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integer n. • Recognize patterns and write the th term of a sequence. • Find the sums of powers of n integers. • Find finite differences of sequences. Why you should learn it Finite differences can be used to determine what type of model can be used to represent a sequence. For instance, in Exercise 61 on page 682, you will use finite differences to find a model that represents the number of individual income tax returns filed in the United States from 1998 to 2003. Introduction In this section, you will study a form of mathematical proof called mathematical induction. It is important that you see clearly the logical need for it, so take a closer look at the problem discussed in Example 5 in Section 9.2. S1 S2 S3 S4 S5 1 12 1 3 22 1 3 5 32 1 3 5 7 42 1 3 5 7 9 52 Judging from the pattern formed by these first five sums, it appears that the sum of the first odd integers is n 1 3 5 7 9... 2n 1 n2. Sn Although this particular formula is valid, it is important for you to see that recognizing a pattern and then simply jumping to the conclusion that the pattern n is not a logically valid method of proof. There are must be true for all values of many examples in which a pattern appears to be developing for small values of n and then at some point the pattern fails. One of the most famous cases of this was the conjecture by the French mathematician Pierre de Fermat (1601–1665), who speculated that all numbers of the form 22n 1, n 0, 1, 2,... n 0, 1, 2, 3, and 4, the conjecture is true. are prime. For Fn Mario Tama/Getty Images F0 F1 F2 F3 F4 3 5 17 257 65,537 The size of the next Fermat number is so great that it was difficult for Fermat to determine whether it was prime or not. However, another later found the well-known mathematician, Leonhard Euler (1707–1783), factorization 4,294,967,297 F5 F5 4,294,967,297 6416,700,417 which proved that F5 is not prime and therefore Fermat’s conjecture was false. Just because a rule, pattern, or formula seems to work for several values of you cannot simply decide that it is valid for all values of without going n, through a legitimate proof. |
Mathematical induction is one method of proof. n 333202_0904.qxd 12/5/05 11:35 AM Page 674 674 Chapter 9 Sequences, Series, and Probability It is important to recognize that in order to prove a statement by induction, both parts of the Principle of Mathematical Induction are necessary. The Principle of Mathematical Induction n. be a statement involving the positive integer Let Pn If 1. P1 is true, and 2. for every positive integer k, the truth of Pk implies the truth of Pk1 then the statement must be true for all positive integers n. Pn To apply the Principle of Mathematical Induction, you need to be able to Pk1, To determine Pk1 Pk. determine the statement substitute the quantity k 1 for a given statement k Pk. in the statement for Example 1 A Preliminary Example Find the statement for each given statement Pk. Pk1 k 2k 12 4 a. Pk : Sk b. c. d. 1 5 9... 4k 1 3 4k 3 Pk : Sk Pk : k 3 < 5k2 Pk : 3k ≥ 2k 1 Solution a. Pk1 : Sk1 k 12k 1 12 4 k 12k 22 4 Replace by k k 1. Simplify. b. Pk1 : Sk1 1 5 9... 4k 1 1 3 4k 1 3 1 5 9... 4k 3 4k 1 c. Pk1: k 1 3 < 5k 12 k 4 < 5k2 2k 1 d. Pk1 : 3k1 ≥ 2k 1 1 3k1 ≥ 2k 3 Now try Exercise 1. A well-known illustration used to explain why the Principle of Mathematical Induction works is the unending line of dominoes shown in Figure 9.6. If the line actually contains infinitely many dominoes, it is clear that you could not knock the entire line down by knocking down only one domino at a time. However, suppose it were true that each domino would knock down the next one as it fell. Then you could knock them all down simply by pushing the first one and starting a chain reaction. Mathematical induction works in the same way. If the truth of Pk is true, the chain reaction proceeds as implies follows: implies the truth of and so on. implies implies and if Pk1 P |
1 P3, P3 P2, P2 P4, P1 FIGURE 9.6 333202_0904.qxd 12/5/05 11:35 AM Page 675 Section 9.4 Mathematical Induction 675 When using mathematical induction to prove a summation formula (such as the one in Example 2), it is helpful to think of Sk1 as Sk1 Sk ak1 where ak1 is the k 1 th term of the original sum. Example 2 Using Mathematical Induction Use mathematical induction to prove the following formula. Sn 1 3 5 7... 2n 1 n2 Solution Mathematical induction consists of two distinct parts. First, you must show that the formula is true when n 1, 1 12. the formula is valid, because 1. When n 1. S1 The second part of mathematical induction has two steps. The first step is to assume that the formula is valid for some integer The second step is to use this assumption to prove that the formula is valid for the next integer, k 1. k. 2. Assuming that the formula Sk 1 3 5 7... 2k 1 k2 is true, you must show that the formula Sk1 k 12 is true. Sk1 1 3 5 7... 2k 1 2k 1 1 1 3 5 7... 2k 1 2k 2 1 2k 1 Sk k2 2k 1 k 12 Group terms to form Replace Sk. by k 2. Sk Combining the results of parts (1) and (2), you can conclude by mathematical induction that the formula is valid for all positive integer values of n. Now try Exercise 5. k 1 positive integers but is true for all values of It occasionally happens that a statement involving natural numbers is not true for the first In these instances, you use a slight variation of the Principle of Mathematical Induction This variation is called the extended P1. in which you verify principle of mathematical induction. To see the validity of this, note from Figure dominoes can be knocked down by knocking over 9.6 that all but the first k the th domino. This suggests that you can prove a statement to be true for n ≥ k In Exercises 17–22 of this section, you are asked to apply this extension of mathematical induction. by showing that is true and that rather than implies n ≥ k. k 1 Pk1. Pn Pk Pk Pk 333202_0904.qxd 12 |
/5/05 11:35 AM Page 676 676 Chapter 9 Sequences, Series, and Probability Example 3 Using Mathematical Induction Use mathematical induction to prove the formula 12 22 32 42... n2 Sn nn 12n 1 6 for all integers n ≥ 1. Solution 1. When n 1, 12 123. S1 6 the formula is valid, because 2. Assuming that 12 22 32 42... k2 Sk kk 12k 1 6 k2 ak you must show that Sk1 k 1k 1 12k 1 1 6 k 1k 22k 3 6. To do this, write the following. Sk1 ak1 Sk 12 22 32 42... k 2 k 12 kk 12k 1 6 k 12 kk 12k 1 6k 12 6 k 1k2k 1 6k 1 6 k 12k 2 7k 6 6 k 1k 22k 3 6 Substitute for Sk. By assumption Combine fractions. Factor. Simplify. Sk implies Sk1. Combining the results of parts (1) and (2), you can conclude by mathematical induction that the formula is valid for all integers n ≥ 1. Now try Exercise 11. Remember that when adding rational expressions, you must first find the least common denominator (LCD). In Example 3, the LCD is 6. When proving a formula using mathematical induction, the only statement that you need to verify is As a check, however, it is a good idea to try verifying some of the other statements. For instance, in Example 3, try verifying P2 and P3. P1. 333202_0904.qxd 12/5/05 11:35 AM Page 677 To check a result that you have proved by mathematical induction, it helps to list the n. statement for several values of For instance, in Example 4, you could list 1 < 21 2, 2 < 23 8, 5 < 25 32, 2 < 22 4, 4 < 24 16, 6 < 26 64, From this list, your intuition confirms that the statement n < 2n is reasonable. Section 9.4 Mathematical Induction 677 Example 4 Proving an Inequality by Mathematical Induction Prove that n < 2n for all positive integers n. Solution 1. For n 1 and n 2, the statement is true because 1 < 21 and 2 < 22. 2. Assuming that k < 2k you need to show that k 1 < 2k1. For |
n k, you have 2k1 22k > 2k 2k. 2k k k > k 1 Because 2k1 > 2k > k 1 or By assumption for all it follows that k > 1, k 1 < 2k1. Combining the results of parts (1) and (2), you can conclude by mathematical n ≥ 1. induction that for all integers n < 2n Now try Exercise 17. Example 5 Proving Factors by Mathematical Induction Prove that 3 is a factor of 4n 1 for all positive integers n. Solution 1. For n 1, 41 1 3. the statement is true because So, 3 is a factor. 2. Assuming that 3 is a factor of 4k 1, you must show that 3 is a factor of 4k1 1. To do this, write the following. 4k. Subtract and add 4k1 1 4k1 4k 4k 1 4k4 1 4k 1 4k 3 4k 1 4k 3 it follows that 3 is Combining the results of parts (1) and (2), you can for all positive Because 3 is a factor of 4k1 1. a factor of conclude by mathematical induction that 3 is a factor of integers and 3 is also a factor of Regroup terms. 4k 1, 4n 1 Simplify. n. Now try Exercise 29. Pattern Recognition Although choosing a formula on the basis of a few observations does not guarantee the validity of the formula, pattern recognition is important. Once you have a pattern or formula that you think works, you can try using mathematical induction to prove your formula. 333202_0904.qxd 12/5/05 11:35 AM Page 678 678 Chapter 9 Sequences, Series, and Probability Finding a Formula for the nth Term of a Sequence To find a formula for the th term of a sequence, consider these guidelines. n 1. Calculate the first several terms of the sequence. It is often a good idea to write the terms in both simplified and factored forms. n 2. Try to find a recognizable pattern for the terms and write a formula for the th term of the sequence. This is your hypothesis or conjecture. You might try computing one or two more terms in the sequence to test your hypothesis. 3. Use mathematical induction to prove your hypothesis. Example 6 Finding a Formula for a Finite Sum Find a formula for the finite sum and prove its validity. 2 3 1 1 1 |
1 1 2 Solution Begin by writing out the first few sums. 3 4 4 5... 1 nn 1 S1 S2 S3 S4 12 1 2 1 3 4 48 60 From this sequence, it appears that the formula for the th sum is k Sk kk 1 k k 1. To prove the validity of this hypothesis, use mathematical induction. Note that so you can begin by assuming you have already verified the formula for that the formula is valid for and trying to show that it is valid for n k 1. n 1, n k Sk1 kk 1 1 k 1k 2 1 k 1k 2 k k 1 kk 2 1 k 1k 2 By assumption k 2 2k 1 k 1k 2 k 12 k 1k 2 k 1 k 2 So, by mathematical induction, you can conclude that the hypothesis is valid. Now try Exercise 35. 333202_0904.qxd 12/5/05 11:35 AM Page 679 Section 9.4 Mathematical Induction 679 Sums of Powers of Integers The formula in Example 3 is one of a collection of useful summation formulas. n This and other formulas dealing with the sums of various powers of the first positive integers are as follows. Sums of Powers of Integers 1. 2. 3. 4. 1 2 3 4... n nn 1 2 12 22 32 42... n2 nn 12n 1 6 13 23 33 43... n3 n2n 12 4 14 24 34 44... n4 nn 12n 13n2 3n 1 30 5. 15 25 35 45... n5 n2n 122n2 2n 1 12 Example 7 Finding a Sum of Powers of Integers Find each sum. a. 7 i1 i3 13 23 33 43 53 63 73 b. 4 i1 6i 4i 2 Solution a. Using the formula for the sum of the cubes of the first positive integers, you n obtain 7 i1 i3 13 23 33 43 53 63 73 4964 4 784. Formula 3 b. 4 i1 727 12 4 6i 4i2 4 64 i1 6i 4 i 44 i1 4i2 i2 i1 i1 644 1 2 444 18 1 6 Formula 1 and 2 610 430 60 120 60 Now try Exercise 47. 333202_0904.qxd 12/5/05 11:35 AM Page 680 680 Chapter 9 Sequ |
ences, Series, and Probability For a linear model, the first differences should be the same nonzero number. For a quadratic model, the second differences are the same nonzero number. Finite Differences The first differences of a sequence are found by subtracting consecutive terms. The second differences are found by subtracting consecutive first differences. are as The first and second differences of the sequence follows. 3, 5, 8, 12, 17, 23,... n: an: 1 3 First differences: 2 Second differences: 2 5 1 3 8 1 3 4 12 5 17 6 23 4 5 6 1 1 For this sequence, the second differences are all the same. When this happens, the sequence has a perfect quadratic model. If the first differences are all the same, the sequence has a linear model. That is, it is arithmetic. Example 8 Finding a Quadratic Model Find the quadratic model for the sequence 3, 5, 8, 12, 17, 23,.... Solution You know from the second differences shown above that the model is quadratic and has the form an an2 bn c. By substituting 1, 2, and 3 for tions in three variables. n, you can obtain a system of three linear equa- a12 b1 c 3 a22 b2 c 5 a32 b3 c 8 a1 a2 a3 Substitute 1 for n. Substitute 2 for n. Substitute 3 for n. You now have a system of three equations in a, b, and c. a b c 3 4a 2b c 5 9a 3b c 8 Equation 1 Equation 2 Equation 3 Using the techniques discussed in Chapter 7, you can find the solution to be So, the quadratic model is c 2. and a 1 2, 1 2 b 1 2, n2 1 2 an n 2. Try checking the values of a1, a2, and a3. Now try Exercise 57. 333202_0904.qxd 12/5/05 11:35 AM Page 681 Section 9.4 Mathematical Induction 681 9.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The first step in proving a formula by ________ ________ is to show that the formula is true when n 1. 2. The ________ differences of a sequence are found by subtracting consecutive terms. 3. A sequence is |
an ________ sequence if the first differences are all the same nonzero number. 4. If the ________ differences of a sequence are all the same nonzero number, then the sequence has a perfect quadratic model. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, find Pk1 for the given Pk. 1. Pk 3. Pk 5 kk 1 k 2k 1 2 4 2. Pk 1 2k 2 4. Pk k 3 2k 1 In Exercises 5–16, use mathematical induction to prove the formula for every positive integer n. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 2 4 6 8... 2n nn 1 3 7 11 15... 4n 1 n2n 1 2 7 12 17... 5n 3 n 2 1 4 7 10... 3n 2 n 2 3n 1 5n 1 1 2 22 23... 2n1 2n 1 21 3 32 33... 3n1 3n 1 1 2 3 4... n nn 1 2 13 23 33 43... n3 n2n 1 2 4 i5 n2n 122n2 2n 1 12 i 4 nn 12n 13n2 3n 1 30 ii 1 nn 1n 2 n i1 n i1 n i1 n i1 1 2i 12i 1 3 n 2n 1 In Exercises 17–22, prove the inequality for the indicated integer values of n. 17. 19. n! > 2n 18. n 4 3 > n, n ≥ 2 n1 n < x x, y y 1 an ≥ na, 2n2 > n 12, 20. 21. 22. n ≥ 1 and 0 < x < y and a > 0 n ≥ 1 n ≥ 3 In Exercises 23–34, use mathematical induction to prove n. the property for all positive integers n a b abn an bn an bn 24. 23. 25. If 0, x2 x1 x1 x 2 x3... xn 0,..., xn 1 x1 1x2 0, then 1... xn 1. 1x3 26. If x1 > |
0, x2 > 0,..., xn > 0, ln ln x2... xn lnx1x2 x1 then... ln xn. 27. Generalized Distributive Law: xy1... yn a bin and xy2... xyn are complex conjugates for all 28. xy1 y2 a bin n ≥ 1. 29. A factor of 30. A factor of 31. A factor of 32. A factor of 33. A factor of 34. A factor of is 3. is 3. is 2. n3 3n2 2n n3 n 3 n4 n 4 22n1 1 24n2 1 is 5. 22n1 32n1 is 3. is 5. In Exercises 35– 40, find a formula for the sum of the first terms of the sequence. n 35. 37. 39. 40. 1, 5, 9, 13,... 1, 9 100, 729 10, 81 1000 40 24 12,..., 36. 38. 25, 22, 19, 16,... 3, 9 4, 81 8,... 2, 27 1 2nn 1 1n 2,... 333202_0904.qxd 12/5/05 11:35 AM Page 682 682 Chapter 9 Sequences, Series, and Probability In Exercises 41–50, find the sum using the formulas for the sums of powers of integers. Model It (co n t i n u e d ) 41. 43. 45. 47. 49. 15 n n1 6 n1 n2 5 n1 n4 6 n1 n2 n 6 i1 6i 8i 3 42. 44. 46. 48. 50. 30 n n1 10 n1 n3 8 n1 n5 20 n1 n3 n 3 1 2 j 1 2 j 2 10 j1 (a) Find the first differences of the data shown in the table. (b) Use your results from part (a) to determine whether a linear model can be used to approximate the data. n If so, find a model algebraically. Let represent the year, with corresponding to 1998. n 8 (c) Use the regression feature of a graphing utility to find a linear model for the data. Compare this model with the one from part (b). (d) |
Use the models found in parts (b) and (c) to estimate the number of individual tax returns filed in 2008. How do these values compare? In Exercises 51–56, write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. Synthesis 51. 53. 55. a1 an a1 an a0 an 0 an1 3 an1 2 an1 2 3 n 2 52. 54. 56. a1 an a2 an a0 an 2 an1 3 2an1 0 an1 n In Exercises 57–60, find a quadratic model for the sequence with the indicated terms. 57. 58. 59. 60. a0 a0 a0 a0 3, a1 7, a1 3, a2 3, a2 3, a4 6, a3 1, a4 0, a6 15 10 9 36 Model It 61. Data Analysis: Tax Returns The table shows the (in millions) of individual tax returns filed in (Source: number the United States from 1998 to 2003. Internal Revenue Service) an Year 1998 1999 2000 2001 2002 2003 Number of returns, an 120.3 122.5 124.9 127.1 129.4 130.3 62. Writing In your own words, explain what is meant by a proof by mathematical induction. True or False? statement is true or false. Justify your answer. In Exercises 63–66, determine whether the 63. If the statement P1 P7 imply that the statement true for all positive integers is true but the true statement is true, then n. Pn P6 does not is not necessarily 64. If the statement Pk is true and Pk implies Pk1, then P1 is also true. 65. If the second differences of a sequence are all zero, then the sequence is arithmetic. 66. A sequence with n terms has n 1 second differences. Skills Review In Exercises 67–70, find the product. 67. 69. 2x2 12 5 4x3 68. 70. 2x y2 2x 4y3 In Exercises 71–74, (a) state the domain of the function, (b) identify all intercepts, (c) find any vertical and horizontal asymptotes, and (d) plot additional solution points as needed to |
