{40}{t}$$
(ii) 40 t
(iii) $$\frac{t}{40}$$
(iv) t + 40
Solution:
(iii) $$\frac{t}{40}$$
(e) x = 6, y = 2, z = 3, what is the value of the expression $$\frac{x y-x z}{z^2}$$?
(i) $$\frac{-2}{3}$$
(ii) $$\frac{2}{3}$$
(iii) 3$$\frac{1}{3}$$
(iv) 5
Solution:
(i) $$\frac{-2}{3}$$
Put x = 6, y = 2, z = 3 in the expression $$\frac{x y-x z}{z^2}$$, we get
$$\frac{(6)(2)-(6)(3)}{(3)^2}$$
= $$\frac{12-18}{9}$$
= $$\frac{-6}{9}=-\frac{2}{3}$$
(a) Your Mom has a sum of ₹ x. She spent ₹ 3000 on grocery and ₹ 2000 on milk. Write the algebraic expression for the amount left with her.
Solution:
₹ (x – 3000 – 2000) = (x – 5000)
(b) The digit in the ones place of a 2-digit number is twice the digit at the tens place. Write the number.
Solution:
T O
x 2x
Number = 10 × x + 1 × 2x
= 10x + 2x
= 12x
(c) If a notebook costs ₹ p and a pencil costs ₹ s, then what will be the cost of two notebooks and three pencils?
Solution:
Cost of 1 notebook = ₹ p
Cost of 2 notebooks = ₹ 2p …………(1)
Cost of 1 pencil = ₹ s
Cost of 3 pencils = ₹ 3x …(2)
On adding (1) and 2) we get
Cost of 2 notebooks + cost of 3 pencils = ₹ 2p + ₹ 3s
= ₹ (2p + 3s)
(d) What will you get if 3 times x is subtracted from the largest two-digit number?
Solution:
Largest 2-digit number = 99
3 times x = 3x
When 3 times x is subtracted from 99,
we get = 99 – 3x
(e) Rohit travels by car with the speed of x km/hr for 2 hours and 3 hours with the speed of y km/hr. Write the expression for total distance travelled by him.
Solution:
Distance = speed × time
Distance = x × 2 = 2x km and
Distance = y × 3 = 3y km
Total distance = (2x + 3y) km
Question 2.
Write the number which is
(a) 6 less than two-third of number x
(b) 3 times the sum of z and 5
Solution:
(a) 6 less than $$\frac{2}{3}$$ x = $$\frac{2}{3}$$ x – 6
(b) 3 times the sum of z and 5 = 3 (z + 5)
Question 3.
Write statements for:
(a) $$\frac{4}{5}$$ (a + 2)
(b) $$\frac{4}{5}$$ a + 2
Solution:
(a) $$\frac{4}{5}$$ (a + 2) = four-fifth the sum of 5 a and 2
(b) $$\frac{4}{5}$$ a + 2 = 2 more that four-fifth of 5 number a.
Question 4.
Write the following expressions using literals, numbers and arithmetic operations:
(a) diameter of a circle is twice its radius
(b) twice of x subtracted from one-third of y
(c) total cost of 3 tables at ₹ x each and 4 chairs at ₹ y each
(d) cost of one book if the cost of 10 books is ₹ x.
Solution:
(a) If d = diameter and
∴ d = 2r
(b) $$\frac{1}{3}$$ y – 2x
(c) ₹ (3x + 4y)
(d) ₹ $$\frac{x}{10}$$
Question 5.
Write the product form of the following:
(a) – 3p3q2r
(b) (- y)27
Solution:
(a) – 3p3q2r
= – 3 × p × p × p × q × q × r
(b) (- y)21 = – y × – y × – y × ……….. 27 times
Question 6.
Write a statement to represent a constant.
Solution:
Number of paise in a rupee.
Question 7.
Write a statement to represent a variable.
Solution:
They are not same.
Question 8.
Write the product form of ($$\frac{2}{3}$$x)3 and $$\frac{2}{3}$$x3. Are they same?
Solution:
($$\frac{2}{3}$$x)3 = $$\frac{2}{3}$$x × $$\frac{2}{3}$$x × $$\frac{2}{3}$$
= $$\frac{8}{27}$$x3
$$\frac{2}{3}$$x3 = $$\frac{2}{3}$$ × x × x × x
= $$\frac{2}{3}$$x3
They are not same.
Question 9.
Write one example each of a:
(а) Monomial expression
(b) Trinomial expression
(d) Binomial expression
Solution:
(a) 3x
(b) 2a + b + 3
(c) a + b + c + 5
(d) 2x + 3y
Question 10.
Separate the terms of:
3p3 – 2p2 + p – 6.
Solution:
3p3 – 2p2, p, – 6
Question 11.
Encircle the like terms:
a2bc, a2b2c, b2a2c2, – ba2c, c2ab, bca2, cba2 – a2bc.
Solution:
a2bc, – ba2c, – a2bc, bca2, cba2
Question 12.
Fill in the blanks:
(a) An algebraic expression whose terms are – 2a, 3b, – 5 is ________.
Solution:
– 2a + 3b – 5
(b) The numerical coefficient in is ________.
Solution:
– $$\frac{3}{2}$$
(c) The coefficient of y in 2.5 x2y is ________.
Solution:
2.5 x2
(d) The coefficient of p in – p is ________.
Solution:
– 1
(e) Is 2x – 3y – x a binomial or a trinomial expression?
Solution:
binomial
(f) The exponential form of (- p) × (- p) × (- p) × …………. 17 times is ________.
Solution:
(- p)17
(g) Sum of – 7x, 8x, x, – 9x is ________.
Solution:
– 7x
(h) If we subtract – 9y from – 2y, we get ________.
Solution:
7y
Question 13.
Evaluate a – 4b + 2 if a = 2, b = 1.
Solution:
Here a = 2, b = 1
a – 4b + 2 = 2 – 4(1) + 2
= 4 – 4 = 0
Question 14.
In a bus x passengers have taken, tickets of ₹ 5 each and y passengers have taken tickets of ₹ 10 each. Find the total number of passengers and total fare received by the conductor.
Solution:
Fare of x passengers = ₹ 5x
Fare of y passengers = ₹ 10y
Total fare = ₹ 5x + ₹ 10y = ₹ (5x + 10y)
Total number of passengers = x + y.
Question 15.
On a particular day x ladies were travelling in ladies compartment of metro rail and y people in other compartments from Station A to Station B. If 10 passengers from other compartments and 15 passengers (ladies) from ladies compartment got down in between, write the algebraic expression for the number of passengers left at Station B.
Ans.
No. of people travelling in other compartments = y
∴ Total no. passengers = (x + y)
No. of ladies got down =15
No. of people got down = 10
∴ Total number of passengers got down = 15 + 10 = 25
∴ No. of passengers left = Total no. of passengers – Total no. of passengers got down
= (x + y) – 25.
### DAV Class 8 Maths Chapter 1 HOTS
Question 1.
If two cubes whose volumes are given by (x3 + 6y2 + 12z + 8) cubic units and (x3 – 2z + 10 – y2) cubic units respectively are melted to form a new cube, find the expression representing the volume of the new cube thus formed.