sketch the graph of the rational function. 71. 72. 73. 74. f x x x 3 gx x2 x2 4 ht t 7 t f x 5 x 1 x 333202_0905.qxd 12/5/05 11:37 AM Page 683 9.5 The Binomial Theorem Section 9.5 The Binomial Theorem 683 What you should learn • Use the Binomial Theorem to calculate binomial coefficients. • Use Pascal’s Triangle to calculate binomial coefficients. • Use binomial coefficients to write binomial expansions. Why you should learn it You can use binomial coefficients to model and solve real-life problems. For instance, in Exercise 80 on page 690, you will use binomial coefficients to write the expansion of a model that represents the amounts of child support collected in the U. S. Binomial Coefficients Recall that a binomial is a polynomial that has two terms. In this section, you will study a formula that gives a quick method of raising a binomial to a power. To begin, look at the expansion of for several values of x yn n. x y0 1 x y1 x y x y2 x 2 2xy y 2 x y3 x3 3x 2y 3xy 2 y3 x y4 x4 4x 3y 6x 2y 2 4xy 3 y4 x y5 x 5 5x 4y 10x 3y 2 10x 2y 3 5xy4 y 5 There are several observations you can make about these expansions. n 1 terms. 1. In each expansion, there are 2. In each expansion, y x by 1 in successive terms, whereas the powers of and have symmetrical roles. The powers of decrease y increase by 1. x 3. The sum of the powers of each term is n. For instance, in the expansion of x y5, the sum of the powers of each term is 5. 4 1 5 3 2 5 x y5 x 5 5x4y1 10x3y 2 10x 2y 3 5x1y4 y 5 4. The coefficients increase and then decrease in a symmetric pattern. The coefficients of a binomial expansion are called binomial coefficients. To © Vince Streano/Corbis find them, you can use the Binomial Theorem. The Binomial Theorem In the expansion of x yn x yn x n nxn1y... xnr y |
r the coefficient of is nCr x n1y r... nxyn1 yn nCr. n! n r!r! n r The symbol is often used in place of nCr to denote binomial coefficients. For a proof of the Binomial Theorem, see Proofs in Mathematics on page 724. 333202_0905.qxd 12/5/05 11:37 AM Page 684 684 Chapter 9 Sequences, Series, and Probability Te c h n o l o g y Most graphing calculators are programmed to evaluate nC r. Consult the user’s guide for your calculator and then evaluate 8C5. You should get an answer of 56. Example 1 Finding Binomial Coefficients Find each binomial coefficient. b. 10 3 c. 7C0 d. 8 8 a. 8C2 Solution a. b. 8C2 10 3 c. 7C0 8! 6! 2! 10! 7! 3! 7! 7! 0! 1 8 7 2 1 8 7 6! 6! 2! 10 9 8 7! 7! 3! 8 8 d. 8! 0! 8! 1 28 10 9 8 3 2 1 120 Now try Exercise 1. r 0 r n, and When as in parts (a) and (b) above, there is a simple pattern for evaluating binomial coefficients that works because there will always be factorial terms that divide out from the expression. 2 factors 3 factors 8C2 8 7 2 1 and 10 3 10 9 8 3 2 1 2 factors 3 factors Example 2 Finding Binomial Coefficients Find each binomial coefficient. b. 7 4 c. 12C1 d. 12 11 a. 7C3 Solution a. b. c. d. 7 6 5 35 7C3 12 12 1 12! 1! 11! 12C1 12 11 35 12 11! 1! 11! 12 1 12 Now try Exercise 7. It is not a coincidence that the results in parts (a) and (b) of Example 2 are the same and that the results in parts (c) and (d) are the same. In general, it is true that nCr nCnr. This shows the symmetric property of binomial coefficients that was identified earlier. 333202_0905.qxd 12/5/05 11:37 AM Page 685 Exploration Complete the table and describe the result. n 9 7 12 6 10 r 5 1 |
4 0 7 nCr nCnr What characteristic of Pascal’s Triangle is illustrated by this table? Section 9.5 The Binomial Theorem 685 Pascal’s Triangle There is a convenient way to remember the pattern for binomial coefficients. By arranging the coefficients in a triangular pattern, you obtain the following array, which is called Pascal’s Triangle. This triangle is named after the famous French mathematician Blaise Pascal (1623–1662). 1 2 6 20 1 3 10 35 1 4 15 1 4 15 1 3 10 35 1 5 21 1 6 1 6 1 5 21 1 7 1 1 1 7 1 4 6 10 1 15 6 21 The first and last numbers in each row of Pascal’s Triangle are 1. Every other number in each row is formed by adding the two numbers immediately above the number. Pascal noticed that numbers in this triangle are precisely the same numbers that are the coefficients of binomial expansions, as follows. 0th row x y0 1 x y1 1x 1y x y2 1x 2 2xy 1y 2 x y3 1x3 3x 2y 3xy 2 1y3 x y4 1x4 4x3y 6x 2y 2 4xy 3 1y4 x y5 1x5 5x4y 10x3y 2 10x 2y 3 5xy4 1y 5 x y6 1x6 6x5y 15x4y 2 20x3y 3 15x 2y4 6xy5 1y6 x y7 1x7 7x 6y 21x5y 2 35x4y 3 35x3y4 21x 2y5 7xy6 1y7 3rd row 2nd row 1st row The top row in Pascal’s Triangle is called the zeroth row because it Similarly, the next row is corresponds to the binomial expansion called the first row because it corresponds to the binomial expansion x y1 1x 1y. In general, the nth row in Pascal’s Triangle gives the x yn. coefficients of x y0 1. Example 3 Using Pascal’s Triangle Use the seventh row of Pascal’s Triangle to find the binomial coefficients. 8C0, 8C1, 8C2, 8C3, 8C4, 8C5, 8C6, 8C7, 8C8 Solution 1 7 21 35 35 21 7 1 1 8C0 8 |
8C1 28 8C2 56 8C3 70 8C4 56 8C5 28 8C6 8 8C7 1 8C8 Now try Exercise 11. 333202_0905.qxd 12/5/05 11:37 AM Page 686 686 Chapter 9 Sequences, Series, and Probability Historical Note Precious Mirror “Pascal’s”Triangle and forms of the Binomial Theorem were known in Eastern cultures prior to the Western “discovery” of the theorem. A Chinese text entitled Precious Mirror contains a triangle of binomial expansions through the eighth power. Binomial Expansions As mentioned at the beginning of this section, when you write out the coefficients for a binomial that is raised to a power, you are expanding a binomial. The formulas for binomial coefficients give you an easy way to expand binomials, as demonstrated in the next four examples. Example 4 Expanding a Binomial Write the expansion for the expression x 13. Solution The binomial coefficients from the third row of Pascal’s Triangle are 1, 3, 3, 1. So, the expansion is as follows. x 13 1x3 3x 21 3x12 113 x3 3x 2 3x 1 Now try Exercise 15. To expand binomials representing differences rather than sums, you alternate signs. Here are two examples. x 13 x3 3x 2 3x 1 x 14 x4 4x3 6x 2 4x 1 Example 5 Expanding a Binomial Write the expansion for each expression. a. 2x 34 b. x 2y4 Solution The binomial coefficients from the fourth row of Pascal’s Triangle are 1, 4, 6, 4, 1. Therefore, the expansions are as follows. a. 2x 34 12x4 42x33 62x232 42x33 134 16x4 96x3 216x 2 216x 81 b. x 2y4 1x4 4x32y 6x22y2 4x2y3 12y4 x 4 8x3y 24x 2y2 32xy3 16y4 Now try Exercise 19. 333202_0905.qxd 12/5/05 11:37 AM Page 687 Section 9.5 The Binomial Theorem 687 Te c h n o l o g y You can use a graphing utility to check the expansion in Example 6. Graph the original binomial expression and the expansion in |
the same viewing window. The graphs should coincide as shown below. 200 Example 6 Expanding a Binomial Write the expansion for x2 43. Solution Use the third row of Pascal’s Triangle, as follows. x2 43 1x23 3x224 3x242 143 x 6 12x 4 48x2 64 Now try Exercise 29. −5 5 Sometimes you will need to find a specific term in a binomial expansion. Instead of writing out the entire expansion, you can use the fact that, from the Binomial Theorem, the r 1th term is nCr xnr yr. −100 Example 7 Finding a Term in a Binomial Expansion a 2b8. a. Find the sixth term of b. Find the coefficient of the term a6b5 in the expansion of 3a 2b11. Solution a. Remember that the formula is for the is one less than the number of the term you are looking for. So, to find the sixth term in this binomial expansion, use as shown. term, so n 8, r 5, r r 1th 8C5a 852b5 56 a3 2b5 x 3a, n 11, r 5, b. In this case, x a, and 5625a3b5 y 2b. and y 2b, 1792a3b5. Substitute these values to obtain nCr x nr y r 11C5 3a62b5 462729a632b5 10,777,536a6b5. So, the coefficient is 10,777,536. Now try Exercise 41. W RITING ABOUT MATHEMATICS Error Analysis You are a math instructor and receive the following solutions from one of your students on a quiz. Find the error(s) in each solution. Discuss ways that your student could avoid the error(s) in the future. a. Find the second term in the expansion of 2x 3y5. 52x43y 2 720x 4y 2 b. Find the fourth term in the expansion of 2 x27y4 9003.75x 2y 4 1 6C4 2 x 7y6 1. 333202_0905.qxd 12/5/05 11:37 AM Page 688 688 Chapter 9 Sequences, Series, and Probability 9.5 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The coefficients of a binomial |
expansion are called ________ ________. 2. To find binomial coefficients, you can use the ________ ________ or ________ ________. 3. The notation used to denote a binomial coefficient is ________ or ________. 4. When you write out the coefficients for a binomial that is raised to a power, you are ________ a ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–10, calculate the binomial coefficient. 1. 3. 5. 7. 9. 5C3 12C0 20C15 10 4 100 98 2. 4. 6. 8. 10. 8C6 20C20 12C5 10 6 100 2 In Exercises 11–14, evaluate using Pascal’s Triangle. 8 5 7C4 11. 13. 8 7 6C3 12. 14. In Exercises 15–34, use the Binomial Theorem to expand and simplify the expression. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 34. x 14 a 64 y 43 x y5 r 3s6 3a 4b5 2x y3 x 2 y24 y5 1 x 2x 34 5x 3 2 3x 15 4x 13 16. 18. 20. 22. 24. 26. 28. 30. 32. x 16 a 55 y 25 c d3 x 2y4 2x 5y5 7a b3 x 2 y 26 2y6 1 x In Exercises 35–38, expand the binomial by using Pascal’s Triangle to determine the coefficients. 35. 37. 2t s5 x 2y5 36. 38. 3 2z4 2v 36 In Exercises 39– 46, find the specified expansion of the binomial. n th term in the 39. 41. 43. 45. x y10, x 6y5, 4x 3y9, 10x 3y12, n 4 n 3 n 8 n 9 40. 42. 44. 46. x y6, x 10z7, 5a 6b5, 7x 2y15, n 7 n 4 n 5 n 7 In Exercises 47–54, find the coefficient a of the term in the expansion of the binomial. Binomial x 312 x 2 312 x 2y10 4 |
x y10 3x 2y9 2x 3y8 x 2 y10 z 2 t10 47. 48. 49. 50. 51. 52. 53. 54. Term ax5 ax8 ax8y 2 ax 2y8 ax4y5 ax 6y 2 ax8y 6 az4t8 In Exercises 55–58, use the Binomial Theorem to expand and simplify the expression. 55. 56. 57. 58. x 34 2t 13 x 23 y133 u35 25 In Exercises 59–62, expand the expression in the difference quotient and simplify. f x h f x h 59. f x x3 61. f x x Difference quotient 60. 62. f x x4 f x 1 x 333202_0905.qxd 12/5/05 11:37 AM Page 689 In Exercises 63–68, use the Binomial Theorem to expand the complex number. Simplify your result. 63. 65. 67. 1 i4 2 3i6 1 2 3 2 64. 66. 68. 2 i5 5 93 5 3 i4 i3 Section 9.5 The Binomial Theorem 689 78. To find the probability that the sales representative in Exercise 77 makes four sales if the probability of a sale with any one customer is evaluate the term 1 2, 1 2 41 2 4 8C4 in the expansion of 1 2 8. 1 2 Approximation In Exercises 69–72, use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 69, use the expansion 1.028 1 0.028 1 80.02 280.022.... 69. 71. 1.028 2.9912 70. 72. 2.00510 1.989 Model It 79. Data Analysis: Water Consumption The table f t shows the per capita consumption of bottled water (in gallons) in the United States from 1990 through (Source: Economic Research Service, U.S. 2003. Department of Agriculture) Year Consumption, f t Graphical Reasoning In Exercises 73 and 74, use a graphf ing utility to graph and in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function in standard form. g g 73. 74. f x x3 4x, f x x4 4x 2 1, gx f x 4 g |
x f x 3 n In Exercises 75–78, consider Probability independent trials of an experiment in which each trial has two possible outcomes: “success” or “failure.” The probability of a success on each trial is and the probability of a failure is q 1 p. in the successes expansion of in the n In this context, the term trials of the experiment. gives the probability of nC k p k q n k p qn p, k 75. A fair coin is tossed seven times. To find the probability of obtaining four heads, evaluate the term 1 2 4 1 2 3 7C4 in the expansion of 1 2 7. 1 2 1 4. 76. The probability of a baseball player getting a hit during any given time at bat is To find the probability that the player gets three hits during the next 10 times at bat, evaluate the term 1 33 7 10C3 4 4 in the expansion of 1 4 10. 3 4 1 3. 77. The probability of a sales representative making a sale with any one customer is The sales representative makes eight contacts a day. To find the probability of making four sales, evaluate the term 4 42 3 8C4 1 3 in the expansion of 1 3 8. 2 3 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 8.0 8.0 9.7 10.3 11.3 12.1 13.0 13.9 15.0 16.4 17.4 18.8 20.7 22.0 (a) Use the regression feature of a graphing utility to find a cubic model for the data. Let represent the year, with corresponding to 1990. t 0 t (b) Use a graphing utility to plot the data and the model in the same viewing window. (c) You want to adjust the model so that corresponds to 2000 rather than 1990. To do this, you 10 units to the left to obtain shift the graph of gt f t 10. in standard form. f Write gt t 0 (d) Use a graphing utility to graph g in the same viewing window as f. (e) Use both models to estimate the per capita consumption of bottled water in 2008. Do you obtain the same answer? (f) Describe the overall trend in the data. What factors do you think may have contributed to the increase in the per capita consumption of bottled water? 333202_0905.qxd 12/5/05 11 |
:37 AM Page 690 690 Chapter 9 Sequences, Series, and Probability 80. Child Support The amounts (in billions of dollars) of child support collected in the United States from 1990 to 2002 can be approximated by the model f t 0.031t 2 0.82t 6.1, 0 ≤ t ≤ 12 f t represents the year, with corresponding to (Source: U.S. Department of Health t 0 t where 1990 (see figure). and Human Services) t 92. The sum of the numbers in the th row of Pascal’s Triangle n 88. Graphical Reasoning Which two functions have identical graphs, and why? Use a graphing utility to graph the functions in the given order and in the same viewing window. Compare the graphs. (a) (b) (c) (d) (e) f x 1 x3 gx 1 x3 hx 1 3x 3x 2 x3 kx 1 3x 3x 2 x 3 px 1 3x 3x 2 x 3 Proof integers and where In Exercises 89–92, prove the property for all r 0 ≤ r ≤ n. n 89. 90. 91. nCr nC0 n1Cr nCn r nC1 nCr... ± nCn 0 nC2 nCr 1 is 2n. Skills Review In Exercises 93–96, the graph of and use the graph to write an equation for the graph of is shown. Graph f g. y gx f x x2 y 6 5 4 3 2 1 −1 1 32 4 5 6 952 −1 1 2 3 x x 94. f x x2 y 4 3 2 x 1 2 −3 −2 −1 −2 −3 96. f x x y 1 −1 1 2 3 4 x −4 −5 f(t ( 27 24 21 18 15 12 10 11 12 13 Year (0 ↔ 1990) (a) You want to adjust the model so that corresponds to 2000 rather than 1990. To do this, you shift the graph gt f t 10. f of Write 10 units to the left and obtain gt in standard form. t 0 (b) Use a graphing utility to graph f and g in the same 93. viewing window. (c) Use the graphs to estimate when the child support col- lections will exceed $30 billion. Synthesis True or False? statement is true or false. Just |
ify your answer. In Exercises 81– 83, determine whether the 81. The Binomial Theorem could be used to produce each row of Pascal’s Triangle. 82. A binomial that represents a difference cannot always be accurately expanded using the Binomial Theorem. 83. The x10 x2 312 -term and the x14 -term of the expansion of have identical coefficients. 84. Writing In your own words, explain how to form the rows of Pascal’s Triangle. 85. Form rows 8–10 of Pascal’s Triangle. 86. Think About It How many terms are in the expansion of x yn? 87. Think About It How do the expansions of x yn differ? x yn and In Exercises 97 and 98, find the inverse of the matrix. 97. 6 5 5 4 98. 1.2 2 2.3 4 333202_0906.qxd 12/5/05 11:39 AM Page 691 9.6 Counting Principles Section 9.6 Counting Principles 691 What you should learn • Solve simple counting problems. • Use the Fundamental Counting Principle to solve counting problems. • Use permutations to solve counting problems. • Use combinations to solve counting problems. Why you should learn it You can use counting principles to solve counting problems that occur in real life. For instance, in Exercise 65 on page 700, you are asked to use counting principles to determine the number of possible ways of selecting the winning numbers in the Powerball lottery. © Michael Simpson/FPG/Getty Images Simple Counting Problems This section and Section 9.7 present a brief introduction to some of the basic counting principles and their application to probability. In Section 9.7, you will see that much of probability has to do with counting the number of ways an event can occur. The following two examples describe simple counting problems. Example 1 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. Then, a second piece of paper is drawn from the box, and its number is written down. Finally, the two numbers are added together. How many different ways can a sum of 12 be obtained? Solution To solve this problem, count the different ways that a sum of 12 can be obtained using two numbers from 1 to 8. First number Second number From |
this list, you can see that a sum of 12 can occur in five different ways. Now try Exercise 5. Example 2 Selecting Pairs of Numbers at Random Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces of paper are drawn from the box at the same time, and the numbers on the pieces of paper are written down and totaled. How many different ways can a sum of 12 be obtained? Solution To solve this problem, count the different ways that a sum of 12 can be obtained using two different numbers from 1 to 8. First number Second number 4 8 5 7 7 5 8 4 So, a sum of 12 can be obtained in four different ways. Now try Exercise 7. The difference between the counting problems in Examples 1 and 2 can be described by saying that the random selection in Example 1 occurs with replacement, whereas the random selection in Example 2 occurs without replacement, which eliminates the possibility of choosing two 6’s. 333202_0906.qxd 12/5/05 11:39 AM Page 692 692 Chapter 9 Sequences, Series, and Probability The Fundamental Counting Principle Examples 1 and 2 describe simple counting problems in which you can list each possible way that an event can occur. When it is possible, this is always the best way to solve a counting problem. However, some events can occur in so many different ways that it is not feasible to write out the entire list. In such cases, you must rely on formulas and counting principles. The most important of these is the Fundamental Counting Principle. Fundamental Counting Principle E1 E1 be two events. The first event Let and E2 m2 ways. After can occur in has occurred, of ways that the two events can occur is m1 m2. E2 E1 m1 can occur in different ways. The number different The Fundamental Counting Principle can be extended to three or more can events. For instance, the number of ways that three events occur is E1, E2, m1 m2 m3. and E3 Example 3 Using the Fundamental Counting Principle How many different pairs of letters from the English alphabet are possible? Solution There are two events in this situation. The first event is the choice of the first letter, and the second event is the choice of the second letter. Because the English alphabet contains 26 letters, it follows that the number of two-letter pairs is 26 26 676. Now try Exercise 13. Example 4 Using the Fundamental Count |
ing Principle Telephone numbers in the United States currently have 10 digits. The first three are the area code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (Note that at this time, a local telephone number cannot begin with 0 or 1.) Solution Because the first digit of a local telephone number cannot be 0 or 1, there are only eight choices for the first digit. For each of the other six digits, there are 10 choices. Area Code Local Number 8 10 10 10 10 10 10 So, the number of local telephone numbers that are possible within each area code is 8 10 10 10 10 10 10 8,000,000. Now try Exercise 19. 333202_0906.qxd 12/5/05 11:39 AM Page 693 Section 9.6 Counting Principles 693 Permutations One important application of the Fundamental Counting Principle is in determining the number of ways that elements can be arranged (in order). An ordering of elements is called a permutation of the elements. n n Definition of Permutation A permutation of different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on. n Example 5 Finding the Number of Permutations of n Elements How many permutations are possible for the letters A, B, C, D, E, and F? Solution Consider the following reasoning. First position: Any of the six letters Second position: Any of the remaining five letters Third position: Any of the remaining four letters Fourth position: Any of the remaining three letters Fifth position: Any of the remaining two letters Sixth position: The one remaining letter So, the numbers of choices for the six positions are as follows. Permutations of six letters 6 5 4 3 2 1 The total number of permutations of the six letters is 6! 6 5 4 3 2 1 720. Now try Exercise 39. Number of Permutations of n Elements The number of permutations of elements is!. n In other words, there are n! different ways that elements can be ordered. n 333202_0906.qxd 12/5/05 11:39 AM Page 694 694 Chapter 9 Sequences, Series, and Probability Eleven thoroughbred racehorses hold the title of Triple Crown winner for winning the Kentucky Derby, the Preakness, and the Belmont Stakes in the same year. Forty-nine horses have won two out of the three races |
. Example 6 Counting Horse Race Finishes Eight horses are running in a race. In how many different ways can these horses come in first, second, and third? (Assume that there are no ties.) Solution Here are the different possibilities. Win (first position): Eight choices Place (second position): Seven choices Show (third position): Six choices Using the Fundamental Counting Principle, multiply these three numbers together to obtain the following. Different orders of horses 8 7 6 So, there are 8 7 6 336 different orders. Now try Exercise 43. It is useful, on occasion, to order a subset of a collection of elements rather than the entire collection. For example, you might want to choose and order r elements. Such an ordering is called a permutation of n elements taken r at a time. elements out of a collection of n Te c h n o l o g y Most graphing calculators are programmed to evaluate Consult the user’s guide for your calculator and then evaluate 8P5. You should get an answer of 6720. nPr. Permutations of n Elements Taken r at a Time The number of permutations of elements taken at a time is n r nPr n! n r! nn 1n 2... n r 1. Using this formula, you can rework Example 6 to find that the number of permutations of eight horses taken three at a time is 8P3 8! 8 3! 8! 5! 8 7 6 5! 5! 336 which is the same answer obtained in the example. 333202_0906.qxd 12/5/05 11:39 AM Page 695 Section 9.6 Counting Principles 695 Remember that for permutations, order is important. So, if you are looking at the possible permutations of the letters A, B, C, and D taken three at a time, the permutations (A, B, D) and (B, A, D) are counted as different because the order of the elements is different. Suppose, however, that you are asked to find the possible permutations of the letters A, A, B, and C. The total number of permutations of the four letters would be However, not all of these arrangements would be distinguishable because there are two A’s in the list. To find the number of distinguishable permutations, you can use the following formula. 4!. 4P4 Distinguishable Permutations Suppose a set of objects has n |
3 n n1 of a third kind, and so on, with n n1... nk. n2 n 3 of one kind of object, n2 of a second kind, Then the number of distinguishable permutations of the objects is n n! n1! n2! n3!... nk!. Example 7 Distinguishable Permutations In how many distinguishable ways can the letters in BANANA be written? Solution This word has six letters, of which three are A’s, two are N’s, and one is a B. So, the number of distinguishable ways the letters can be written is n! n1! n2! n3! 6! 3! 2! 1! 6 5 4 3! 3! 2! 60. The 60 different distinguishable permutations are as follows. AAABNN AANABN ABAANN ANAABN ANBAAN BAAANN BNAAAN NAABAN NABNAA NBANAA AAANBN AANANB ABANAN ANAANB ANBANA BAANAN BNAANA NAABNA NANAAB NBNAAA AAANNB AANBAN ABANNA ANABAN ANBNAA BAANNA BNANAA NAANAB NANABA NNAAAB AABANN AANBNA ABNAAN ANABNA ANNAAB BANAAN BNNAAA NAANBA NANBAA NNAABA AABNAN AANNAB ABNANA ANANAB ANNABA BANANA NAAABN NABAAN NBAAAN NNABAA AABNNA AANNBA ABNNAA ANANBA ANNBAA BANNAA NAAANB NABANA NBAANA NNBAAA Now try Exercise 45. 333202_0906.qxd 12/5/05 11:39 AM Page 696 696 Chapter 9 Sequences, Series, and Probability Combinations When you count the number of possible permutations of a set of elements, order is important. As a final topic in this section, you will look at a method of selecting subsets of a larger set in which order is not important. Such subsets are called combinations of n elements taken r at a time. For instance, the combinations A, B, C and B, A, C are equivalent because both sets contain the same three elements, and the order in which the |
elements are listed is not important. So, you would count only one of the two sets. A common example of how a combination occurs is a card game in which the player is free to reorder the cards after they have been dealt. Example 8 Combinations of n Elements Taken r at a Time In how many different ways can three letters be chosen from the letters A, B, C, D, and E? (The order of the three letters is not important.) Solution The following subsets represent the different combinations of three letters that can be chosen from the five letters. A, B, C A, B, E A, C, E B, C, D B, D, E A, B, D A, C, D A, D, E B, C, E C, D, E From this list, you can conclude that there are 10 different ways that three letters can be chosen from five letters. Now try Exercise 55. Combinations of n Elements Taken r at a Time The number of combinations of elements taken at a time is n r nCr n! n r!r! which is equivalent to nCr nPr r!. Note that the formula for is the same one given for binomial coefficients. To see how this formula is used, solve the counting problem in Example 8. In that problem, you are asked to find the number of combinations of five elements taken three at a time. So, and the number of combinations is n 5, r 3, nCr 5C3 5! 2!3! 2 5 4 3! 2 1 3! 10 which is the same answer obtained in Example 8. 333202_0906.qxd 12/5/05 11:39 AM Page 697 A A A A Example 9 Counting Card Hands Section 9.6 Counting Principles 697 2 3 4 5 6 7 8 9 10 10 10 10 J Q K FIGURE 9.7 cards Standard deck of playing A standard poker hand consists of five cards dealt from a deck of 52 (see Figure 9.7). How many different poker hands are possible? (After the cards are dealt, the player may reorder them, and so order is not important.) Solution You can find the number of different poker hands by using the formula for the number of combinations of 52 elements taken five at a time, as follows. 52C5 52! 52 5!5! 52! 47!5! 52 51 50 49 48 47! 5 4 3 2 1 47! 2 |
,598,960 Now try Exercise 63. Example 10 Forming a Team You are forming a 12-member swim team from 10 girls and 15 boys. The team must consist of five girls and seven boys. How many different 12-member teams are possible? Solution There are boys. By the Fundamental Counting Principal, there are choosing five girls and seven boys. ways of choosing five girls. The are 15C7 10C5 ways of choosing seven ways of 10C5 15C7 10C5 15C7 15! 8! 7! 10! 5! 5! 252 6435 1,621,620 So, there are 1,621,620 12-member swim teams possible. Now try Exercise 65. When solving problems involving counting principles, you need to be able to distinguish among the various counting principles in order to determine which is necessary to solve the problem correctly. To do this, ask yourself the following questions. 1. Is the order of the elements important? Permutation 2. Are the chosen elements a subset of a larger set in which order is not important? Combination 3. Does the problem involve two or more separate events? Fundamental Counting Principle 333202_0906.qxd 12/5/05 11:39 AM Page 698 698 Chapter 9 Sequences, Series, and Probability 9.6 Exercises VOCABULARY CHECK: Fill in the blanks. 1. The ________ ________ ________ states that if there are ways for one event to occur and ways m2 m1 m2 m1 ways for both events to occur. for a second event to occur, there are 2. An ordering of elements is called a ________ of the elements. n 3. The number of permutations of elements taken at a time is given by the formula ________. n r 4. The number of ________ ________ of objects is given by n n! n1!n2!n3!... nk!. 5. When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of elements taken at a time. n r PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. Random Selection In Exercises 1– 8, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. 1. An odd integer 3. A prime integer |
2. An even integer 4. An integer that is greater than 9 5. An integer that is divisible by 4 6. An integer that is divisible by 3 7. Two distinct integers whose sum is 9 8. Two distinct integers whose sum is 8 9. Entertainment Systems A customer can choose one of three amplifiers, one of two compact disc players, and one of five speaker models for an entertainment system. Determine the number of possible system configurations. 10. Job Applicants A college needs two additional faculty members: a chemist and a statistician. In how many ways can these positions be filled if there are five applicants for the chemistry position and three applicants for the statistics position? 11. Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of two mathematics courses, one of three science courses, and one of five courses from the social sciences and humanities. How many schedules are possible? 12. Aircraft Boarding Eight people are boarding an aircraft. Two have tickets for first class and board before those in the economy class. In how many ways can the eight people board the aircraft? 13. True-False Exam In how many ways can a six-question true-false exam be answered? (Assume that no questions are omitted.) 14. True-False Exam In how many ways can a 12-question true-false exam be answered? (Assume that no questions are omitted.) 15. License Plate Numbers In the state of Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct license plate numbers can be formed in Pennsylvania? 16. License Plate Numbers In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between “O” and “zero” and between “I” and “one,” the letters “O” and “I” are not used. How many distinct license plate numbers can be formed in this state? 17. Three-Digit Numbers How many three-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be a multiple of 5. (d) The number is at least 400. 18. Four-Digit Numbers How many four-digit numbers can be formed |
under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than 5000. (d) The leading digit cannot be zero and the number must be even. 19. Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 40, inclusive) is selected. How many different lock combinations are possible? 333202_0906.qxd 12/5/05 11:39 AM Page 699 20. Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 50, inclusive) is selected. How many different lock combinations are possible? 21. Concert Seats Four couples have reserved seats in a row for a concert. In how many different ways can they be seated if (a) there are no seating restrictions? (b) the two members of each couple wish to sit together? 22. Single File In how many orders can four girls and four boys walk through a doorway single file if (a) there are no restrictions? (b) the girls walk through before the boys? In Exercises 23–28, evaluate n Pr. 23. 25. 27. 4P4 8P3 5P4 24. 26. 28. 5 P5 20 P2 7P4 In Exercises 29 and 30, solve for n. 29. 14 nP3 n2P4 30. nP5 18 n2P4 In Exercises 31–36, evaluate using a graphing utility. 31. 33. 35. 20 P5 100 P3 20C5 32. 34. 36. 100 P5 10 P8 10C7 37. Posing for a Photograph In how many ways can five children posing for a photograph line up in a row? 38. Riding in a Car In how many ways can six people sit in a six-passenger car? 39. Choosing Officers From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled? 40. Assembly Line Production There are four processes involved in assembling a product, and these processes can be performed in any order. The management wants to test each order to determine which is the least time-consuming. How many different orders will have to be tested? In Exercises 41– |
44, find the number of distinguishable permutations of the group of letters. 41. A, A, G, E, E, E, M 42. B, B, B, T, T, T, T, T 43. A, L, G, E, B, R, A 44. M, I, S, S, I, S, S, I, P, P, I 45. Write all permutations of the letters A, B, C, and D. 46. Write all permutations of the letters A, B, C, and D if the letters B and C must remain between the letters A and D. Section 9.6 Counting Principles 699 47. Batting Order A baseball coach is creating a nine-player batting order by selecting from a team of 15 players. How many different batting orders are possible? 48. Athletics Six sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet. In how many ways can the sprinters come in first, second, and third? (Assume there are no ties.) 49. Jury Selection From a group of 40 people, a jury of 12 people is to be selected. In how many different ways can the jury be selected? 50. Committee Members As of January 2005, the U.S. Senate Committee on Indian Affairs had 14 members. Assuming party affiliation was not a factor in selection, how many different committees were possible from the 100 U.S. senators? 51. Write all possible selections of two letters that can be formed from the letters A, B, C, D, E, and F. (The order of the two letters is not important.) 52. Forming an Experimental Group In order to conduct an experiment, five students are randomly selected from a class of 20. How many different groups of five students are possible? 53. Lottery Choices In the Massachusetts Mass Cash game, a player chooses five distinct numbers from 1 to 35. In how many ways can a player select the five numbers? 54. Lottery Choices In the Louisiana Lotto game, a player chooses six distinct numbers from 1 to 40. In how many ways can a player select the six numbers? 55. Defective Units A shipment of 10 microwave ovens contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units? 56 |