Solution:
Vol. of first cube = (x3 + 6y2 + 12z + 8) cubic units
Vol. of second cube = (x3 – 2z + 10 – y2) cubic units
∴ Volume of new cube = Vol. of first cube + Vol. of second cube
= (x3 + 6y2 + 12z + 8) + (x2 – 2z + 10 – y2)
= (x3 + x3) + (6y2 – y2) + (12z – 2z) + (8 + 10)
= 2x3 + 5y2 + 10z + 18 cubic units
Question 1.
Identify the like terms from the following:
a2b, – ba2, 3a2b2, 7ab2, – a2b2, 7a2b
Solution:
Like terms are a2b, 7a2b.
Question 2.
Fill in the blanks:
(a) 2p + 3q = ____ + 2p
(b) x + 2y = ____ + x
(c) 2x × ____ = 2x
(d) 6x + ____ = 6x
Solution:
(a) 3q
(b) 2y
(c) 1
(d) 0
Question 3.
Write the following in exponential form:
(a) (2x) × (2x) × (2x) …………… 11 times
(b) $$\left(-\frac{1}{2} x\right) \times\left(-\frac{1}{2} x\right) \times\left(-\frac{1}{2} x\right)$$
(c) $$\left(\frac{3 y}{2}\right) \times\left(\frac{3}{2} y\right) \times\left(\frac{3}{2} y\right)$$
(d) x × x × x × x × x × x
Solution:
(a) (2x) × (2x) × (2x) × ……………. 11 times
= (2x)11
(b) $$\left(-\frac{1}{2} x\right) \times\left(-\frac{1}{2} x\right) \times\left(-\frac{1}{2} x\right)=\left(-\frac{1}{2} x\right)^3$$
(c) $$\left(\frac{3 y}{2}\right) \times\left(\frac{3}{2} y\right) \times\left(\frac{3}{2} y\right)=\left(\frac{3}{2} y\right)^3$$
(d) x × x × x × x × x × x = (x)6
Question 4.
Write the product form of the following:
(a) $$\left(-\frac{2}{7}\right)^{23}$$
(b) $$\left(\frac{5}{6} x y\right)^3$$
(c) (- 2x)7
(d)$$\left(\frac{1}{7} x^2 y z\right)^3$$
Solution:
(a) ($$\frac{- 2}{7}$$)23
= $$\frac{- 2}{7}$$ × $$\frac{- 2}{7}$$ × $$\frac{- 2}{7}$$ ……………. 23 times
(b) ($$\frac{5}{6}$$ xy)3
= $$\frac{5}{6}$$ xy × $$\frac{5}{6}$$ xy × $$\frac{5}{6}$$ xy
(e) (- 2x)7 = (- 2x) × (- 2x) × (- 2x) × (- 2x) × (- 2x) × (- 2x) × (- 2x)
(d) ($$\frac{1}{7}$$ x2yz)3
= $$\frac{1}{7}$$ x2yz × $$\frac{1}{7}$$ x2yz × $$\frac{1}{7}$$ x2yz
Question 5.
Identify the statements as constants and variables.
(a) Two and two make four
Solution:
Constant
(b) Temperature in a day
Solution:
Variable
(c) Number of days in a year
Solution:
Variable
(d) Number of days in a leap year
Solution:
Constant
(e) Number of students in a school
Solution:
Variable
(f) Number of players in a cricket team
Solution:
Constant
(g) Number of books in a library
Solution:
Variable
(h) Number of workers in a factory
Solution:
Variable
Question 6.
Write the numerical coefficients in the following terms:
(a) – $$\frac{2}{3}$$ y
Solution:
– $$\frac{2}{3}$$
(b) $$\frac{5}{6}$$ x
Solution:
$$\frac{5}{6}$$
(c) $$\frac{1}{2}$$ xy2
Solution:
$$\frac{1}{2}$$
(d) – 2y
Solution:
– 2
(e) – p
Solution:
– 1
(f) 8x2y
Solution:
8
(g) – 2yz2
Solution:
– 2
(h) – 3m
Solution:
– 3
(i) 2n
Solution:
2
Question 7.
Find the monomial, binomial, trinomial and quadrinomial terms.
3, a + b, 3x + 2y, a + b – 2c, x + 2y – y2 + 7, 0, 5yz2, 3x + 7y, – 3 + 2y + 3z
Solution:
Monomial = 3, 0, 5yz2
Binomials = a + b, 3x + 2y, 3x + 7y
Trinomials = a + b – 2e, – 3 + 2y + 3z
Quadrinomials = x + 2y – y2 + 7
Question 8.
Add the following using column method:
(a) x3 + y3 + 3x2y, 3x3 – y3 + 2x2y and 6x3 + y3 – 3x2y
(b) – p – q, 3p + q and p + 3q
(c) 5x2 – y2 – 8z and 3x2 +y2 + 8z
(d) x + y – z and – x – y + z
Solution:
(a) x3 + y3 + 3x2y, 3x3 – y3 + 2x2y and 6x3 + y3 – 3x2y
(b) – p – q, 3p + q and p + 3q
(c) 5x2 – y2 – 8z and 3x2 +y2 + 8z
= 8x2
(d) x + y – z and – x – y + z
= 0
Question 9.
Subtract: 8x3 + 3x2 + 6x – 5 from x3 – 5x2 + 8x + 9.
Solution:
Question 10.
From the sum of 7x3 + 2x + 7 and x3 + 8x2 – 3 subtract x3 – x2 – x + 5.
Solution:
sum of 7x3 + 2x + 7 and x3 + 8x2 – 3
Now we have to subtract x3 – x2 – x + 5 from 8x3 + 8x2 + 2x + 4 | 4,153 | 10,174 | {"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} | 4.8125 | 5 | CC-MAIN-2024-22 | latest | en | 0.791298 |
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# The diagonals of a rectangle ABCD intersect at O, if $\angle BOC = 70^\circ$ . Find $\angle ODA$
Last updated date: 24th Feb 2024
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Hint:
Here, we have to find the measure of a given angle. We will use the properties of angles in a rectangle and triangle. We will first find the measure of the opposite angle of the given angle using the property vertically opposite angle. Then we will use the property of isosceles triangle and sum property of a triangle to find the measure of the required angle.
Complete step by step solution:
Let ABCD be a rectangle. The diagonals of a rectangle ABCD intersect at O.
We are given that $\angle BOC = 70^\circ$.
We know that in a rectangle, vertically opposite angles are equal.
By using this property, we get
$\angle BOC = 70^\circ = \angle AOD$ ………………………………………………………………………….$\left( 1 \right)$
We know that the diagonals of a rectangle are equal and bisect each other.
So, in $\Delta AOD$ , we get
$AO = OD$
We know that angles opposite to equal sides of an isosceles triangle are equal.
By using this property, we get
$\angle OAD = \angle ODA$ ……………………………………………………………………………………………$\left( 2 \right)$
We know that the sum of the angles of a triangle is $180^\circ$.