. Interpersonal Relationships The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3, (b) 8, (c) 12, and (d) 20. 57. Poker Hand You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.) 58. Job Applicants A toy manufacturer interviews eight people for four openings in the research and development department of the company. Three of the eight people are women. If all eight are qualified, in how many ways can the employer fill the four positions if (a) the selection is random and (b) exactly two selections are women? 333202_0906.qxd 12/5/05 11:39 AM Page 700 700 Chapter 9 Sequences, Series, and Probability 59. Forming a Committee A six-member research committee at a local college is to be formed having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 20 students in contention for the committee. How many six-member committees are possible? 60. Law Enforcement A police department uses computer imaging to create digital photographs of alleged perpetrators from eyewitness accounts. One software package contains 195 hairlines, 99 sets of eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheek structures. (a) Find the possible number of different faces that the software could create. (b) A eyewitness can clearly recall the hairline and eyes and eyebrows of a suspect. How many different faces can be produced with this information? Geometry In Exercises 61–64, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) (d) Number of two-scoop ice cream cones created from 31 different flavors Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 67 and 68, determine whether 67. The number of letter pairs that can be formed in any order from any of the first 13 letters in |
the alphabet (A–M) is an example of a permutation. 68. The number of permutations of n elements can be deter- mined by using the Fundamental Counting Principle. 69. What is the relationship between nCr and nCnr? 70. Without calculating the numbers, determine which of the following is greater. Explain. (a) The number of combinations of 10 elements taken six at a time (b) The number of permutations of 10 elements taken six at a time 61. Pentagon 63. Octagon 62. Hexagon 64. Decagon (10 sides) Proof In Exercises 71– 74, prove the identity. Model It 65. Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in 27 states, Washington D.C., and the U.S. Virgin Islands. The game is played by drawing five white balls out of a drum of 53 white balls (numbered 1–53) and one red powerball out of a drum of 42 red balls (numbered 1– 42). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of 53 balls. 66. Permutations or Combinations? Decide whether each scenario should be counted using permutations or combinations. Explain your reasoning. (a) Number of ways 10 people can line up in a row for con- cert tickets 71. n Pn 1 n Pn 73. nCn 1 nC1 72. nCn 74. nCr nC0 n Pr r! 75. Think About It Can your calculator evaluate 100P80? If not, explain why. 76. Writing Explain in words the meaning of n Pr. Skills Review In Exercises 77– 80, evaluate the function at each specified value of the independent variable and simplify. 77. 78. 79. (b) f 0 f 3 f x 3x2 8 (a) gx x 3 2 g7 (a) f x x 5 6 (a) f 5 g3 (b) (b) f 1 80. f x x2 2x |
5, x2 2, (b) f 1 (a) f 4 (c) f 5 (c) gx 1 (c) x ≤ 4 x > 4 (c) f 11 f 20 In Exercises 81– 84, solve the equation. Round your answer to two decimal places, if necessary. (b) Number of different arrangements of three types of flowers from an array of 20 types (c) Number of three-digit pin numbers for a debit card 81. x 3 x 6 83. log2 x 3 5 82. 4 t 3 2t 1 84. e x3 16 333202_0907.qxd 12/5/05 11:41 AM Page 701 9.7 Probability What you should learn • Find the probabilities of events. • Find the probabilities of mutually exclusive events. • Find the probabilities of independent events. • Find the probability of the complement of an event. Why you should learn it Probability applies to many games of chance. For instance, in Exercise 55, on page 712, you will calculate probabilities that relate to the game of roulette. Section 9.7 Probability 701 The Probability of an Event Any happening for which the result is uncertain is called an experiment. The possible results of the experiment are outcomes, the set of all possible outcomes of the experiment is the sample space of the experiment, and any subcollection of a sample space is an event. For instance, when a six-sided die is tossed, the sample space can be represented by the numbers 1 through 6. For this experiment, each of the outcomes is equally likely. To describe sample spaces in such a way that each outcome is equally likely, you must sometimes distinguish between or among various outcomes in ways that appear artificial. Example 1 illustrates such a situation. Example 1 Finding a Sample Space Find the sample space for each of the following. a. One coin is tossed. b. Two coins are tossed. c. Three coins are tossed. Solution a. Because the coin will land either heads up (denoted by H ) or tails up (denoted Hank de Lespinasse/The Image Bank by ), the sample space is T S H, T. b. Because either coin can land heads up or tails up, the possible outcomes are as follows. HH HT TH T T heads up on both coins heads up on first coin and tails up on second coin tails up on first coin and heads up on second coin tails up on both coins So, the sample space is S |
HH, HT, TH, TT. Note that this list distinguishes between the two cases though these two outcomes appear to be similar. HT and TH, even c. Following the notation of part (b), the sample space is S HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Note that this list distinguishes among the cases HHT, HTH, and THH, and among the cases HTT, THT, and TTH. Now try Exercise 1. 333202_0907.qxd 12/5/05 11:41 AM Page 702 702 Chapter 9 Sequences, Series, and Probability Exploration Toss two coins 100 times and write down the number of heads that occur on each toss (0, 1, or 2). How many times did two heads occur? How many times would you expect two heads to occur if you did the experiment 1000 times? Increasing likelihood of occurrence 0.0 0.5 1.0 Impossible event (cannot occur) The occurrence of the event is just as likely as it is unlikely. Certain event (must occur) FIGURE 9.8 You can write a probability as a fraction, decimal, or percent. For instance, in Example 2(a), the probability of getting two heads can be written as or 25%. 1 4, 0.25, To calculate the probability of an event, count the number of outcomes in the is denoted by nS. The event and in the sample space. The number of outcomes in event nE, E probability that event will occur is given by and the number of outcomes in the sample space S nEnS. is denoted by E The Probability of an Event If an event has nS nE E equally likely outcomes, the probability of event PE nE nS. E is equally likely outcomes and its sample space has S Because the number of outcomes in an event must be less than or equal to the number of outcomes in the sample space, the probability of an event must be a number between 0 and 1. That is, 0 ≤ PE ≤ 1 as indicated in Figure 9.8. If impossible event. If event. PE 1, PE 0, event E E event must occur, and cannot occur, and E is called an is called a certain E Example 2 Finding the Probability of an Event a. Two coins are tossed. What is the probability that both land heads up? b. A card is drawn from a standard deck of playing |
cards. What is the probability that it is an ace? Solution a. Following the procedure in Example 1(b), let E HH and S HH, HT, TH, TT. The probability of getting two heads is PE nE nS 1 4. b. Because there are 52 cards in a standard deck of playing cards and there are four aces (one in each suit), the probability of drawing an ace is PE nE nS 4 52 1 13. Now try Exercise 11. 333202_0907.qxd 12/5/05 11:41 AM Page 703 Section 9.7 Probability 703 Example 3 Finding the Probability of an Event Two six-sided dice are tossed. What is the probability that the total of the two dice is 7? (See Figure 9.9.) Solution Because there are six possible outcomes on each die, you can use the Fundamental Counting Principle to conclude that there are or 36 different outcomes when two dice are tossed. To find the probability of rolling a total of 7, you must first count the number of ways in which this can occur. 6 6 FIGURE 9.9 First die Second die You could have written out each sample space in Examples 2 and 3 and simply counted the outcomes in the desired events. For larger sample spaces, however, you should use the counting principles discussed in Section 9.6. So, a total of 7 can be rolled in six ways, which means that the probability of rolling a 7 is PE nE nS 6 36 1 6. Now try Exercise 15. Example 4 Finding the Probability of an Event Twelve-sided dice, as shown in Figure 9.10, can be constructed (in the shape of regular dodecahedrons) such that each of the numbers from 1 to 6 appears twice on each die. Prove that these dice can be used in any game requiring ordinary six-sided dice without changing the probabilities of different outcomes. Solution For an ordinary six-sided die, each of the numbers 1, 2, 3, 4, 5, and 6 occurs only once, so the probability of any particular number coming up is PE nE nS 1 6. For one of the 12-sided dice, each number occurs twice, so the probability of any particular number coming up is FIGURE 9.10 Now try Exercise 17. PE nE nS 2 12 1 6. 333202_0907.qxd 12/5/05 11:41 AM Page 704 704 Chapter 9 Sequences |
, Series, and Probability Example 5 The Probability of Winning a Lottery In the Arizona state lottery, a player chooses six different numbers from 1 to 41. If these six numbers match the six numbers drawn (in any order) by the lottery commission, the player wins (or shares) the top prize. What is the probability of winning the top prize if the player buys one ticket? Solution To find the number of elements in the sample space, use the formula for the number of combinations of 41 elements taken six at a time. nS 41C6 41 40 39 38 37 36 6 5 4 3 2 1 4,496,388 If a person buys only one ticket, the probability of winning is PE nE nS 1 4,496,388. Now try Exercise 21. Example 6 Random Selection The numbers of colleges and universities in various regions of the United States in 2003 are shown in Figure 9.11. One institution is selected at random. What is the probability that the institution is in one of the three southern regions? (Source: National Center for Education Statistics) Solution From the figure, the total number of colleges and universities is 4163. Because there are colleges and universities in the three southern regions, the probability that the institution is in one of these regions is 700 284 386 1370 PE nE nS 1370 4163 0.329. Mountain 274 West North Central 441 East North Central 630 Pacific 563 New England 261 Middle Atlantic 624 South Atlantic 700 West South Central 386 East South Central 284 FIGURE 9.11 Now try Exercise 33. 333202_0907.qxd 12/5/05 11:41 AM Page 705 Mutually Exclusive Events Section 9.7 Probability 705 B A and (from the same sample space) are mutually exclusive if A Two events B and have no outcomes in common. In the terminology of sets, the intersection and of PA B 0. is the empty set, which is written as B A B For instance, if two dice are tossed, the event of rolling a total of 6 and the event of rolling a total of 9 are mutually exclusive. To find the probability that one or the other of two mutually exclusive events will occur, you can add their individual probabilities. A Probability of the Union of Two Events B If occurring is given by and A are events in the same sample space, the probability of or A B PA B PA PB PA B. If A B are mutually exclusive, then and PA B PA PB. Example 7 The |
Probability of a Union of Events One card is selected from a standard deck of 52 playing cards. What is the probability that the card is either a heart or a face card? Hearts Solution Because the deck has 13 hearts, the probability of selecting a heart (event A ) is PA 13 52. Similarly, because the deck has 12 face cards, the probability of selecting a face B card (event ) is PB 12 52. Because three of the cards are hearts and face cards (see Figure 9.12), it follows that PA B 3 52. Finally, applying the formula for the probability of the union of two events, you can conclude that the probability of selecting a heart or a face card is PA B PA PB PA B 13 52 12 52 3 52 22 52 0.423. Now try Exercise 45. 2♥ 4♥ 6♥ 8♥ A♥ 3♥ 5♥ 7♥ 9♥ 10♥ Face cards FIGURE 9.12 n(A ∩ B) = 3 K♣ K♥ Q♥ J♥ Q♣ K♦ J♣ J♦ Q♦ K♠ Q♠ J♠ 333202_0907.qxd 12/5/05 11:41 AM Page 706 706 Chapter 9 Sequences, Series, and Probability Example 8 Probability of Mutually Exclusive Events The personnel department of a company has compiled data on the numbers of employees who have been with the company for various periods of time. The results are shown in the table. Years of service Number of employees 0–4 5–9 10–14 15–19 20–24 25–29 30–34 35–39 40–44 157 89 74 63 42 38 37 21 8 If an employee is chosen at random, what is the probability that the employee has (a) 4 or fewer years of service and (b) 9 or fewer years of service? Solution a. To begin, add the number of employees to find that the total is 529. Next, let represent choosing an employee with 0 to 4 years of service. Then the event probability of choosing an employee who has 4 or fewer years of service is A PA 157 529 0.297. b. Let event represent choosing an employee with 5 to 9 years of service. Then B PB 89 529. Because event can conclude that these two events are mutually exclusive and that from part (a) and event have no outcomes in common, you A B 89 529 PA B PA PB 157 529 246 529 0.465. So, the probability |
of choosing an employee who has 9 or fewer years of service is about 0.465. Now try Exercise 47. 333202_0907.qxd 12/5/05 11:41 AM Page 707 Section 9.7 Probability 707 Independent Events Two events are independent if the occurrence of one has no effect on the occurrence of the other. For instance, rolling a total of 12 with two six-sided dice has no effect on the outcome of future rolls of the dice. To find the probability that two independent events will occur, multiply the probabilities of each. Probability of Independent Events A and If occur is B are independent events, the probability that both A B and will PA and B PA PB. Example 9 Probability of Independent Events A random number generator on a computer selects three integers from 1 to 20. What is the probability that all three numbers are less than or equal to 5? Solution The probability of selecting a number from 1 to 5 is PA 5 20 1 4. So, the probability that all three numbers are less than or equal to 5 is 1 4 1 4 PA PA PA 1 4 1 64. Now try Exercise 48. Example 10 Probability of Independent Events In 2004, approximately 20% of the adult population of the United States got their news from the Internet every day. In a survey, 10 people were chosen at random from the adult population. What is the probability that all 10 got their news from the Internet every day? (Source: The Gallup Poll) Solution A represent choosing an adult who gets the news from the Internet every day. Let The probability of choosing an adult who got his or her news from the Internet every day is 0.20, the probability of choosing a second adult who got his or her news from the Internet every day is 0.20, and so on. Because these events are independent, you can conclude that the probability that all 10 people got their news from the Internet every day is PA10 0.2010 0.0000001. Now try Exercise 49. 333202_0907.qxd 12/5/05 11:41 AM Page 708 708 Chapter 9 Sequences, Series, and Probability Exploration You are in a class with 22 other people. What is the probability that at least two out of the 23 people will have a birthday on the same day of the year? The complement of the probability that at least two people have the same birthday is the probability that all 23 birthdays are different. So, first find the probability that all 23 |
people have different birthdays and then find the complement. Now, determine the proba- bility that in a room with 50 people at least two people have the same birthday. The Complement of an Event The complement of an event space that are not in PA A 1 PA PA 1. The complement of event A A So, the probability of is the collection of all outcomes in the sample Because is denoted by are mutually exclusive, it follows that A A. and because and A. or is A A PA 1 PA. For instance, if the probability of winning a certain game is PA 1 4 the probability of losing the game is PA 1 1 4 3 4. Probability of a Complement A Let be an event and let PA, the probability of the complement is PA 1 PA. A be its complement. If the probability of A is Example 11 Finding the Probability of a Complement A manufacturer has determined that a machine averages one faulty unit for every 1000 it produces. What is the probability that an order of 200 units will have one or more faulty units? Solution To solve this problem as stated, you would need to find the probabilities of having exactly one faulty unit, exactly two faulty units, exactly three faulty units, and so on. However, using complements, you can simply find the probability that all units are perfect and then subtract this value from 1. Because the probability that any given unit is perfect is 999/1000, the probability that all 200 units are perfect is 200 PA 999 1000 0.819. So, the probability that at least one unit is faulty is PA 1 PA 1 0.819. 0.181 Now try Exercise 51. 333202_0907.qxd 12/5/05 11:41 AM Page 709 Section 9.7 Probability 709 9.7 Exercises VOCABULARY CHECK: In Exercises 1–7, fill in the blanks. 1. An ________ is an event whose result is uncertain, and the possible results of the event are called ________. 2. The set of all possible outcomes of an experiment is called the ________ ________. 3. To determine the ________ of an event, you can use the formula where nE is the number of outcomes in the event and nS PE nE nS, is the number of outcomes in the sample space. 4. If PE 0, then E is an ________ event, and if PE 1, then E is a ________ event |
. 5. If two events from the same sample space have no outcomes in common, then the two events are ________ ________. 6. If the occurrence of one event has no effect on the occurrence of a second event, then the events are ________. 7. The ________ of an event A is the collection of all outcomes in the sample space that are not in A. 8. Match the probability formula with the correct probability name. (a) Probability of the union of two events (b) Probability of mutually exclusive events (c) Probability of independent events (d) Probability of a complement (i) (ii) (ii) (iv) PA B PA PB PA 1 PA PA B PA PB PA B PA and B PA PB PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, determine the sample space for the experiment. 1. A coin and a six-sided die are tossed. 2. A six-sided die is tossed twice and the sum of the points is recorded. Drawing a Card In Exercises 11–14, find the probability for the experiment of selecting one card from a standard deck of 52 playing cards. 11. The card is a face card. 12. The card is not a face card. 3. A taste tester has to rank three varieties of yogurt, A, B, 13. The card is a red face card. and C, according to preference. 4. Two marbles are selected from a bag containing two red marbles, two blue marbles, and one yellow marble. The color of each marble is recorded. 5. Two county supervisors are selected from five supervisors, A, B, C, D, and E, to study a recycling plan. 6. A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by S) or there may be no sale (denote by F). 14. The card is a 6 or lower. (Aces are low.) Tossing a Die the experiment of tossing a six-sided die twice. In Exercises 15–20, find the probability for 15. The sum is 4. 16. The sum is at least 7. 17. The sum is less than 11. 18. The sum is 2, 3, or 12. 19. The sum is odd and no |
more than 7. 20. The sum is odd or prime. Tossing a Coin In Exercises 7–10, find the probability for the experiment of tossing a coin three times. Use S {HHH, HHT, HTH, H T T, THH, TH T, the sample space T TH, T T T}. Drawing Marbles In Exercises 21–24, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and three red marbles. 7. The probability of getting exactly one tail 21. Both marbles are red. 8. The probability of getting a head on the first toss 22. Both marbles are yellow. 9. The probability of getting at least one head 23. Neither marble is yellow. 10. The probability of getting at least two heads 24. The marbles are of different colors. 333202_0907.qxd 12/5/05 11:41 AM Page 710 710 Chapter 9 Sequences, Series, and Probability In Exercises 25–28, you are given the probability that an event will happen. Find the probability that the event will not happen. A person is selected at random from the sample. Find the probability that the described person is selected. (a) A person who doesn’t favor the amendment 26. PE 0.36 (b) A Republican 25. 27. 28. PE 0.7 PE 1 4 PE 2 3 In Exercises 29–32, you are given the probability that an event will not happen. Find the probability that the event will happen. 29. 30. 31. 32. PE 0.14 PE 0.92 PE 17 35 PE 61 100 33. Data Analysis A study of the effectiveness of a flu vaccine was conducted with a sample of 500 people. Some participants in the study were given no vaccine, some were given one injection, and some were given two injections. The results of the study are listed in the table. No vaccine One injection Two injections Total Flu No flu Total 7 149 156 2 52 54 13 277 290 22 478 500 A person is selected at random from the sample. Find the specified probability. (a) The person had two injections. (b) The person did not get the flu. (c) The person got the flu and had one injection. 34. Data Analysis One hundred college students were interviewed to determine their political party affiliations and whether they favored a balanced- |
budget amendment to the Constitution. The results of the study are listed in the table, represents Republican. where represents Democrat and D R Favor Not Favor Unsure Total D R Total 23 32 55 25 9 34 7 4 11 55 45 100 (c) A Democrat who favors the amendment 35. Graphical Reasoning The figure shows the results of a recent survey in which 1011 adults were asked to grade U.S. public schools. (Source: Phi Delta Kappa/Gallup Poll) Grading Public Schools A 2% Don’t know 7% D 12% C 52% B 24% Fail 3% (a) Estimate the number of adults who gave U.S. public schools a B. (b) An adult is selected at random. What is the probabilty that the adult will give the U.S. public schools an A? (c) An adult is selected at random. What is the probabilty the adult will give the U.S. public schools a C or a D? 36. Graphical Reasoning The figure shows the results of a survey in which auto racing fans listed their favorite type of racing. (Source: ESPN Sports Poll/TNS Sports) Favorite Type of Racing NHRA drag racing 13% Motorcycle 11% Other 11% Formula One 6% NASCAR 59% (a) What is the probability that an auto racing fan selected at random lists NASCAR racing as his or her favorite type of racing? (b) What is the probability that an auto racing fan selected at random lists Formula One or motorcycle racing as his or her favorite type of racing? (c) What is the probability that an auto racing fan selected at random does not list NHRA drag racing as his or her favorite type of racing? 333202_0907.qxd 12/5/05 11:41 AM Page 711 Section 9.7 Probability 711 44. Card Game The deck of a card game is made up of 108 cards. Twenty-five each are red, yellow, blue, and green, and eight are wild cards. Each player is randomly dealt a seven-card hand. (a) What is the probability that a hand will contain exactly two wild cards? (b) What is the probability that a hand will contain two wild cards, two red cards, and three blue cards? 45. Drawing a Card One card is selected at random from an ordinary deck of 52 playing cards. Find the probabilities that (a) the card is an even-numbered card, ( |
b) the card is a heart or a diamond, and (c) the card is a nine or a face card. 46. Poker Hand Five cards are drawn from an ordinary deck of 52 playing cards. What is the probability that the hand drawn is a full house? (A full house is a hand that consists of two of one kind and three of another kind.) 47. Defective Units A shipment of 12 microwave ovens contains three defective units. A vending company has ordered four of these units, and because each is identically packaged, the selection will be random. What are the probabilities that (a) all four units are good, (b) exactly two units are good, and (c) at least two units are good? 48. Random Number Generator Two integers from 1 through 40 are chosen by a random number generator. What are the probabilities that (a) the numbers are both even, (b) one number is even and one is odd, (c) both numbers are less than 30, and (d) the same number is chosen twice? 49. Flexible Work Hours In a survey, people were asked if they would prefer to work flexible hours—even if it meant slower career advancement—so they could spend more time with their families. The results of the survey are shown in the figure. Three people from the survey were chosen at random. What is the probability that all three people would prefer flexible work hours? Flexible Work Hours Flexible hours 78% Don’t know 9% Rigid hours 13% 37. Alumni Association A college sends a survey to selected members of the class of 2006. Of the 1254 people who graduated that year, 672 are women, of whom 124 went on to graduate school. Of the 582 male graduates, 198 went on to graduate school. An alumni member is selected at random. What are the probabilities that the person is (a) female, (b) male, and (c) female and did not attend graduate school? 38. Education In a high school graduating class of 202 students, 95 are on the honor roll. Of these, 71 are going on to college, and of the other 107 students, 53 are going on to college. A student is selected at random from the class. What are the probabilities that the person chosen is (a) going to college, (b) not going to college, and (c) on the honor roll, but not going to college? 39. Winning an Election Taylor, Moore, and |
Jenkins are candidates for public office. It is estimated that Moore and Jenkins have about the same probability of winning, and Taylor is believed to be twice as likely to win as either of the others. Find the probability of each candidate winning the election. 40. Winning an Election Three people have been nominated for president of a class. From a poll, it is estimated that the first candidate has a 37% chance of winning and the second candidate has a 44% chance of winning. What is the probability that the third candidate will win? In Exercises 41–52, the sample spaces are large and you should use the counting principles discussed in Section 9.6. 41. Preparing for a Test A class is given a list of 20 study problems, from which 10 will be part of an upcoming exam. A student knows how to solve 15 of the problems. Find the probabilities that the student will be able to answer (a) all 10 questions on the exam, (b) exactly eight questions on the exam, and (c) at least nine questions on the exam. 42. Payroll Mix-Up Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What are the probabilities that (a) exactly one paycheck will be inserted in the correct envelope and (b) at least one paycheck will be inserted in the correct envelope? 43. Game Show On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, you win the car. What is the probability of winning, given the following conditions? (a) You guess the position of each digit. (b) You know the first digit and guess the positions of the other digits. 333202_0907.qxd 12/5/05 11:41 AM Page 712 712 Chapter 9 Sequences, Series, and Probability 50. Consumer Awareness Suppose that the methods used by shoppers to pay for merchandise are as shown in the circle graph. Two shoppers are chosen at random. What is the probability that both shoppers paid for their purchases only in cash? How Shoppers Pay for Merchandise Mostly credit 7% Mostly cash 27% Half cash, half credit 30% Only credit 4% Only cash 32% 51. Backup System A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What are the probabilities that during a given |
flight (a) both systems function satisfactorily, functions (b) at satisfactorily, and (c) both systems fail? least one system 52. Backup Vehicle A fire company keeps two rescue vehicles. Because of the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is 90%. The availability of one vehicle is independent of the availability of the other. Find the probabilities that (a) both vehicles are available at a given time, (b) neither vehicle is available at a given time, and (c) at least one vehicle is available at a given time. 53. A Boy or a Girl? Assume that the probability of the birth of a child of a particular sex is 50%. In a family with four children, what are the probabilities that (a) all the children are boys, (b) all the children are the same sex, and (c) there is at least one boy? 54. Geometry You and a friend agree to meet at your favorite fast-food restaurant between 5:00 and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is the probability that the two of you will actually meet, assuming that your arrival times are random within the hour ( You meet You meet You don’t meet 60 45 30 15 Y ou arrive first Y our friend arrives first 45 30 15 Your arrival time (in minutes past 5:00 P.M.) 60 Model It 55. Roulette American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 1–36, of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. 10 25 29 12 8 2 7 00 1 19 1 3 13 36 20 35 14 2 0 8 2 11 30 26 9 (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a |
red pocket on three consecutive spins. (f) European roulette does not contain the 00 pocket. Repeat parts (a)–(e) for European roulette. How do the probabilities for European roulette compare with the probabilities for American roulette? 333202_0907.qxd 12/5/05 11:41 AM Page 713 56. Estimating is dropped onto a paper that contains a grid of squares units on a side (see figure). d A coin of diameter d (a) Find the probability that the coin covers a vertex of one of the squares on the grid. (b) Perform the experiment 100 times and use the results to approximate. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether 57. If A B and A ties, then are independent events with nonzero probabilican occur when occurs. B 58. Rolling a number less than 3 on a normal six-sided die has a probability of. The complement of this event is to roll a number greater than 3, and its probability is 1 3 1 2. 59. Pattern Recognition and Exploration Consider a group of people. n (a) Explain why the following pattern gives the probabili- ties that the people have distinct birthdays. n n 2: n 3: 365 365 365 365 364 365 364 365 365 364 3652 365 364 363 3653 363 365 (b) Use the pattern in part (a) to write an expression for the people have distinct birthdays. probability that n 4 (c) Let Pn be the probability that the people have distinct birthdays. Verify that this probability can be obtained recursively by n P1 1 and Pn 365 n 1 365 Pn1. (d) Explain why gives the probability that at least two people in a group of people have the same birthday. Qn 1 Pn n Section 9.7 Probability 713 (e) Use the results of parts (c) and (d) to complete the table. n Pn Qn 10 15 20 23 30 40 50 (f) How many people must be in a group so that the probability of at least two of them having the same birthday is greater than Explain. 1 2? 60. Think About It A weather forecast indicates that the probability of rain is 40%. What does this mean? Skills Review In Exercises 61–70, find all real solutions of the equation. 61. 62. 63 |
. 64. 65. 66. 67. 68. 69. 70. 6x2 8 0 4x2 6x 12 0 x 3 x2 3x 0 x 5 x 3 2x 0 12 x 3 32 x 2x 2 x 5 3 2x 3 3 x 2 2 x 4 1 2x 3 1 4 x x 2 5 x 2 13 x2 2x In Exercises 71–74, sketch the graph of the solution set of the system of inequalities. 71. 72 5x 2y ≥ 10 x2 y ≥ 2 y ≥ x 4 74. x2 y2 ≤ 4 x y ≥ 2 73. 333202_090R.qxd 12/5/05 11:43 AM Page 714 714 Chapter 9 Sequences, Series, and Probability 9 Chapter Summary What did you learn? Section 9.1 Use sequence notation to write the terms of sequences (p. 642). Use factorial notation (p. 644). Use summation notation to write sums (p. 646). Find the sums of infinite series (p. 647). Use sequences and series to model and solve real-life problems (p. 648). Section 9.2 Recognize, write, and find the nth terms of arithmetic sequences (p. 653). Find nth partial sums of arithmetic sequences (p. 656). Use arithmetic sequences to model and solve real-life problems (p. 657). Section 9.3 Recognize, write, and find the nth terms of geometric sequences (p. 663). Find nth partial sums of geometric sequences (p. 666). Find sums of infinite geometric series (p. 667). Use geometric sequences to model and solve real-life problems (p. 668). Section 9.4 Use mathematical induction to prove statements involving a positive integer n (p. 673). Recognize patterns and write the nth term of a sequence (p. 677). Find the sums of powers of integers (p. 679). Find finite differences of sequences (p. 680). Section 9.5 Use the Binomial Theorem to calculate binomial coefficients (p. 683). Use Pascal’s Triangle to calculate binomial coefficients (p. 685). Use binomial coefficients to write binomial expansions (p. 686). Section 9.6 Solve simple counting problems (p. 691). Use the Fundamental Counting Principle to solve counting problems (p. 692). Use |
permutations to solve counting problems (p. 693). Use combinations to solve counting problems (p. 696). Section 9.7 Find the probabilities of events (p. 701). Find the probabilities of mutually exclusive events (p. 705). Find the probabilities of independent events (p. 707). Find the probability of the complement of an event (p. 708). Review Exercises 1–8 9–12 13–20 21–24 25, 26 27–40 41–46 47, 48 49–60 61–70 71–76 77, 78 79–82 83–86 87–90 91–94 95–98 99–102 103–108 109, 110 111, 112 113, 114 115, 116 117, 118 119, 120 121, 122 123, 124 333202_090R.qxd 12/5/05 11:43 AM Page 715 9 Review Exercises Review Exercises 715 In Exercises 1–4, write the first five terms of the 9.1 sequence. (Assume that begins with 1.) n 1. an 2. an 2 6 n 1n 5n 2n 1 3. an 72 n! 4. an nn 1 In Exercises 5–8, write an expression for the apparent th term of the sequence. (Assume that begins with 1.) n n 5. 6. 7. 2, 2, 2, 2, 2,... 1, 2, 7, 14, 23,... 4, 2, 4 3, 1, 4 5,... 8. 1, 1 2, 1 3, 1 4, 1 5,... 25. Compound Interest A deposit of $10,000 is made in an account that earns 8% interest compounded monthly. The n balance in the account after months is given by n, n 1, 2, 3,... 10,0001 0.08 12 An (a) Write the first 10 terms of this sequence. (b) Find the balance in this account after 10 years by find- ing the 120th term of the sequence. 26. Education The enrollment (in thousands) in Head Start programs in the United States from 1994 to 2002 can be approximated by the model an an 1.07n2 6.1n 693, n 4, 5,..., 12 n is the year, with n 4 where corresponding to 1994. Find the terms of this finite |