By using this property in $\Delta AOD$, we get
$\angle OAD + \angle AOD + \angle ODA = 180^\circ$
By substituting equation $\left( 1 \right)$ in the above equation, we get
$\Rightarrow \angle ODA + \angle AOD + \angle ODA = 180^\circ$
By adding the equal angles, we get
$\Rightarrow 2\angle ODA + \angle AOD = 180^\circ$
By substituting equation $\left( 1 \right)$ in the above equation, we get
$\Rightarrow 2\angle ODA + 70^\circ = 180^\circ$
Subtracting $70^\circ$ from both side, we get
$\Rightarrow 2\angle ODA = 180^\circ - 70^\circ$
$\Rightarrow 2\angle ODA = 110^\circ$
Dividing by 2 on both the sides, we get
$\Rightarrow \angle ODA = \dfrac{{110^\circ }}{2}$
$\Rightarrow \angle ODA = 55^\circ$
Therefore, $\angle ODA$ is $55^\circ$
Note:
We know that the sum of all interior angles of a rectangle is 360 degrees. The diagonals of a rectangle bisect each other so that the lengths of the diagonals are equal in length. Since the diagonal is a straight angle the diagonals of a rectangle bisect each other at different angles where one angle is an acute angle and the other angle is an obtuse angle. We should also remember that if the diagonals bisect each other at right angles, then it is a square. | 704 | 2,546 | {"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} | 4.875 | 5 | CC-MAIN-2024-10 | longest | en | 0.75042 |
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Visit one of the following newspapers’ websites: USA Today, New York Times, Wall Street Journal, or Washington Post. Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. The chosen article must have a publication date during this quarter.
The article should use one of the following categories of descriptive statistics:
• Measures of Frequency - Counting Rules, Percent, Frequency, Frequency Distributions
• Measures of Central Tendency - Mean, Median, Mode
• Measures of Dispersion or Variation - Range, Variance, Standard Deviation
• Measures of Position - Percentile, Quartiles
Write a two to three (2-3) page paper in which you:
1. Write a summary of the article.
2. Explain how the article uses descriptive statistics.
3. Explain how the article applies to the real world, your major, your current job, or your future career goal.
4. Analyze the reasons why the article chose to use the various types of data shared in the article.
5. Format your paper according to the Strayer Writing Standards. Please take a moment to review the SWS documentation for details.
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### Assignment: Descriptive Statistics
Assignment: Descriptive Statistics
Due Week 7 and worth 140 points
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, …
• Rated 1 times
### assignment 1: math 300
Course: MAT 300 Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. …
• Rated 1 times
### Descriptive Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. …
• Rated 1 times
### Assignment: Descriptive Statistics
Assignment: Descriptive Statistics
Due Week 7 and worth 140 points
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current …
• Rated 1 times
### Statistics
Course: MAT 300 Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career …
• Not rated
### Assignment: Descriptive Statistics
Course: MAT 300 Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. The …
• Not rated
### Descriptive Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. The …
• Not rated
### Descriptive Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. …
• Not rated
### can you help me
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. …
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### Statistics
Visit one of the following newspapers’ websites: , , , or . Select an article that uses statistical data related to a current event, your major, your current field, or your future career goal. … | 832 | 3,793 | {"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} | 2.78125 | 3 | CC-MAIN-2019-22 | longest | en | 0.862015 |
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# \$0.25Two cars are 238 mile apart and...
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Two cars are 238 mile apart and traveling toward each other along the same road. They meet in 2 hour. One car is traveling 5 miles per hour slower than the other. What is the speed of each car?
Please write the problem solution in the solution text box; do not attach files.
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olivier May 12, 2008 09:44
"turbulent viscosity limited to viscosity ratio"
Hi, I do a calculation with species transport. For a calculation for first order all is ok but for second order I have the following sentence after each iteration "turbulent viscosity limited to viscosity ratio of 1e5 in 3(or 4) cells." Where does this problem come from (mesh,under-relaxation factors...)? Thanks for yours answers...
wyzh May 12, 2008 10:23
Re: "turbulent viscosity limited to viscosity rati
You can modify the Maximum Tub.Viscosity Ratio in FLUENT: Solve---Controls---Limits
olivier May 13, 2008 05:31
Re: "turbulent viscosity limited to viscosity rati
Thank you, but do you think is ok to obtain a turbulent viscosity ratio greater than 10000 ? Actually my calculation don't converge.
kbs May 13, 2008 07:06
Re: "turbulent viscosity limited to viscosity rati
hey oliver I have same problem as yours what turbulence model you tried ? is it k-epsilon ?
olivier May 13, 2008 07:48
Re: "turbulent viscosity limited to viscosity rati
Yes it's the k-epsilon model.
bohis May 13, 2008 09:28
Re: "turbulent viscosity limited to viscosity rati
either bad mesh or k-epsilon is a traitor
wyzh May 13, 2008 11:58
Re: "turbulent viscosity limited to viscosity rati
I am a beginner of FLUENT. I think that your calculation is convergent or not has nothing to do with the "turbulent viscosity limited to viscosity ratio" . And the result may be credible ,although "turbulent viscosity limited to viscosity ratio" is displayed during your iteration.
Ant May 14, 2008 02:03
Re: "turbulent viscosity limited to viscosity rati
Oliver,
The message you receive is more often due to bad mesh. If you are sure that the mesh quality is good just check if there is any jumps in the mesh size (particularly in regions of high gradients).
One way if to create a point using maximum value of tubulent viscosity ratio (TVR). Then create planes passing through the point and see how the mesh and TVR look like on these planes. The jump in mesh usually happens when you have a multi-block structured mesh.
Regards, Ant
olivier May 14, 2008 04:11
Re: "turbulent viscosity limited to viscosity rati
Thank you, I think the problem is probably due to the mesh. But I have this message for a Second Order calculation and not for a First Order. If the mesh is bad, will the message have to appear in every case?
Ant May 14, 2008 09:48
Re: "turbulent viscosity limited to viscosity rati
Oliver,
The message need not appear in each case. This could be because the absolute quality of mesh in your simulation could be good (Equiangle Skew), but there could be huge jump in the mesh in critical regions. With second order the error in gradients in these regions would be magnified.
Suggest you plot planes and see where there is jump in mesh. Probably near inlets/outlets. You could have tiny inlet pipe connected to a huge domain. The mesh along the length for the inlet pipe could be fine and could turn out to be coarse immediately into the domain.
Regards, Ant
hfla October 10, 2015 05:39
Hello everybody.
I have a similar problem. I'm analysing a simple geometry by using an axisymmetric model with a k-epslilon turbulence model by using star cmm+. I imposed mass flow at the inlet and outflow with a target mass flow rate at the outlet. When I do a simulation with a low mass flow rate the solution is OK and it converges without warnings, but when I increase the mass flow rate I receive the warning "turbulent viscosity ratio limited in 4 cells". Even in this case my solution converges but I receive the warning message. I tried to decrease the URF but without results. Do you think that an increase of the viscosity ratio could be a good idea?
Thanks a lot.
hfla October 10, 2015 05:49
Hello everybody.
I have a similar problem. I'm analysing a geometry by using an axisymmetric model with a k-epslilon turbulence model by using star cmm+. I imposed mass flow at the inlet and outflow with a target mass flow rate at the outlet. When I do a simulation with a low mass flow rate the solution is OK and it converges without warnings, but when I increase the mass flow rate I receive the warning "turbulent viscosity ratio limited in 4 cells". Even in this case my solution converges but I receive the warning message. I tried to decrease the URF but without results. Do you think that an increase of the viscosity ratio could be a good idea? | 1,170 | 4,732 | {"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} | 2.90625 | 3 | CC-MAIN-2017-34 | latest | en | 0.882148 |
https://math.stackexchange.com/questions/2585483/interesting-olympiad-style-problem-about-invariance | 1,653,667,767,000,000,000 | text/html | crawl-data/CC-MAIN-2022-21/segments/1652662658761.95/warc/CC-MAIN-20220527142854-20220527172854-00418.warc.gz | 462,664,854 | 67,251 | Problem: The following operations are permitted with the quadratic polynomial $ax^2 +bx +c:$ (a) switch $a$ and $c$, (b) replace $x$ by $x + t$ where $t$ is any real. By repeating these operations, can you transform $x^2 − x − 2$ into $x^2 − x − 1?$
My Attempt: Notice that the sum of coefficients $S\equiv a+b+c\pmod{t}$ is invariant. This is clear if we switch $a$ and $c.$ If we replace $x$ with $x+t$ then we have $ax^2+(2at+b)x+(at^2+bt+c)$ and so $S\equiv a+2at+b+at^2+bt+c\equiv a+b+c\pmod{t}.$ Now for $x^2-x-2$ we have $S\equiv -2\pmod{t}$ and at the end we want $S\equiv -1\pmod{t}$, which is impossible. I am not sure whether this is correct because $t\in \mathbb{R}.$ So any inputs will be much appreciated.
• You can use different values of $t$ on different occasions, so you have to prove the choices are compatible. Dec 30, 2017 at 13:48
The two operations preserve the discriminant $b^2-4ac$, now the discriminant of $x^2-x-2$ is $9$ while the discriminant of $x^2-x-1$ is $7$. So ...
• This is interesting, for the same answer I din't get any up vote. And I even answer before. Dec 30, 2017 at 13:55
• @JohnWatson 1) You answered before but you deleted your answer for some time. 2) I provide more details than you do 3) Your first paragraph is confusing, I find Mark Benett's comment much clearer 4) You did get some upvotes (4 as of now). Dec 30, 2017 at 15:04
• I don't know why are you reacting so nervously. My comment (obviously) wasn't directed at you. And, yes I did delete it because my answer (as yours!) is not an answer on his question. Dec 30, 2017 at 15:29
• -1: this answer doesn't even address the OP's question. Dec 31, 2017 at 2:33
• @MartinArgerami It is not clear whether the OP's question is more about the problem itself or the attempted solution, and the OP also says "any input appreciated" Dec 31, 2017 at 7:30
I don't understand. Is it $S = a+b+c\pmod{t}$ or $S= a+b+c$. Because I don't understand how you get $S\equiv a+b+c\pmod{t}$ in the second case.
Anyway, just calculate the discriminant and show that it doesn't change:
Mark new polynomial with $a'x^2+b'x+c'$
Case 1. If we change only we get from $ax^2+bx+c$ this $cx^2+bx+a$ so $a'=c$, $b'=b$ and $c'=a$ so $$D' = b'^2 -4a'c' = b^2-4ac = D$$
Case 2. If we replace $x$ with $x+t$ we get $$a(x+t)^2+b(x+t)+c = ax^2+(2at+b)x +at^2+bt+c$$ so $a' = a$, $b' = 2at+b$ and $c'=at^2+bt+c$ so $$D' = (2at+b)^2-4a(at^2+bt+c) = 4a^2t^2+4abt+b^2 -4at^2-4abt-4ac = D$$
So since the discriminant at begining is different from the end it is impossible. | 890 | 2,548 | {"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} | 3.859375 | 4 | CC-MAIN-2022-21 | latest | en | 0.920554 |
https://converter.ninja/volume/metric-teaspoons-to-us-customary-teaspoons/529-brteaspoon-to-usteaspoon/ | 1,685,448,493,000,000,000 | text/html | crawl-data/CC-MAIN-2023-23/segments/1685224645595.10/warc/CC-MAIN-20230530095645-20230530125645-00275.warc.gz | 222,346,784 | 5,493 | # 529 metric teaspoons in US customary teaspoons
## Conversion
529 metric teaspoons is equivalent to 536.628540278248 US customary teaspoons.[1]
## Conversion formula How to convert 529 metric teaspoons to US customary teaspoons?
We know (by definition) that: $1\mathrm{brteaspoon}\approx 1.01442068105529\mathrm{usteaspoon}$
We can set up a proportion to solve for the number of US customary teaspoons.
$1 brteaspoon 529 brteaspoon ≈ 1.01442068105529 usteaspoon x usteaspoon$
Now, we cross multiply to solve for our unknown $x$:
$x\mathrm{usteaspoon}\approx \frac{529\mathrm{brteaspoon}}{1\mathrm{brteaspoon}}*1.01442068105529\mathrm{usteaspoon}\to x\mathrm{usteaspoon}\approx 536.6285402782485\mathrm{usteaspoon}$
Conclusion: $529 brteaspoon ≈ 536.6285402782485 usteaspoon$
## Conversion in the opposite direction
The inverse of the conversion factor is that 1 US customary teaspoon is equal to 0.00186348642485822 times 529 metric teaspoons.
It can also be expressed as: 529 metric teaspoons is equal to $\frac{1}{\mathrm{0.00186348642485822}}$ US customary teaspoons.
## Approximation
An approximate numerical result would be: five hundred and twenty-nine metric teaspoons is about five hundred and thirty-six point six two US customary teaspoons, or alternatively, a US customary teaspoon is about zero times five hundred and twenty-nine metric teaspoons.
## Footnotes
[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).
Results may contain small errors due to the use of floating point arithmetic. | 415 | 1,576 | {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "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-2023-23 | latest | en | 0.716285 |
https://www.businessinsider.com/starbucks-and-dunkin-donuts-coffee-to-ice-ratio-to-test-2016-9 | 1,560,858,863,000,000,000 | text/html | crawl-data/CC-MAIN-2019-26/segments/1560627998716.67/warc/CC-MAIN-20190618103358-20190618125358-00057.warc.gz | 697,916,475 | 44,752 | # I measured the coffee-to-ice ratio at Starbucks and Dunkin Donuts to figure out where you get the best deal
Oh, Starbucks. The coffee chain we love to hate but continue to form lines around the block for. They've received backlash for everything from rising drink prices and unnecessarily unhealthy products, to new rewards program, and now, for the amount of ice they put in their cold drinks.
Back in May, a Chicago woman filed a lawsuit against Starbucks claiming that the amount of advertised iced coffee you pay for is not what actually ends up in your cup. The suit specifically cites that the Venti drinks, which are advertised as a 24-ounce drink, contain only 14 ounces of liquid after ice is added.
Is she just being unreasonable, or are your baristas seriously underfilling your drink? I had to get to the bottom of this in the most scientific way possible. So I drove all around campus and started pouring cups of coffee into my measuring cup.
I bought a 16-ounce drink everywhere I went. After getting the drink, I would immediately pour the coffee into my measuring cup. I always ordered an unsweetened iced coffee, for the record. I visited three Starbucks locations and one Dunkin' Donuts.
My goals for the experiment were to see how much liquid actually comes in a cup, how it compares to the amount the lawsuit is claiming they're leaving out, and to see if most Starbucks stores are consistent with their pours. The Dunkin' stop was to see if there was a comparable difference between Starbucks and what other coffee chains are giving you.