sequence. Use a graphing utility to construct a bar graph that represents the sequence. (Source: U.S. Administration for Children and Families) In Exercises 9–12, simplify the factorial expression. In Exercises 27–30, determine whether the sequence 9.2 is arithmetic. If so, find the common difference. 9. 5! 11. 3! 5! 6! 10. 12. 3! 2! 7! 6! 6! 8! In Exercises 13–18, find the sum. 13. 15. 17. 6 i1 5 6 j 2 2k3 4 j 1 10 k1 14. 16. 18. 5 k2 8 i1 4k i i 1 4 j 0 j 2 1 In Exercises 19 and 20, use sigma notation to write the sum. 19. 20.... 1 220 1 21 1 2 2 3 1 22 3 4 1 23... 9 10 In Exercises 21–24, find the sum of the infinite series. 21. 23. i1 k1 5 10i 2 100 k 22. 24. i1 k2 3 10i 27. 5, 3, 1, 1 3 2, 2, 29. 1, 1, 5 2, 2,... 3,... 28. 0, 1, 3, 6, 10,... 6 9,... 30. 7 9, 8 9, 5 9, 9 9, In Exercises 31–34, write the first five terms of the arithmetic sequence. 31. 33. 34. a1 a1 a1 4, d 3 25, ak1 4.2, ak1 ak ak 3 0.4 32. a1 6, d 2 In Exercises 35–40, find a formula for sequence. an for the arithmetic 35. 37. 39. a1 a1 a2 7, y, 93, d 12 d 3y a6 65 36. 38. 40. a1 a1 a 7 25, 2x, 8, a13 d 3 d x 6 In Exercises 41–44, find the partial sum. 8 2j 3 10 42. 41. j1 20 3j j1 43. 3k 4 2 11 k1 44. 25 k1 3k 1 4 9 10 k 45. Find the sum of the first 100 positive multiples of 5. 46. Find the sum of the integers from 20 to |
80 (inclusive). 333202_090R.qxd 12/5/05 11:43 AM Page 716 716 Chapter 9 Sequences, Series, and Probability 47. Job Offer The starting salary for an accountant is $34,000 with a guaranteed salary increase of $2250 per year. Determine (a) the salary during the fifth year and (b) the total compensation through 5 full years of employment. 48. Baling Hay In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Because each round gets shorter, the farmer estimates that the same pattern will continue. Estimate the total number of bales made if the farmer takes another six trips around the field. 75. k1 42 3 k1 76. k1 1.3 1 10 k1 77. Depreciation A paper manufacturer buys a machine for $120,000. During the next 5 years, it will depreciate at a rate of 30% per year. (That is, at the end of each year the depreciated value will be 70% of what it was at the beginning of the year.) (a) Find the formula for the th term of a geometric sequence that gives the value of the machine full years after it was purchased. t n In Exercises 49–52, determine whether the sequence 9.3 is geometric. If so, find the common ratio. (b) Find the depreciated value of the machine after 5 full years. 49. 51. 5, 10, 20, 40,... 1 3, 8 3, 2 3,... 3, 4 50. 52. 54, 18, 6, 2,... 1 4, 2 7,... 5, 3 6, 4 In Exercises 53–56, write the first five terms of the geometric sequence. 53. 55. a1 a1 4, r 1 4 9, a3 4 54. 56. a1 a1 2, r 2 2, a3 12 In Exercises 57–60, write an expression for the th term of the geometric sequence. Then find the 20th term of the sequence. n 57. 59. a1 a1 16, 8 a2 100, r 1.05 58. 60. a3 a1 6, 5, 1 a4 r 0.2 In Exercises 61–66, find the sum of the finite |
geometric sequence. 7 3i1 2i1 5 61. 62. i1 i1 63. 65. 4 i1 5 i1 i 1 2 2i1 64. 66. i1 1 3 6 i1 4 i1 63i In Exercises 67–70, use a graphing utility to find the sum of the finite geometric sequence. 67. 69. 10 i1 25 i1 103 5 i1 1001.06i1 68. 70. 15 i1 200.2i1 i1 86 5 20 i1 In Exercises 71–76, find the sum of the infinite geometric series. 71. 73. i1 i1 i1 7 8 0.1i1 72. 74. i1 i1 i1 1 3 0.5i1 78. Annuity You deposit $200 in an account at the beginning of each month for 10 years. The account pays 6% compounded monthly. What will your balance be at the end of 10 years? What would the balance be if the interest were compounded continuously? In Exercises 79–82, use mathematical induction to 79. 80. 9.4 n. prove the formula for every positive integer 3 5 7... 2n 1 nn ari a1 rn 1 r a kd n 2 2 5 2 2a n 1d n1 n1 81. 82. i0 k0 n 3 In Exercises 83–86, find a formula for the sum of the first terms of the sequence. n 83. 9, 13, 17, 21,... 9 25, 85. 1, 27 125,... 3 5, 84. 68, 60, 52, 44,... 86. 12, 1, 1 12, 1 144,... In Exercises 87–90, find the sum using the formulas for the sums of powers of integers. 87. 89. 30 n1 7 n1 n n 4 n 88. 90. 10 n1 6 n1 n2 n 5 n2 In Exercises 91–94, write the first five terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a linear model, a quadratic model, or neither. 91. 93. a1 an a1 an 5 an1 16 an1 5 1 92. 94. a1 an a0 an 2n 3 an1 0 n an1 |
333202_090R.qxd 12/5/05 11:43 AM Page 717 In Exercises 95–98, use the Binomial Theorem to 9.5 calculate the binomial coefficient. 95. 97. 6C4 8C5 96. 98. 10C7 12C3 In Exercises 99–102, use Pascal’s Triangle to calculate the binomial coefficient. 101. 102. 100. 99. 9 4 5 3 7 3 8 6 In Exercises 103–108, use the Binomial Theorem to expand and simplify the expression. (Remember that i 1. ) 103. 104. 105. 106. 107. 108. x 44 x 36 a 3b5 3x y 27 5 2i 4 4 5i 3 Review Exercises 717 116. Menu Choices A local sub shop offers five different breads, seven different meats, three different cheeses, and six different vegetables. Find the total number of combinations of sandwiches possible. 9.7 117. Apparel A man has five pairs of socks, of which no two pairs are the same color. He randomly selects two socks from a drawer. What is the probability that he gets a matched pair? 118. Bookshelf Order A child returns a five-volume set of books to a bookshelf. The child is not able to read, and so cannot distinguish one volume from another. What is the probability that the books are shelved in the correct order? 119. Students by Class At a particular university, the numbers of students in the four classes are broken down by percents, as shown in the table. Class Percent Freshmen Sophomores Juniors Seniors 31 26 25 18 9.6 109. Numbers in a Hat Slips of paper numbered 1 through 14 are placed in a hat. In how many ways can you draw two numbers with replacement that total 12? 110. Home Theater Systems A customer in an electronics store can choose one of six speaker systems, one of five DVD players, and one of six plasma televisions to design a home theater system. How many systems can be designed? 111. Telephone Numbers The same three-digit prefix is used for all of the telephone numbers in a small town. How many different telephone numbers are possible by changing only the last four digits? 112. Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of three mathematics courses, one of four science courses, and one of six |
history courses. How many schedules are possible? 113. Bike Race There are 10 bicyclists entered in a race. In how many different ways could the top three places be decided? 114. Jury Selection A group of potential jurors has been narrowed down to 32 people. In how many ways can a jury of 12 people be selected? 115. Apparel You have eight different suits to choose from to take on a trip. How many combinations of three suits could you take on your trip? A single student is picked randomly by lottery for a cash scholarship. What is the probability that the scholarship winner is (a) a junior or senior? (b) a freshman, sophomore, or junior? 120. Data Analysis A sample of college students, faculty, and administration were asked whether they favored a proposed increase in the annual activity fee to enhance student life on campus. The results of the study are listed in the table. Students Faculty Admin. Total Favor Oppose Total 237 163 400 37 38 75 18 7 25 292 208 500 A person is selected at random from the sample. Find each specified probability. (a) The person is not in favor of the proposal. (b) The person is a student. (c) The person is a faculty member and is in favor of the proposal. 333202_090R.qxd 12/5/05 11:43 AM Page 718 718 Chapter 9 Sequences, Series, and Probability 121. Tossing a Die A six-sided die is tossed three times. What is the probability of getting a 6 on each roll? 122. Tossing a Die A six-sided die is tossed six times. What is the probability that each side appears exactly once? 123. Drawing a Card You randomly select a card from a 52-card deck. What is the probability that the card is not a club? 124. Tossing a Coin Find the probability of obtaining at least one tail when a coin is tossed five times. Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 125–129, determine whether n 2n 1 i3 5 i1 2i i1 125. 126. 127. 128. n 2! n! i 3 2i 5 5 i1 8 k1 3k 3 8 k1 k 6 j1 2 j 8 j3 2 j2 129. The value of nCr. 130. Think About It An infinite sequence is a function. What is always greater than the value of |
nPr is the domain of the function? 131. Think About It How do the two sequences differ? (a) an (b) an 1n n 1n1 n 132. Graphical Reasoning The graphs of two sequences are shown below. Identify each sequence as arithmetic or geometric. Explain your reasoning. (a) an 4 −2 −8 −12 −16 −20 2 86 10 n (b) an 100 80 60 40 20 −20 Graphical Reasoning In Exercises 135–138, match the sequence or sum of a sequence with its graph without doing any calculations. Explain your reasoning. [The graphs are labeled (a), (b), (c), and (d).] (b) 10 (d) 0 0 5 0 0 10 10 10 10 (a) 6 0 −4 (c) 5 0 0 135. 136. 137. an an an n1 2 41 41 n 41 2 2 k1 n1 k1 138. an n k1 41 2 k1 139. Population Growth Consider an idealized population with the characteristic that each member of the population produces one offspring at the end of every time period. If each member has a life span of three time periods and the population begins with 10 newborn members, then the following table shows the population during the first five time periods. Age Bracket 0–1 1–2 2–3 Total Time Period 1 10 2 10 10 10 20 3 20 10 10 40 4 40 20 10 70 5 70 40 20 130 2 864 10 n Sn Sn1 Sn2 Sn3, n > 3. The sequence for the total population has the property that 133. Writing Explain what is meant by a recursion formula. 134. Writing Explain why the terms of a geometric sequence decrease when 0 < r < 1. Find the total population during the next five time periods. 140. The probability of an event must be a real number in what interval? Is the interval open or closed? 333202_090R.qxd 12/5/05 11:43 AM Page 719 9 Chapter Test Chapter Test 719 Take this test as you would take a test in class. When you are finished, check your work against the answers given in the back of the book. 1. Write the first five terms of the sequence 1n 3n 2 2. Write an expression for the th term of the sequence. an n. n (Assume that begins with 1.) 3 1!, 4 2!, 5 3 |
!, 6 4!, 7 5!,... 3. Find the next three terms of the series. Then find the fifth partial sum of the series. 6 17 28 39... 4. The fifth term of an arithmetic sequence is 5.4, and the 12th term is 11.0. Find the n th term. 5. Write the first five terms of the sequence an 52n1. n (Assume that begins with 1.) i1 41 2 i. In Exercises 6 –8, find the sum. 6. 50 i1 2i 2 5. 7. 7 n1 8n 5 8. 9. Use mathematical induction to prove the formula. 5 10 15... 5n 5nn 1 2 10. Use the Binomial Theorem to expand the expression x 2y4. 11. Find the coefficient of the term a3 b5 in the expansion of 2a 3b8. In Exercises 12 and 13, evaluate each expression. 12. (a) 13. (a) 9 P2 11C4 (b) (b) 70 P3 66C4 14. How many distinct license plates can be issued consisting of one letter followed by a three-digit number? 15. Eight people are going for a ride in a boat that seats eight people. The owner of the boat will drive, and only three of the remaining people are willing to ride in the two bow seats. How many seating arrangements are possible? 16. You attend a karaoke night and hope to hear your favorite song. The karaoke song book has 300 different songs (your favorite song is among the 300 songs). Assuming that the singers are equally likely to pick any song and no song is repeated, what is the probability that your favorite song is one of the 20 that you hear that night? 17. You are with seven of your friends at a party. Names of all of the 60 guests are placed in a hat and drawn randomly to award eight door prizes. Each guest is limited to one prize. What is the probability that you and your friends win all eight of the prizes? 18. The weather report calls for a 75% chance of snow. According to this report, what is the probability that it will not snow? 333202_090R.qxd 12/8/05 10:55 AM Page 720 720 Chapter 9 Sequences, Series, and Probability 9 Cumulative Test for Chapters 7–9 Take this test to review the material |
from earlier chapters. When you are finished, check your work against the answers given in the back of the book. In Exercises 1– 4, solve the system by the specified method. 1. Substitution y 2 y 2 3 x2 x 1 3. Elimination 2x 4y z x 2y 2z x 3y z 3 6 1 2. Elimination x 3y 2x 4y 1 0 4. Gauss-Jordan Elimination 7 5 3 x 3y 2z 2x y z 4x y z In Exercises 5 and 6, sketch the graph of the solution set of the system of inequalities. 5. 2x y ≥ x 3y ≤ 3 2 6. x y > 5x 2y < 6 10 7. Sketch the region determined by the constraints. Then find the minimum and z 3x 2y, maximum values, and where they occur, of the objective function subject to the indicated constraints. x 4y ≤ 20 2x y ≤ 12 x ≥ 0 y ≥ 0 8. A custom-blend bird seed is to be mixed from seed mixtures costing $0.75 per pound and $1.25 per pound. How many pounds of each seed mixture are used to make 200 pounds of custom-blend bird seed costing $0.95 per pound? x 2x 3x 2y y 3y z 2z 4z 9 9 7 SYSTEM FOR 10 AND 11 8 1 2 0 3 6 5 1 4 MATRIX FOR 16 9. Find the equation of the parabola 6, 4. 3, 1, and y ax2 bx c passing through the points 0, 4, In Exercises 10 and 11, use the system of equations at the left. 10. Write the augmented matrix corresponding to the system of equations. 11. Solve the system using the matrix found in Exercise 10 and Gauss-Jordan elimination. In Exercises 12–15, use the following matrices to find each of the following, if possible. A [ 4 1 A B A 2B 14. 12. 0 2], B [1 1 3 0] 13. 15. 2B AB 16. Find the determinant of the matrix at the left. 17. Find the inverse of the matrix (if it exists): 1 3 5 2 7 7. 1 10 15 333202_090R.qxd 12/8/05 10:56 AM Page 721 Age group 14 17 18 24 2534 0.09 0 |
.06 0.12 Gym Jogging Walking shoes shoes 0.09 0.10 0.25 shoes 0.03 0.05 0.12 MATRIX FOR 18 y 6 5 2 1 − ( 2, 3) (1, 5) (4, 1) −2 −1 1 2 3 4 x FIGURE FOR 21 Cumulative Test for Chapters 7–9 721 18. The percents (by age group) of the total amounts spent on three types of footwear in a recent year are shown in the matrix. The total amounts (in millions) spent by each age group on the three types of footwear were $442.20 (14–17 age group), $466.57(18–24 age group), and $1088.09 (25–34 age group). How many dollars worth of gym shoes, jogging shoes, and walking shoes were sold that year? (Source: National Sporting Goods Association) In Exercises 19 and 20, use Cramer’s Rule to solve the system of equations. 19. 20. 52 5 8x 3y 3x 5y 5x 4y 3z 3x 8y 7z 7x 5y 6z 7 9 53 21. Find the area of the triangle shown in the figure. 22. Write the first five terms of the sequence 1n1 2n 3 23. Write an expression for the th term of the sequence. an n n (assume that begins with 1). 2! 4, 3! 5, 4! 6, 5! 7, 6! 8,... 24. Find the sum of the first 20 terms of the arithmetic sequence 8, 12, 16, 20,.... 25. The sixth term of an arithmetic sequence is 20.6, and the ninth term is 30.2. (a) Find the 20th term. (b) Find the th term. n 26. Write the first five terms of the sequence 32n1 an n (assume that begins with 1). 27. Find the sum: i0 1.3 1 10 i1. 28. Use mathematical induction to prove the formula 3 7 11 15... 4n 1 n2n 1. 29. Use the Binomial Theorem to expand and simplify z 34. In Exercises 30–33, evaluate the expression. 30. 7P3 31. 25P2 32. 8 4 33. 10C3 In Exercises 34 and 35, find the |
number of distinguishable permutations of the group of letters. 34. B, A, S, K, E, T, B, A, L, L 35. A, N, T, A, R, C, T, I, C, A 36. A personnel manager at a department store has 10 applicants to fill three different sales positions. In how many ways can this be done, assuming that all the applicants are qualified for any of the three positions? 37. On a game show, the digits 3, 4, and 5 must be arranged in the proper order to form the price of an appliance. If the digits are arranged correctly, the contestant wins the appliance. What is the probability of winning if the contestant knows that the price is at least $400? 333202_090R.qxd 12/5/05 11:43 AM Page 722 Proofs in Mathematics Properties of Sums (p. 647) 1. 2. 3. n i1 c cn, c is a constant. n i1 cai cn ai, i1 c is a constant. n i1 ai bi n i1 ai n i1 bi 4. n i1 ai bi n ai i1 n i1 bi Proof Each of these properties follows directly from the properties of real numbers. 1. n i1 c c c c... c cn n terms The Distributive Property is used in the proof of Property 2. 2. n i1 cai ca1 ca2 ca3... can ca1 a2 a3... an cn i1 ai The proof of Property 3 uses the Commutative and Associative Properties of Addition. 3. n i1 ai bi a1 a1 n i1 b1 a2 b2 a3... an bn b3 b1 b2 b3... bn a2 a3... an ai n i1 bi The proof of Property 4 uses the Commutative and Associative Properties of Addition and the Distributive Property. 4. n i1 ai bi b1 a2 b2 a3... an bn a2 a2 a3 a3... an... an b2 b3 b2 b3... bn... bn b3 b1 b1 a1 a1 a1 n i1 ai n i1 bi Infinite |