And now for some quick math before we get to the real results. The lawsuit claims that a Starbucks Venti drink only comes 58% full. Being the broke college student that I am, I decided to stick with the Grande and 16-ounce drinks to save a little cash. Still, using this math, these drinks at Starbucks would have to contain about 9.5 ounces of coffee to be on par. Now let's get started.
1/
## Starbucks location one
Armed with my measuring cup shoved inside my purse, I was ready to conduct my experiment. The first Starbucks location I visited was Purdue's nearest off-campus store. It was definitely nowhere near as crowded as the on campus locations. The lack of people inside drove me to the outside tables to measure the drink. I didn't want the baristas to know that I was onto the potential scandal.
The drink came with about 11 ounces of coffee. That means it had 69% of the advertised amount of coffee. It's a big enough discrepency to be a little upset about, but nothing like what the suit claims. More tests had to be conducted for consistency's sake.
2/
## Starbucks location two
I ventured to the store in our residential area for the next test. This store was decently crowded enough that I could hide in the back corner to pour my order into a measuring cup, only to emerge with a cup of ice.
This cup had almost exactly 10 ounces of coffee. That's 63% of the advertised ounce-age. Less than the first location but just a tad more than the suit claims.
3/
## Starbucks location three
This was the last Starbucks store on my quest. Purdue's notoriously busiest store with the longest line, this place was easy to hide amongst the crowd as I scrutinized my drink contents.
Another 10 ounce drink for me. I'm sad about the potential six ounces of coffee I'm missing out on, but applaud the consistency between stores. And now for the ultimate test—how do Starbucks pours compare to another chain?
4/
## Dunkin' Donuts
This store is so intimate and was so not crowded when I visited that I was driven to conduct my experiment in my car. Probably unnecessary, but I was really trying to avoid being asked why I was carrying around a giant measuring cup.
The results surprised me. The Dunkin drink also came with 10 ounces of coffee. No differences between the East coast classicand West coast favorite.
5/
## So what does it all mean?
Is Starbucks intentionally giving you the short pour? Eh, probably not. All three locations were fairly consistent and on par with another popular coffee chain. If you wanna sue Starbucks, you'd probably have to sue most other chains as well.
Of course, there was plenty of room for error in this experiment. A 24 ounce drink has the potential to store even more ice, so there's more potential for liquid coffee loss. And they may tend to put in more ice in other types of drinks when compared to the iced coffee.
Ice in your iced coffee is probably something you'll just have to deal with. I don't believe they're giving you more ice to make you pay more for less—I think they're just filling the cup with ice (hence the name "iced coffee").
If paying for all that ice in your drink is something that really gets to you, try ordering your drink with less or no ice, or look for stores that size up on iced drinks. As for the Starbucks ice debate, I say case closed.
Read the original article on Spoon University. Copyright 2019. Follow Spoon University on Twitter.
More: Contributor Spoon University Starbucks Food | 1,078 | 5,053 | {"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} | 2.921875 | 3 | CC-MAIN-2019-26 | latest | en | 0.972462 |
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• Subject: Math
### TXRCFP: Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013
The Texas Response to Curriculum Focal Points Revised 2013 was created from the 2012 revision of the TEKS as a guide for implementation of effective mathematics instruction by identifying critical areas of content at each grade level.
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• Subject: Math
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• Resource ID: R4SCI0058 | 945 | 4,414 | {"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} | 3.484375 | 3 | CC-MAIN-2020-05 | latest | en | 0.77258 |
https://brainmass.com/math/vector-calculus/hyperplane-27983 | 1,477,166,729,000,000,000 | text/html | crawl-data/CC-MAIN-2016-44/segments/1476988719041.14/warc/CC-MAIN-20161020183839-00026-ip-10-171-6-4.ec2.internal.warc.gz | 837,664,851 | 18,730 | Share
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# Hyperplane
For both 1 and 2, could you tell me whether or not there is a hyperplane that strictly separates the given sets A,B. If there is, find one. If there is not, prove so please.
1) A={(x,y):abs(x) + abs(y) <=1}, B={(1,1)}
2) A={(x,y):xy >= 4}, B={(x,y):x^2+y^2 <= 1}
where abs = absolute value
#### Solution Preview
The two problems are all in x-y plane. So a hyperplane is actually a line.
(1) Yes.
We can select the line: x+y=1.5
For element (1,1) in B, we know 1+1=2>1.5
But for any point (x,y) in A, we know |x|+|y|<=1, then we have ...
#### Solution Summary
This is a proof regarding hyperplanes with given characteristics.
\$2.19 | 222 | 679 | {"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-2016-44 | longest | en | 0.797092 |
https://www.calcsforcash.com/4-expert-psat-calculator-tips-approved-calculators/ | 1,701,637,498,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100508.53/warc/CC-MAIN-20231203193127-20231203223127-00526.warc.gz | 803,214,257 | 31,471 | # 4 Expert PSAT Calculator Tips+ Approved Calculators
### Is it allowed to use a calculator on the PSAT?
Just like the SAT, there are four sections on the PSAT — Math (no calculator allowed), Math (calculator allowed), writing section, and reading section. The first math section in which the calculator is not allowed contains 17 questions and lasts for 25 minutes. The second section in which calculators are allowed contains 11 questions and lasts for 45 minutes.
The College Board says that almost all the questions on the calculator Math section (second section) can be solved without the help of a calculator. But for some questions, using a calculator will be helpful. To be more precise, the college board’s official website says that you can use a calculator for this section when it can make your life easier, but you must not overcomplicate questions by taking help of a calculator for psat when it is better to solve those manually. For example, using a calculator to multiply 34 x 174 is better as you can get an error-free answer quite fast. However, if you solve the equation 3x + 4y = 17, 4x + 8y = 26, it will be better to solve it manually than taking help from a graphing calculator.
### What calculators are allowed on the PSAT?
Three kinds of calculators are allowed by the college board for the PSAT exam— scientific calculators, four-function calculators, and graphing calculators.
## Four function calculators
These PSAT calculators are named so as they can only perform four functions— subtract, add, divide, and multiply. Even though theoretically you could use this type of calculator on the PSAT, it is not recommended mainly because it lacks helpful features like entry lines and parentheses.
The only benefit of a four function calculator for psat is that it tends to be reasonably priced and takes up less space. But to be honest, the cons of not being able to visualize what you just entered into the calculator outweigh the pros to a great extent.
## Scientific calculators
You are allowed to use all types of scientific calculators on the PSAT exam. These calculators have all the features that a four-function calculator has, in addition to a few other useful functions. These functions include pi and trigonometric functions and parentheses. Having parentheses is particularly useful as you can actually see what you just entered into the calculator to ensure your calculations happen in the correct order. Suppose solving a complex calculation by entering it into the calculator all at once with parentheses is much easier than having to break it up into different simpler parts that then calculate those separately. The larger the process of calculation, the more the chances of errors.
## Graphing calculator
These calculators are very helpful as they have a lot of features. Apart from having entry lines and all the other features that a scientific calculator possesses, these calculators are also useful in working out solutions to linear equations. For example, if you want to get the coordinates to determine where the two equations cross, a graphing calculator can help you with that.
The only disadvantage of these calculators is that students can get overwhelmed with their multi-functionality. The students can get tricked into using a graphing calculator for PSAT when performing the calculation manually could have been faster. You must avoid this problem by challenging yourself to determine if it is really useful to use these calculators for all the questions during your practice tests.
PSAT calculators you must avoid
Some calculators, such as the Voyage 200 and TI-92, go against the guidelines of the College Board; therefore, they have been banned from being used on the PSAT. The guidelines say that you can’t use a calculator that can access the internet or be noisy. Also, you are not allowed to use a calculator that requires a connection to an electrical outlet.
You can’t use a calculator that meets the criteria given below on the PSAT:
• It is a part of a handheld or a portable laptop, computer, pocket organizer, or electronic writing pad. You can also not use your computer while taking the test.
• Its hardware or software has the QWERTY keypad. You are also not allowed to use any hardware peripherals with a calculator that is otherwise allowed.
• It is equipped with a stylus, pen input, or touch-screen capability.
• It has wireless or Bluetooth connectivity feature
• It makes noise or paper tapes
• It can connect to the internet or requires an electrical outlet.
• It can work like a cell phone and has video/audio recording features. Cellphones are not allowed on the PSAT as well.
• It is equipped with a digital video/audio player.
• It has a scanning or camera feature.
### Four PSAT calculator tips
Now that you are aware of which calculators you can and cannot bring to the PSAT let us look at a few important tips that can help you use your calculator efficiently on the exam day.
Tip 1: Bring a calculator that is allowed
Look at the list of calculators that are allowed on the PSAT and get one that relates to the list. Don’t get a calculator if that is not allowed. Bringing with you a calculator that has not been approved for the test is of no use, and you won’t be allowed to carry it to the examination hall.
Tip 2: Be familiar with the way your calculator works
A calculator is used in an exam to enhance your accuracy and speed. However, if you bring a calculator you have never used before, your speed will be decreased, and you will also be more likely to make errors, no matter how good quality or fancy the calculator is.
Tip 3: Check the entry line feature of the calculator beforehand
When you are in the middle of the exam and have a lot to finish, it is easy to accidentally hit the wrong numbers and end up wasting your time. To reduce the incidence of these errors, make sure to check out if the entry line feature of your calculator is working fine. Also, double-check the numbers you have entered before clicking on entering to ensure that you have entered the right numbers. Doing this is especially important if you are using a calculator that does not keep a running log of all your previous calculations.
Tip 4: Use a calculator only when it is necessary
This is particularly important when using a graphing calculator. With these calculators, it seems easier to make the calculator perform all the calculations for you. What you don’t realize is that entering a lengthy equation into the calculator to get it solved can take longer time than it would take to perform the calculation manually. However, this does not mean you should skip using calculators altogether. Just use them to supplement your skills rather than replace them.
### Conclusion
Calculators are great tools to increase speed and accuracy on exams. We hope that this blog helped you understand what calculators are allowed on the PSAT and also helped you with some tips on how to use them. | 1,431 | 6,996 | {"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} | 3.171875 | 3 | CC-MAIN-2023-50 | longest | en | 0.920502 |
http://acm.zju.edu.cn/onlinejudge/showProblem.do?problemCode=2567 | 1,566,563,133,000,000,000 | text/html | crawl-data/CC-MAIN-2019-35/segments/1566027318375.80/warc/CC-MAIN-20190823104239-20190823130239-00131.warc.gz | 10,788,966 | 2,788 | Welcome to ZOJ
Problem Sets Information Select Problem Runs Ranklist
ZOJ Problem Set - 2567
Time Limit: 5 Seconds Memory Limit: 32768 KB Special Judge
In the Middle Ages m European cities imported many goods from n Arabian cities. Due to continous feudal wars, European cities did not trade with each other, so is some European city needed some Arabian goods, the special trade route was established for this particular trade.
Studying the manuscripts historians have found out that each European city imported goods from at least two Arabian cities, and each Arabian city exported goods to at least two European cities. They have also investigated different factors and identified all potential trade routes (trade routes between some pairs of cities were impossible due to various reasons).
Now historians wonder, what is the minimal possible number of trade routes, that could have existed. Help them to find that out.
Input
The first line of the input file contains m, n, and p - the number of European and Arabian cities respectively, and the number of potential trade routes (1 <= m, n <= 300, 1 <= p <= nm). The following p lines describe potential trade routes, each description consists of two numbers - the European and the Arabian city connected by the route.
Output
On the first line of the output file print k - the minimal possible number of trade routes that could have existed. After that output k numbers - some minimal set of routes that might have existed to satisfy all conditions. Routes are numbered starting from 1 as they are given in the input file.
If historians must have made a mistake and it is impossible to satisfy the specified conditions, print -1 on the first and the only line of the output file.
Sample Input
```5 5 14
1 2
1 3
1 4
1 5
2 1
2 5
3 1
3 5
4 1
4 5
5 1
5 2
5 3
5 4
```
Sample Output
```12
1 2 3 5 6 7 8 9 10 12 13 14
```
Author: Andrew Stankevich
Source: Andrew Stankevich's Contest #4
Submit Status | 476 | 1,973 | {"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} | 2.734375 | 3 | CC-MAIN-2019-35 | latest | en | 0.926388 |
https://live.midifan.com/viewer/squeeze-theorem-worksheet.html | 1,701,342,886,000,000,000 | text/html | crawl-data/CC-MAIN-2023-50/segments/1700679100184.3/warc/CC-MAIN-20231130094531-20231130124531-00348.warc.gz | 415,874,650 | 8,276 | # Squeeze Theorem Worksheet
Squeeze Theorem Worksheet - Web squeeze theorem examples squeeze theorem. Squeeze theorem (1)determine if each sequence is convergent or divergent. “sandwich theorem” or “pinching theorem” if 𝒈 :𝑥 ; Web in other words, the squeeze theorem is a proof that shows the value of a limit by smooshing a tricky function between two equal and known values. Evaluate this limit using the squeeze theorem. Web this quiz and attached worksheet will help gauge your understanding of using the squeeze theorem. Web use the squeeze theorem: Web one helpful tool in tackling some of the more complicated limits is the squeeze theorem:. L𝑳 and lim ℎ :𝑥 ; Theorem 1 (the squeeze theorem) if f, g, and h are functions and for.
Web lesson worksheet nagwa is an educational technology startup aiming to help teachers teach and students learn. Web this quiz and attached worksheet will help gauge your understanding of using the squeeze theorem. Web the next theorem, called the squeeze theorem, proves very useful for establishing basic trigonometric limits. And if lim 𝑔 :𝑥 ; If two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same. If f(x) g(x) h(x) when x is near a (but not necessarily at a [for instance, g(a) may be unde ned]) and lim x!a f(x) = lim. “sandwich theorem” or “pinching theorem” if 𝒈 :𝑥 ; Using the polar coordinates centered at the origin and radius r: Web in other words, the squeeze theorem is a proof that shows the value of a limit by smooshing a tricky function between two equal and known values. Web the squeeze theorem let f (x), g(x) f ( x), g ( x), and h(x) h ( x) be defined for all x≠ a x ≠ a over an open interval containing a a. Determine the limit of the. Web use the squeeze theorem: The squeeze theorem (1) lim x!0 x 2 sin ˇ x. Evaluate this limit using the squeeze theorem. Squeeze theorem (1)determine if each sequence is convergent or divergent. L𝑳 and lim ℎ :𝑥 ; Web explore and practice nagwa’s free online educational courses and lessons for math and physics across different grades. Web one helpful tool in tackling some of the more complicated limits is the squeeze theorem:. Yes, luke's suggestion seems to be correct. Theorem 1 (the squeeze theorem) if f, g, and h are functions and for.