Series The study of infinite series was considered a novelty in the fourteenth century. Logician Richard Suiseth, whose nickname was Calculator, solved this problem. If throughout the first half of a given time interval a variation continues at a certain intensity; throughout the next quarter of the interval at double the intensity; throughout the following eighth at triple the intensity and so ad infinitum; The average intensity for the whole interval will be the intensity of the variation during the second subinterval (or double the intensity). This is the same as saying that the sum of the infinite series 2n... is 2. 722 333202_090R.qxd 12/5/05 11:43 AM Page 723 The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with terms is n (p. 656) Sn n 2 a1 an. Proof Begin by generating the terms of the arithmetic sequence in two ways. In the first way, repeatedly add a1 a1 to the first term to obtain an1 d a3 d a1... an2 a2 a1 n 1d. Sn In the second way, repeatedly subtract from the th term to obtain... a3 d an 2d... a1 n a1 2d... an Sn, d a1... a1 a2 an n 1d. the multiples of subtract out and you obtain an n terms Sn an an an2 an1 d an an If you add these two versions of an an an a1 2Sn 2Sn a1 na1 n 2 a1 Sn an. The Sum of a Finite Geometric Sequence The sum of the finite geometric sequence (p. 666) a1, a1r, a1r 2, a1r 3, a1r 4,..., a1r n1 n is given by Sn r 1 with common ratio i1 a1r i1 a11 r n 1 r. Proof Sn rSn a1r a1r 2... a1rn2 a1r n1 a1 a1r a1r 2 a1r 3... a1rn1 a1r n Subtracting the second equation from the first yields Multiply by r. a1 rSn Sn 1 r a1 Sn a1r n. 1 r n, So, and, because r 1, you have Sn a11 r n 1 r. |
723 333202_090R.qxd 12/5/05 11:43 AM Page 724 The Binomial Theorem (p. 683) In the expansion of x yn x yn xn nx n1y... x nryr the coefficient of is nCr x nry r... nxy n1 y n nCr n! n r!r!. Proof The Binomial Theorem can be proved quite nicely using mathematical induction. The steps are straightforward but look a little messy, so only an outline of the proof is presented. n 1, x y1 x1 y1 and the formula is you have 1. If 1C0x 1C1y, valid. 2. Assuming that the formula is true for kCr k! k r!r! the coefficient of n k, kk 1k 2... k r 1 r!. xkryr is To show that the formula is true for x k1ryr in the expansion of x yk1 x ykx y. n k 1, look at the coefficient of x k1ryr x k1ry r kCr1x k1ry r1y k! k 1 r!r 1! From the right-hand side, you can determine that the term involving is the sum of two products. kCr x kry rx k! k r!r! k 1 rk! k 1 r!r! k!k 1 r r k 1 r!r! k 1! k 1 r!r! k1Cr xk1ryr k!r k 1 r!r! x k1ry r x k1ry r x k1ry r So, by mathematical induction, the Binomial Theorem is valid for all positive integers n. 724 333202_090R.qxd 12/5/05 11:43 AM Page 725 P.S. Problem Solving This collection of thought-provoking and challenging exercises further explores and expands upon concepts learned in this chapter. 1. Let 1 and consider the sequence xn given by x0 1 2 xn xn1 1 xn1, n 1, 2,... Use a graphing utility to compute the first 10 terms of the n sequence and make a conjecture about the value of approaches infinity. as xn 2. Consider the sequence n 1 n2 1 an. (a) Use a graphing utility |
to graph the first 10 terms of the sequence. (b) Use the graph from part (a) to estimate the value of an as approaches infinity. n (c) Complete the table. 1 10 100 1000 10,000 n an (d) Use the table from part (c) to determine (if possible) the value of an as approaches infinity. n 3. Consider the sequence 3 1n. an (a) Use a graphing utility to graph the first 10 terms of the sequence. (b) Use the graph from part (a) to describe the behavior of the graph of the sequence. (c) Complete the table. 1 10 101 1000 10,001 n an (d) Use the table from part (c) to determine (if possible) the value of an as approaches infinity. n 4. The following operations are performed on each term of an arithmetic sequence. Determine if the resulting sequence is arithmetic, and if so, state the common difference. (a) A constant C is added to each term. (b) Each term is multiplied by a nonzero constant C. (c) Each term is squared. 5. The following sequence of perfect squares is not arithmetic. 1, 4, 9, 16, 25, 36, 49, 64, 81,... However, you can form a related sequence that is arithmetic by finding the differences of consecutive terms. (a) Write the first eight terms of the related arithmetic n th term of this sequence described above. What is the sequence? (b) Describe how you can find an arithmetic sequence that is related to the following sequence of perfect cubes. 1, 8, 27, 64, 125, 216, 343, 512, 729,... (c) Write the first seven terms of the related sequence in part (b) and find the th term of the sequence. n (d) Describe how you can find the arithmetic sequence that is related to the following sequence of perfect fourth powers. 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561,... (e) Write the first six terms of the related sequence in part (d) and find the th term of the sequence. n 6. Can the Greek hero Achilles, running at 20 feet per second, ever catch a tortoise, starting 20 feet ahead of Achilles and running at 10 feet per second? The Greek mathematician Zeno said no. When Achilles runs 20 feet, the tortoise will |
be 10 feet ahead. Then, when Achilles runs 10 feet, the tortoise will be 5 feet ahead. Achilles will keep cutting the distance in half but will never catch the tortoise. The table shows Zeno’s reasoning. From the table you can see that both the distances and the times required to achieve them form infinite geometric series. Using the table, show that both series have finite sums. What do these sums represent? Distance (in feet) Time (in seconds) 20 10 5 2.5 1.25 0.625 1 0.5 0.25 0.125 0.0625 0.03125 7. Recall that a fractal is a geometric figure that consists of a pattern that is repeated infinitely on a smaller and smaller scale. A well-known fractal is called the Sierpinski Triangle. In the first stage, the midpoints of the three sides are used to create the vertices of a new triangle, which is then removed, leaving three triangles. The first three stages are shown on the next page. Note that each remaining triangle is similar to the original triangle. Assume that the length of each side of the original triangle is one unit. 725 333202_090R.qxd 12/5/05 11:43 AM Page 726 Write a formula that describes the side length of the triangles n that will be generated in the th stage. Write a formula for the area of the triangles that will be generated in the th stage. n FIGURE FOR 7 8. You can define a sequence using a piecewise formula. The following is an example of a piecewise-defined sequence. a1 7, an an1 2 3an1, if an1 is even 1, if an1 is odd (a) Write the first 10 terms of the sequence. 7. (b) Choose three different values for For each value of a1 other than a1, a1 find the first 10 terms of the sequence. What conclusions can you make about the behavior of this sequence? 9. The numbers 1, 5, 12, 22, 35, 51, are called pentagonal numbers because they represent the numbers of dots used to make pentagons, as shown below. Use mathematical induction to prove that the th pentagonal number Pn... n is given by n3n 1 2. Pn 10. What conclusion can be drawn from the following infor- mation about the sequence of statements Pk implies (a) Pn? Pk |
1. are all true. (b) (c) (d) P3 is true and P1, P2, P3,..., P50 P1, P2, and P3 Pk1 imply that P2 P2k is true and f1, f2,..., fn,... is true. implies P2k2. 11. Let be the Fibonacci sequence. are all true, but the truth of Pk does not (a) Use mathematical induction to prove that f1 f2... fn fn2 1. (b) Find the sum of the first 20 terms of the Fibonacci sequence. 726 12. The odds in favor of an event occurring are the ratio of the probability that the event will occur to the probability that the event will not occur. The reciprocal of this ratio represents the odds against the event occurring. (a) Six marbles in a bag are red. The odds against choosing a red marble are 4 to 1. How many marbles are in the bag? (b) A bag contains three blue marbles and seven yellow marbles. What are the odds in favor of choosing a blue marble? What are the odds against choosing a blue marble? (c) Write a formula for converting the odds in favor of an event to the probability of the event. (d) Write a formula for converting the probability of an event to the odds in favor of the event. 13. You are taking a test that contains only multiple choice questions (there are five choices for each question). You are on the last question and you know that the answer is not B or D, but you are not sure about answers A, C, and E. What is the probability that you will get the right answer if you take a guess? 14. A dart is thrown at the circular target shown below. The dart is equally likely to hit any point inside the target. What is the probability that it hits the region outside the triangle? 6 n V A p1, p2,..., pn. The expected value x1, x2,..., xn. ring are A and their values, 15. An event has possible outcomes, which have the values n The probabilities of the outcomes occurof an event is the sum of the products of the outcomes’ probabilities p2x2 (a) To win California’s Super Lotto Plus game, you must match five |
different numbers chosen from the numbers 1 to 47, plus one Mega number chosen from the numbers 1 to 27. You purchase a ticket for $1. If the jackpot for the next drawing is $12,000,000, what is the expected value for the ticket?... pnxn. V p1x1 (b) You are playing a dice game in which you need to score 60 points to win. On each turn, you roll two sixsided dice. Your score for the turn is 0 if the dice do not show the same number, and the product of the numbers on the dice if they do show the same number. What is the expected value for each turn? How many turns will it take on average to score 60 points? 1010 333202_1000.qxd 12/8/05 8:52 AM Page 727 Topics in Analytic Geometry 10.1 Lines 10.2 Introduction to Conics: Parabolas 10.3 Ellipses 10.4 Hyperbolas 10.5 Rotation of Conics 10.6 Parametric Equations 10.7 Polar Coordinates 10.8 Graphs of Polar Equations 10.9 Polar Equations of Conics The nine planets move about the sun in elliptical orbits. You can use the techniques presented in this chapter to determine the distances between the planets and the center of the sun AT I O N S Analytic geometry concepts have many real-life applications. The applications listed below represent a small sample of the applications in this chapter. • Inclined Plane, Exercise 56, page 734 • Satellite Orbit, Exercise 60, page 752 • Projectile Motion, Exercises 57 and 58, page 777 • Revenue, • LORAN, Exercise 59, page 741 Exercise 42, page 761 • Architecture, Exercise 57, page 751 • Running Path, Exercise 44, page 762 • Planetary Motion, Exercises 51–56, page 798 • Locating an Explosion, Exercise 40, page 802 727 333202_1001.qxd 12/8/05 8:54 AM Page 728 728 Chapter 10 Topics in Analytic Geometry 10.1 Lines What you should learn • Find the inclination of a line. • Find the angle between two lines. • Find the distance between a point and a line. Why you should learn it The inclination of a line can be used to measure heights indirectly. For instance, in Exercise 56 on page 734, the inclination of |
a line can be used to determine the change in elevation from the base to the top of the Johnstown Inclined Plane. Inclination of a Line In Section 1.3, you learned that the graph of the linear equation y mx b 0, b. is a nonvertical line with slope There, the slope of a line x. was described as the rate of change in with respect to In this section, you will look at the slope of a line in terms of the angle of inclination of the line. and -intercept y y m Every nonhorizontal line must intersect the -axis. The angle formed by such an intersection determines the inclination of the line, as specified in the following definition. x Definition of Inclination The inclination of a nonhorizontal line is the positive angle (less than measured counterclockwise from the -axis to the line. (See Figure 10.1.) x ) y y y y = 0θ AP/Wide World Photos =θ π 2 x x θ x θ x Horizontal Line FIGURE 10.1 Vertical Line Acute Angle Obtuse Angle The inclination of a line is related to its slope in the following manner. Inclination and Slope If a nonvertical line has inclination and slope m, then m tan. The HM mathSpace® CD-ROM and Eduspace® for this text contain additional resources related to the concepts discussed in this chapter. For a proof of the relation between inclination and slope, see Proofs in Mathematics on page 806. 333202_1001.qxd 12/8/05 8:54 AM Page 729 y 3 1 FIGURE 10.2 2x + 3y = 6 θ ≈ 146.3° 1 2 3 y θ θ 2= − θ 1 θ θ 1 θ 2 FIGURE 10.3 x x Section 10.1 Lines 729 Example 1 Finding the Inclination of a Line Find the inclination of the line 2x 3y 6. Solution The slope of this line is equation tan 2 3. m 2 3. So, its inclination is determined from the < <. This means that From Figure 10.2, it follows that arctan 2 3 2 0.588 0.588 2.554. The angle of inclination is about 2.554 radians or about 146.3. Now try Exercise 19. The Angle Between Two Lines Two distinct lines in a plane are either parallel or |
intersecting. If they intersect and are nonperpendicular, their intersection forms two pairs of opposite angles. One pair is acute and the other pair is obtuse. The smaller of these angles is called the angle between the two lines. As shown in Figure 10.3, you can use the inclinations of the two lines to find the angle between the two lines. If two lines have inclinations the angle between the and two lines is 1 and 2, 1 < 2, 1 < where 2 2 2 1. You can use the formula for the tangent of the difference of two angles tan tan 2 tan 2 1 tan 1 tan 1 1 tan 2 to obtain the formula for the angle between two lines. Angle Between Two Lines If two nonperpendicular lines have slopes two lines is tan m2 1 m1m2. m1 m1 and m2, the angle between the 333202_1001.qxd 12/8/05 8:54 AM Page 730 730 Chapter 10 Topics in Analytic Geometry y 4 2 1 3x + 4y − 12 = 0 θ ≈ 79.70° 2x − y − 4 = 0 Example 2 Finding the Angle Between Two Lines Find the angle between the two lines. Line 1: 2x y 4 0 Line 2: 3x 4y 12 0 Solution 2 The two lines have slopes of of the angle between the two lines is m1 tan m2 1 m1m2 m1 1 234 34 2 24 114 11 2. Finally, you can conclude that the angle is and m2 3 4, respectively. So, the tangent 1 3 4 x arctan 11 2 1.391 radians 79.70 FIGURE 10.4 as shown in Figure 10.4. Now try Exercise 27. (x1, y1) d y (x2, y2) FIGURE 10.5 y 4 3 2 1 y = 2x + 1 (4, 1) −3 −2 1 2 3 4 5 −2 −3 −4 x x The Distance Between a Point and a Line Finding the distance between a line and a point not on the line is an application of perpendicular lines. This distance is defined as the length of the perpendicular line segment joining the point and the line, as shown in Figure 10.5. Distance Between a Point and a Line The distance between the point By1 A2 B2 C x1, y1 Ax1 d. and the line Ax By C 0 |
is Remember that the values of to the general equation of a line, between a point and a line, see Proofs in Mathematics on page 806. B,A, C Ax By C 0. in this distance formula correspond For a proof of the distance and Example 3 Finding the Distance Between a Point and a Line Find the distance between the point 4, 1 and the line y 2x 1. Solution The general form of the equation is 2x y 1 0. So, the distance between the point and the line is d 24 11 1 22 12 8 5 3.58 units. The line and the point are shown in Figure 10.6. FIGURE 10.6 Now try Exercise 39. 333202_1001.qxd 12/8/05 8:54 AM Page 731 y 6 5 4 2 1 B (0, 4) h C (5, 2) A (−3, 0) 1 2 3 4 5 x point −2 FIGURE 10.7 Section 10.1 Lines 731 Example 4 An Application of Two Distance Formulas Figure 10.7 shows a triangle with vertices A3, 0, B0, 4, and C5, 2. a. Find the altitude b. Find the area of the triangle. from vertex h B to side AC. Solution a. To find the altitude, use the formula for the distance between line AC and the 0, 4. Slope: The equation of line m 2 0 2 5 3 8 AC 1 4 is obtained as follows. Equation: x 3 y 0 1 4 4y x 3 x 4y 3 0 Point-slope form Multiply each side by 4. General form 0, 4 is So, the distance between this line and the point Altitude h 10 44 3 12 42 13 17 units. b. Using the formula for the distance between two points, you can find the length of the base AC to be b 5 32 2 02 82 22 68 217 units. Distance Formula Simplify. Simplify. Simplify. Finally, the area of the triangle in Figure 10.7 is A 1 2 1 2 bh Formula for the area of a triangle 217 13 17 13 square units. Substitute for b and h. Simplify. Now try Exercise 45. W RITING ABOUT MATHEMATICS Inclination and the Angle Between Two Lines Discuss why the inclination of a line but the angle between two lines cannot can be an angle that is larger than 2. be larger than Decide whether |
the following statement is true or false: “The inclination of a line is the angle between the line and the -axis.” Explain. 2, x 333202_1001.qxd 12/8/05 11:05 AM Page 732 732 Chapter 10 Topics in Analytic Geometry 10.1 Exercises The HM mathSpace® CD-ROM and Eduspace® for this text contain step-by-step solutions to all odd-numbered exercises. They also provide Tutorial Exercises for additional help. VOCABULARY CHECK: Fill in the blanks. 1. The ________ of a nonhorizontal line is the positive angle x- counterclockwise from the axis to the line. ) (less than measured 2. If a nonvertical line has inclination and slope m, m ________. 3. If two nonperpendicular lines have slopes m1 and the angle between the two lines is tan ________. 4. The distance between the point x1, y1 and the line Ax By C 0 is given by d ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. then m2, 1. 3. 5. 7. In Exercises 1–8, find the slope of the line with inclination. =θ π 6 x y y 2. 4. y y In Exercises 19–22, find the inclination degrees) of the line. (in radians and =θ π 4 x 19. 20. 21. 22. 6x 2y 8 0 4x 5y 9 0 5x 3y 0 x y 10 0 In Exercises 23–32, find the angle between the lines. (in radians and degrees) 23. 3x y 3 2x y 2 24. x 3y 2 x 2y 3 y π =θ 3 4 x π =θ 2 3 x radians 3 1.27 radians 6. 5 6 radians 8. 2.88 radians In Exercises 9–14, find the inclination degrees) of the line with a slope of m. (in radians and 9. 11. 13. m 1 m 1 m 3 4 10. 12. 14. m 2 m 2 m 5 2 In Exercises 15–18, find the inclination degrees) of the line passing through the points. (in |