## “Sandwich Theorem” Or “Pinching Theorem” If 𝒈 :𝑥 ;
Web use the squeeze theorem: Yes, luke's suggestion seems to be correct. L𝑳 and lim ℎ :𝑥 ; Web the squeeze theorem let f (x), g(x) f ( x), g ( x), and h(x) h ( x) be defined for all x≠ a x ≠ a over an open interval containing a a.
## Web This Quiz And Attached Worksheet Will Help Gauge Your Understanding Of Using The Squeeze Theorem.
Web x!0 x = 0, the theorem below tells us we have lim x!0 g(x) = 0. Web explore and practice nagwa’s free online educational courses and lessons for math and physics across different grades. The squeeze theorem (1) lim x!0 x 2 sin ˇ x. Theorem 1 (the squeeze theorem) if f, g, and h are functions and for.
## Web In Other Words, The Squeeze Theorem Is A Proof That Shows The Value Of A Limit By Smooshing A Tricky Function Between Two Equal And Known Values.
And if lim 𝑔 :𝑥 ; Using the squeeze theorem, compute the limit of the function f(x) = sin(x) x2 as x approaches 0. Determine the limit of the. Squeeze theorem (1)determine if each sequence is convergent or divergent.
## If F(X) G(X) H(X) When X Is Near A (But Not Necessarily At A [For Instance, G(A) May Be Unde Ned]) And Lim X!A F(X) = Lim.
Web the next theorem, called the squeeze theorem, proves very useful for establishing basic trigonometric limits. Since 11 cos x 1 1, we know 2(x + x 2) (x2 + x 2) cos 1 x 1 x2 + x 2 x2 + x 2 = (x 1)(x+ 2), so lim x!1 x2 + x. If f(x) ≤g(x) ≤h(x) f ( x) ≤ g ( x) ≤ h ( x). Web one helpful tool in tackling some of the more complicated limits is the squeeze theorem:. | 1,035 | 3,881 | {"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} | 4.375 | 4 | CC-MAIN-2023-50 | latest | en | 0.830895 |
https://forum.slowtwitch.com/forum/Slowtwitch_Forums_C1/Triathlon_Forum_F1/Fit/Size_Question_and_Plans_for_New_bike_P40626/ | 1,585,811,797,000,000,000 | text/html | crawl-data/CC-MAIN-2020-16/segments/1585370506673.7/warc/CC-MAIN-20200402045741-20200402075741-00240.warc.gz | 487,728,253 | 13,029 | Well my wife has finally acquiesced to my begging and pleading for a new tri bike. Before I can decide what to buy I need to resolve a fit and/or sizing question I have. I have been riding a Kestrel 200EMS in a size 54 set up for tri. This bike has a top tube measurement of 54.6. I am running a Thomson set back post "flipped" 180 degrees with full aerobars and pursuit bars (I know...Mr. E does not like this set-up but it works for me). Saddle is in the middle of the rails, Based on calculation formulae from Engineer Franke, et al, I believe I am riding at about 76 - 76.5 degrees. Therefore I assume my "virtual" TT length is shorter than the bike's stated 54.6.
How can I extrapolate from this to determine what size tri-bike I should be on? Should I just ignore the above and go off my "crotch to notch" measurement in fit charts since I am not sure the above is really proper fit or not? Seems to me I read somewhere that when determining desired TT length, (which I assume is most important dimension in fit) that you take the crotch to notch and then subtract 10-11cm for a stem and that will give you an approximate desired TT length. Does that sound about right?
As for what bikes I am looking at now...Felt S22, QR Caliente, P2K (although seem to be in between the 52 and 55 so it may not work) and am interested to hear about the new Kestrel tri bike coming out. If you have any other suggestions I would love to hear them. I need to keep it at about \$2K or less. Additionally, since I will be keeping my Kestrel as a road bike, would it be a mistake to buy a somewhat shallower tri bike (ie, S22 and Caliente are about 76 degree ST angle) vs going all out steep at 78 degrees. What is downside of these "multisport" geometries assuming you can slide the saddle up the rails to achieve 78? I live in a hilly area so thought a 76 degree tri bike may be more flexible for me.
Many Thanks!
Mike
Had a Caliente for two months now, and I also test rode an S32 extensively - very similar to the S22, just slightly lower-grade alloy and detail around seat stays, for what that's worth!
In comparison, the S32 rode like it was alive, like there was a direct connection between me and the road, whereas the Caliente felt smooth, relaxed, almost unexciting, but veeeery comfortable.
The decision came down to things like: I wanted to train long miles on the bike, so comfort was paramount; the S32/22 only has one set of bottle bosses; the dealer who sold me the QR measured me up professionally; I trusted the QR importer more than the Felt one (for UK); on the different websites I visited, and literature that I got from the importer, the geometry changed about three times in six months, and I felt uneasy about this, though I would trust Jim Felt as a brilliant builder, I'm not sure if it was his 'hand' on the torch, or whether the frames come from the Far East (and this is no bad thing normally)
As for angles, see many posts this site about how angles matter for nothing, it's where you put your saddle on the post that counts, and thus the length of top tube is crucial.
In conclusion, the QR is the 'head' choice, but the Felt is the 'heart' choice. Go ride. | 773 | 3,183 | {"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} | 2.796875 | 3 | CC-MAIN-2020-16 | latest | en | 0.950881 |
http://www.luyixian.cn/news_show_274038.aspx | 1,600,803,629,000,000,000 | text/html | crawl-data/CC-MAIN-2020-40/segments/1600400206763.24/warc/CC-MAIN-20200922192512-20200922222512-00279.warc.gz | 184,583,504 | 8,667 | # codeforces 1294A Collecting Coins
2020/1/23 21:20:57 人评论 次浏览 分类：学习教程
## A. Collecting Coins
time limit per test2 seconds
memory limit per test256 megabytes
inputstandard input
outputstandard output
Polycarp has three sisters: Alice, Barbara, and Cerene. They’re collecting coins. Currently, Alice has a coins, Barbara has b coins and Cerene has c coins. Recently Polycarp has returned from the trip around the world and brought n coins.
He wants to distribute all these n coins between his sisters in such a way that the number of coins Alice has is equal to the number of coins Barbara has and is equal to the number of coins Cerene has. In other words, if Polycarp gives A coins to Alice, B coins to Barbara and C coins to Cerene (A+B+C=n), then a+A=b+B=c+C.
Note that A, B or C (the number of coins Polycarp gives to Alice, Barbara and Cerene correspondingly) can be 0.
Your task is to find out if it is possible to distribute all n coins between sisters in a way described above.
You have to answer t independent test cases.
## Input
The first line of the input contains one integer t (1≤t≤104) — the number of test cases.
The next t lines describe test cases. Each test case is given on a new line and consists of four space-separated integers a,b,c and n (1≤a,b,c,n≤108) — the number of coins Alice has, the number of coins Barbara has, the number of coins Cerene has and the number of coins Polycarp has.
## Output
For each test case, print “YES” if Polycarp can distribute all n coins between his sisters and “NO” otherwise.
5
5 3 2 8
100 101 102 105
3 2 1 100000000
10 20 15 14
101 101 101 3
YES
YES
NO
NO
YES
## 思路：
``````#include <iostream>
#include <cstring>
#include <cstdio>
using namespace std;
int main() {
int n;
scanf("%d", &n);
while (n--) {
int a, b, c, d;
scanf("%d %d %d %d", &a, &b, &c, &d);
int sum = a + b + c + d;
if (sum % 3 == 0) {
int ave = sum / 3;
if (a <= ave && b <= ave && c <= ave) printf("YES\n");
else printf("NO\n");
} else printf("NO\n");
}
return 0;
}
``````
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--> | 603 | 2,034 | {"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} | 2.71875 | 3 | CC-MAIN-2020-40 | latest | en | 0.770665 |
https://grandpaperwriters.com/answered-assume-bit-errors-are-mutually-independent-and-identically-distributed-each-occurring-with-positive-probability-p/ | 1,696,130,940,000,000,000 | text/html | crawl-data/CC-MAIN-2023-40/segments/1695233510734.55/warc/CC-MAIN-20231001005750-20231001035750-00681.warc.gz | 307,937,550 | 11,585 | # Answered! Assume bit errors are mutually independent and identically distributed, each occurring with positive probability P…
Assume bit errors are mutually independent and identically distributed, each occurring with positive probability P <<1.
a) Find the even parity bit of the eight-bit message 11010011.
Don't use plagiarized sources. Get Your Custom Essay on
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b) Derive the probability of error detection of an arbitrary m-bit message using 1 bit parity, where m is even.
c) Find the two-bit checksum of message 11010011 by simple XOR summation of consecutive two-bit message components.
d) Derive the probability of error detection of an arbitrary 2m-bit message using two- bit checksum of (c). You can apply the answer to (b). | 200 | 921 | {"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} | 2.5625 | 3 | CC-MAIN-2023-40 | latest | en | 0.860031 |
https://www.robotlab.com/blog/3-tips-for-creating-math-word-problems-that-boost-critical-thinking | 1,726,669,499,000,000,000 | text/html | crawl-data/CC-MAIN-2024-38/segments/1725700651899.75/warc/CC-MAIN-20240918133146-20240918163146-00337.warc.gz | 912,938,897 | 14,265 | <img alt="" src="https://secure.smart-enterprise-52.com/266730.png" style="display:none;">
## ROBOTLAB BLOG
### Everything You Need To Know About Robotics in Education and Businesses
Photo by Jeswin Thomas on Unsplash
Well-designed word problems give elementary students a tangible context for understanding math concepts.
“Danny has 564 tomatoes. He then buys 623 carrots. How long will it take him to get home?” This scenario is what many people picture when we mention word problems. Many educators think they’re taboo. Some argue that many students can’t access the content because of the reading, and the problems themselves can be culturally inappropriate. All of these ideas can be considered valid if the word problem isn’t created or executed correctly.
As a special educator who has worked in urban and rural schools, I have always had a love-hate relationship with word problems. I used to be one of the teachers who would claim, “I’m trying to assess their math reasoning, not their reading skills.” I had difficulty realizing that the story problems can give our students a tangible context to connect to in order to understand the concept that we’re teaching. This context is essential when introducing our students to more abstract content for the first time.
I remember working with a group of third graders at the beginning of the school year, talking about a simple arithmetic problem, 25 + __ = 50. These students had been shown that 25 + 25 = 50, but they were confused about why the blank was in the middle of the problem. Once I created a story problem about placing hot chocolate packets in containers for a class party, the students immediately had light bulbs flash over their heads. They then drew out a representation of the problem and figured that they needed to keep adding to 25 until they got to 50. I have realized that we give mostly numerical problems rather than conceptual word problems. It’s like we’re offering the students directions on how to get to a store without showing them the landmarks that they need to reach their destination. No wonder they become lost in the steps, and our instruction becomes more of a checklist instead of practicing critical thinking and problem-solving skills.
Creating the just-right word problem is somewhat of an art. Luckily, just like an art project, word problems don’t need to be perfect.
## 3 KEYS TO CREATING COMPELLING WORD PROBLEMS
1. They need to be relatable to the students. This might sound like a no-brainer, but I see so many word problems that are like my example from the very beginning. Too often they talk about excessive quantities of things or about a topic that the students can’t relate to. When crafting a word problem, we need to make sure the students can connect to or picture what’s going on. This comes from knowing your students and their interests. In my classroom, Pokémon cards and bags of slime are the most popular things. So when we create word problems and use these topics, the students have an interest in the story and can visualize what’s happening. They especially love it when you use their names to make them the star of the word problem.
Because we are working on very abstract reasoning topics, the students need to have something real to cling to, helping to ground them to reality as they work through the task. Without being able to relate to the problem, we will likely lose them, and they will just pick an operation at random and try solving it that way.
2. The word problems need to be solvable. The problems don’t need to be overly complicated; a simple problem that requires critical thinking will do the trick. Many standardized tests will create tricky verbiage when giving word problems and claim that this makes the problem harder to solve. In reality, if we just make the wording more complex rather than making the situation more thought-provoking, all we’re doing is frustrating our students and not challenging them to think.
3. The word problems need to be open-ended. The final thing a functional word problem needs is to be open-ended. “John has four cars and then gets four more. How many does he have now?” is boring and quite simple. Though this will challenge some students for a short time, it does not make any student think critically, and there are limited entry points our students can solve.
By simply rephrasing the question, we provide a task that makes our students think critically about what’s going on in this story. “John has four cars. His brother gave him some more, and now he has eight. How many did his brother give him?” This problem allows for many entry points and opportunities to try a variety of strategies while challenging students to think abstractly through mathematical reasoning and number sense.
As educators, we need to be vigilant about the tasks and word problems we create. While reviewing possible word problems, we should ask ourselves, “Is this word problem interesting? Is it solvable? Is it open-ended?” If the answer to any of these questions is no, consider reevaluating the problems through a critical thinking lens, making them more effective and beneficial for your students. | 1,052 | 5,180 | {"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} | 3.265625 | 3 | CC-MAIN-2024-38 | latest | en | 0.958299 |
http://www.aptitudetests4me.com/Basic_Numeracy_255.html | 1,505,931,926,000,000,000 | text/html | crawl-data/CC-MAIN-2017-39/segments/1505818687428.60/warc/CC-MAIN-20170920175850-20170920195850-00702.warc.gz | 379,913,589 | 5,607 | Aptitude Tests 4 Me
Basic Numeracy/Quantitative Aptitude
1063. A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream. If the speed of the boat in still water is 10 mph, the speed of the stream is
(a) 2 mph (b) 2.5 mph (c) 3 mph (d) 4 mph
1064. If 7 spiders make 7 webs in 7 days, then 1 spider will make 1 web in how many days?
(a) 1 (b) 7/2 (c) 7 (d) 49
1065. From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is
(a) 149 m (b) 156 m (c) 173 m (d) 200 m
1066. The captain of a cricket team of 11 members is 26 years old and the wicket keeper is 3 years older. If the ages of these two are excluded, the average age of the remaining players is one year less than the average age of the whole team. What is the average age of the team?
(a) 23 years (b) 24 years (c) 25 years (d) None of these
TOTAL
Detailed Solution | 311 | 987 | {"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-2017-39 | latest | en | 0.876972 |
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