radians and 15. 16. 17. 18. 6, 1, 10, 8 4, 3 12, 8, 2, 20, 10, 0 0, 100, 50, 0 y 2 1 −1 −3 −2 −1 25. 3x 2y 0 3x 2y 1 26. 2x 3y 22 4x 3y 24 y 2 1 θ x 1 2 −2 −1 −1 − 27. 29. x 2y 7 6x 2y 5 x 2y 8 x 2y 2 28. 30. 5x 2y 16 3x 5y 1 3x 5y 3 3x 5y 12 333202_1001.qxd 12/8/05 8:54 AM Page 733 31. 32. 0.05x 0.03y 0.21 0.07x 0.02y 0.16 0.02x 0.05y 0.19 0.03x 0.04y 0.52 Section 10.1 Lines 733 In Exercises 49 and 50, find the distance between the parallel lines. 49. x y 1 x y 5 50. 3x 4y 1 3x 4y 10 Angle Measurement In Exercises 33–36, find the slope of each side of the triangle and use the slopes to find the measures of the interior angles. y 4 33. y 34. y 6 4 2 (−3, 2) −4 −2 −2 36. − ( 3, 4) y 4 (1, 3) (2, 0) 2 4 x 6 4 2 (2, 1) (4, 4) (6, 2) x 2 4 6 35. y 4 2 (3, 2) x (1, 0) 4 (−4, −1) y 2 −2 −4 −4 −2 x 4 −2 −2 x 4 51. Road Grade A straight road rises with an inclination of 0.10 radian from the horizontal (see figure). Find the slope of the road and the change in elevation over a two-mile stretch of the road. − ( 2, 2) (2, 1) −4 −2 2 4 −2 x 2 mi 0.1 radian In Exercises 37–44, find the distance between the point and the line. Point 0, 0 0, 0 2, 3 2, 1 6, 2 10, 8 0, 8 4, 2 37. 38. 39. 40. 41. 42 |
. 43. 44. Line 4x 3y 0 2x y 4 4x 3y 10 x y 2 x 1 0 y 4 0 6x y 0 x y 20 52. Road Grade A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road. 53. Pitch of a Roof A roof has a rise of 3 feet for every horizontal change of 5 feet (see figure). Find the inclination of the roof. 3 ft 5 ft In Exercises 45–48, the points represent the vertices of a ABC in the coordinate plane, triangle. (a) Draw triangle B AC, (b) find the altitude from vertex of the triangle to side and (c) find the area of the triangle. C 4, 0 C 5, 2 C 5 2, 0 C 6, 12 A 0, 0, A 0, 0, A 1 2, 1 A 4, 5, B 1, 4, B 4, 5,, B 3, 10, B 2, 3, 47. 48. 46. 45. 2 333202_1001.qxd 12/8/05 8:54 AM Page 734 734 Chapter 10 Topics in Analytic Geometry 54. Conveyor Design A moving conveyor is built so that it rises 1 meter for each 3 meters of horizontal travel. (a) Draw a diagram that gives a visual representation of the problem. (b) Find the inclination of the conveyor. (c) The conveyor runs between two floors in a factory. The distance between the floors is 5 meters. Find the length of the conveyor. 55. Truss Find the angles and shown in the drawing of the roof truss. α 36 ft 6 ft 6 ft β 9 ft 58. To find the angle between two lines whose angles of m1 are known, substitute, respectively, in the formula for the angle between and and for 1 2 1 2 inclination m2 and two lines. 59. Exploration Consider a line with slope m and -intercept y 0, 4. (a) Write the distance between the origin and the line as d a function of m. (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the origin and the line. (d) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the |
problem. 60. Exploration Consider a line with slope m and -intercept y 0, 4. (a) Write the distance d between the point m. line as a function of 3, 1 and the Model It 56. Inclined Plane The Johnstown Inclined Plane in Johnstown, Pennsylvania is an inclined railway that was designed to carry people to the hilltop community of Westmont. It also proved useful in carrying people and vehicles to safety during severe floods. The railway is 896.5 feet long with a 70.9% uphill grade (see figure). (b) Graph the function in part (a). (c) Find the slope that yields the maximum distance between the point and the line. (d) Is it possible for the distance to be 0? If so, what is the slope of the line that yields a distance of 0? (e) Find the asymptote of the graph in part (b) and interpret its meaning in the context of the problem. Skills Review In Exercises 61– 66, find all -intercepts and -intercepts of the graph of the quadratic function. x y 896.5 ft θ Not drawn to scale (a) Find the inclination of the railway. (b) Find the change in elevation from the base to the top of the railway. (c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find the equation of the line that models the railway track. (d) Sketch a graph of the equation you found in part (c). Synthesis 61. 62. 63. 64. 65. 66. f x x 72 f x x 92 f x x 52 5 f x x 112 12 f x x2 7x 1 f x x2 9x 22 In Exercises 67–72, write the quadratic function in standard form by completing the square. Identify the vertex of the function. 67. 69. 71. 72. f x 3x2 2x 16 f x 5x2 34x 7 f x 6x2 x 12 f x 8x2 34x 21 68. 70. f x 2x2 x 21 f x x2 8x 15 True or False? the statement is true or false. Justify your answer. In Exercises 57 and 58, determine whether 57. A line that has an inclination greater than 2 radians has a negative slope. In Exercises 73–76, graph |
the quadratic function. 73. 75. f x x 42 3 gx 2x2 3x 1 f x 6 x 12 74. 76. gx x2 6x 8 333202_1002.qxd 12/8/05 9:00 AM Page 735 10.2 Introduction to Conics: Parabolas Section 10.2 Introduction to Conics: Parabolas 735 What you should learn • Recognize a conic as the intersection of a plane and a double-napped cone. • Write equations of parabolas in standard form and graph parabolas. • Use the reflective property of parabolas to solve real-life problems. Why you should learn it Parabolas can be used to model and solve many types of real-life problems. For instance, in Exercise 62 on page 742, a parabola is used to model the cables of the Golden Gate Bridge. Cosmo Condina/Getty Images Conics Conic sections were discovered during the classical Greek period, 600 to 300 B.C. The early Greeks were concerned largely with the geometric properties of conics. It was not until the 17th century that the broad applicability of conics became apparent and played a prominent role in the early development of calculus. A conic section (or simply conic) is the intersection of a plane and a doublenapped cone. Notice in Figure 10.8 that in the formation of the four basic conics, the intersecting plane does not pass through the vertex of the cone. When the plane does pass through the vertex, the resulting figure is a degenerate conic, as shown in Figure 10.9. Circle FIGURE 10.8 Basic Conics Ellipse Parabola Hyperbola Point FIGURE 10.9 Degenerate Conics Line Two Intersecting Lines There are several ways to approach the study of conics. You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general seconddegree equation Ax 2 Bxy Cy 2 Dx Ey F 0. However, you will study a third approach, in which each of the conics is defined as a locus (collection) of points satisfying a geometric property. For example, in Section 1.2, you learned that a circle is defined as the collection of all points x, y This leads to the standard form of the equation of a circle that are |
equidistant from a fixed point h, k. x h 2 y k 2 r 2. Equation of circle 333202_1002.qxd 12/8/05 9:00 AM Page 736 736 Chapter 10 Topics in Analytic Geometry Parabolas In Section 2.1, you learned that the graph of the quadratic function f x ax2 bx c is a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola. Definition of Parabola A parabola is the set of all points a fixed line (directrix) and a fixed point (focus) not on the line. x, y in a plane that are equidistant from y d2 Focus Vertex d1 d2 d1 Directrix FIGURE 10.10 Parabola The midpoint between the focus and the directrix is called the vertex, and the line passing through the focus and the vertex is called the axis of the parabola. Note in Figure 10.10 that a parabola is symmetric with respect to its axis. Using the definition of a parabola, you can derive the following standard form of the equation of a parabola whose directrix is parallel to the -axis or to the -axis. y x Standard Equation of a Parabola The standard form of the equation of a parabola with vertex at follows. x h, k is as x h2 4py k, p 0 y k2 4px h, p 0 Vertical axis, directrix: y k p Horizontal axis, directrix: x h p The focus lies on the axis units (directed distance) from the vertex. If the the equation takes one of the following forms. vertex is at the origin p 0, 0, x 2 4py y 2 4px See Figure 10.11. Vertical axis Horizontal axis For a proof of the standard form of the equation of a parabola, see Proofs in Mathematics on page 807. Axis: =x h Focus Axis: x = h Directrix: y = k − p Vertex: (h, k) Directrix: p− =x h p > 0 p > 0 Vertex: )h k (, Directrix: p− k =y Focus: (h, k + p) Axis: y = k = Focus: h p ( +, k |
) Vertex: (, )h k Directrix: x = h − p p < 0 Focus: (h + p, k) Axis: y = k Vertex: (h, k) (a) x h2 4py k p > 0 Vertical axis: (b) x h2 4py k Vertical axis: p < 0 (c) y k2 4px h Horizontal axis: p > 0 (d) y k2 4px h Horizontal axis: p < 0 FIGURE 10.11 333202_1002.qxd 12/8/05 9:00 AM Page 737 Te c h n o l o g y Use a graphing utility to confirm the equation found in Example 1. In order to graph the equation, you may have to use two separate equations: 8x y1 Upper part and 8x. y2 Lower part Section 10.2 Introduction to Conics: Parabolas 737 Example 1 Vertex at the Origin Find the standard equation of the parabola with vertex at the origin and focus 2, 0. Solution The axis of the parabola is horizontal, passing through in Figure 10.12. 0, 0 and 2, 0, as shown y 2 1 −1 −2 2 y x= 8 Focus (2, 0) 2 3 4 x Vertex 1 (0, 0) You may want to review the technique of completing the square found in Appendix A.5, which will be used to rewrite each of the conics in standard form. FIGURE 10.12 So, the standard form is equation is y 2 8x. y 2 4px, where h 0, k 0, and p 2. So, the Now try Exercise 33. Example 2 Finding the Focus of a Parabola Find the focus of the parabola given by y 1 2 x 2 x 1 2. Solution To find the focus, convert to standard form by completing the square. y − 2 Vertex ( 1, 1) 1 2 ) 1 ( 1,− Focus −3 −2 −1 x 1 y = − −21 x 2 x + 1 2 −1 −2 FIGURE 10.13 y 1 2 x 2 x 1 2 Write original equation. 2y x 2 2x 1 Multiply each side by –2. 1 2y x 2 2x 1 1 2y x 2 2x 1 2 2y x2 2x 1 2y 1 x 12 Comparing this equation with x h2 |
4p y k Add 1 to each side. Complete the square. Combine like terms. Standard form you can conclude that is negative, the parabola opens downward, as shown in Figure 10.13. So, the focus of the parabola is h, k p 1, 1 Because and. h 1, k 1, p 1 2. p 2 Now try Exercise 21. 333202_1002.qxd 12/8/05 9:00 AM Page 738 738 Chapter 10 Topics in Analytic Geometry y 8 6 4 (x − 2)2 = 12(y − 1) Focus (2, 4) Vertex (2, 1) −4 −2 2 4 6 8 x −2 −4 FIGURE 10.14 Light source at focus Example 3 Finding the Standard Equation of a Parabola Find the standard form of the equation of the parabola with vertex focus 2, 4. 2, 1 and Solution Because the axis of the parabola is vertical, passing through consider the equation 2, 1 and 2, 4, x h2 4p y k where h 2, k 1, and p 4 1 3. So, the standard form is x 22 12 y 1. You can obtain the more common quadratic form as follows. x 22 12 y 1 x2 4x 4 12y 12 x2 4x 16 12y 1 12 x 2 4x 16 y Write original equation. Multiply. Add 12 to each side. Divide each side by 12. The graph of this parabola is shown in Figure 10.14. Now try Exercise 45. Focus Axis Application Parabolic reflector: Light is reflected in parallel rays. FIGURE 10.15 Axis P α Focus α Tangent line FIGURE 10.16 A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord. The specific focal chord perpendicular to the axis of the parabola is called the latus rectum. Parabolas occur in a wide variety of applications. For instance, a parabolic reflector can be formed by revolving a parabola around its axis. The resulting surface has the property that all incoming rays parallel to the axis are reflected through the focus of the parabola. This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversely, the light rays emanating from the focus of a parabolic reflector used in a flashlight are all |
parallel to one another, as shown in Figure 10.15. A line is tangent to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. Tangent lines to parabolas have special properties related to the use of parabolas in constructing reflective surfaces. Reflective Property of a Parabola The tangent line to a parabola at a point makes equal angles with the following two lines (see Figure 10.16). P 1. The line passing through P and the focus 2. The axis of the parabola 333202_1002.qxd 12/8/05 9:00 AM Page 739 y y = x2 1 d 2 ( 0, 1 4 ) α (1, 1) −1 d 1 α 1 x Section 10.2 Introduction to Conics: Parabolas 739 Example 4 Finding the Tangent Line at a Point on a Parabola Find the equation of the tangent line to the parabola given by 1, 1. y x 2 at the point Solution p 1 and the focus is For this parabola, 4 0, b can find the -intercept two sides of the isosceles triangle shown in Figure 10.17: 0, 1, y as shown in Figure 10.17. You of the tangent line by equating the lengths of the 4 d1 1 4 b (0, b) and d2 FIGURE 10.17 1 02. Note that is important because the distance must be positive. Setting The order of subtraction for the distance d1 rather than produces d2 d1 1 4 b Te c h n o l o g y Use a graphing utility to confirm the result of Example 4. By graphing x 2 y1 and 2x 1 y2 in the same viewing window, you should be able to see that the line touches the parabola at the point 1, 1. 1 4 b 5 4 b 1. So, the slope of the tangent line is m 1 1 1 0 2 and the equation of the tangent line in slope-intercept form is y 2x 1. Now try Exercise 55. W RITING ABOUT MATHEMATICS Television Antenna Dishes Cross sections of television antenna dishes are parabolic in shape. Use the figure shown to write a paragraph explaining why these dishes are parabolic. Amplifier Dish reflector Cable to radio or |
TV 333202_1002.qxd 12/8/05 9:00 AM Page 740 740 Chapter 10 Topics in Analytic Geometry 10.2 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A ________ is the intersection of a plane and a double-napped cone. 2. A collection of points satisfying a geometric property can also be referred to as a ________ of points. x, y 3. A ________ is defined as the set of all points in a plane that are equidistant from a fixed line, called the ________, and a fixed point, called the ________, not on the line. 4. The line that passes through the focus and vertex of a parabola is called the ________ of the parabola. 5. The ________ of a parabola is the midpoint between the focus and the directrix. 6. A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a ________ ________. 7. A line is ________ to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1– 4, describe in words how a plane could intersect with the double-napped cone shown to form the conic section. (e) y 4 (f) y 4 −6 −4 −2 x −4 −2 2 x −4 5. 7. 9. y 2 4x x 2 8y y 12 4x 3 6. 8. 10. x 2 2y y 2 12x x 32 2y 1 In Exercises 11–24, find the vertex, focus, and directrix of the parabola and sketch its graph. 11. 13. 15. 17. 18. 19. 21. 23. 24. y 1 2x 2 y 2 6x x 2 6y 0 x 1 2 8y 4y 2 2 x 2 2x 5 y 1 4 y 2 6y 8x 25 0 y 2 4y 4x 0 12. 14. 16. y 2x 2 y 2 3x x y 2 0 20. 22. 2 4y 1 x 1 2 y2 2y 33 x 1 |
4 In Exercises 25–28, find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. 26. 25. x2 4x 6y 2 0 x 2 2x 8y 9 0 y 2 x y 0 27. 28. y 2 4x 4 0 1. Circle 3. Parabola 2. Ellipse 4. Hyperbola In Exercises 5–10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) y (b) y 4 2 −2 (c) 2 6 y 2 −4 −6 −6 −4 −2 x x 6 4 2 −4 −2 2 4 x (d) y −4 2 −2 −4 x 4 333202_1002.qxd 12/8/05 9:00 AM Page 741 In Exercises 29–40, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. 29. y 30. 6 (3, 6) 4 2 y 8 (−2, 6) −8 −4 4 x −4 −2 2 4 x −8 31. Focus: 32. Focus: 33. Focus: 34. Focus: 2 0, 3 5 2, 0 2, 0 0, 2 35. Directrix: 36. Directrix: 37. Directrix: 38. Directrix: y 1 y 3 x 2 x 3 39. Horizontal axis and passes through the point 40. Vertical axis and passes through the point 4, 6 3, 3 In Exercises 41–50, find the standard form of the equation of the parabola with the given characteristics. 41. y 42. y (2, 0) 2 (3, 1) (4, 0) 2 4 6 −2 − 4 43. y 8 − ( 4, 0) (0, 4) 4 8 4 2 (4.5, 4) (5, 3) 2 4 44. y 12 8 (0, 0) − 4 − 4 8 (3, −3) x x x x 3, 2 − 8 45. Vertex: 46. Vertex: 47. Vertex: 48. Vertex: 49. Focus: 50. Focus: 5, 2; 1, 2; 0, 4; |
2, 1; 2, 2; 0, 0; focus: focus: directrix: 1, 0 y 2 directrix: directrix: directrix: x 1 x 2 y 8 Section 10.2 Introduction to Conics: Parabolas 741 In Exercises 51 and 52, change the equation of the parabola so that its graph matches the description. 51. 52. y 3 2 6x 1; y 1 2 2x 4; upper half of parabola lower half of parabola In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. Parabola y2 8x 0 x2 12y 0 53. 54. Tangent Line x y 2 0 x y 3 0 In Exercises 55–58, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line. x 55. 56. 57. 58. x 2 2y, x 2 2y, y 2x 2, y 2x 2, 4, 8 3, 9 2 1, 2 2, 8 59. Revenue The revenue R (in dollars) generated by the sale of units of a patio furniture set is given by x x 1062 4 5 R 14,045. Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue. 60. Revenue The revenue R (in dollars) generated by the sale of units of a digital camera is given by x x 1352 5 7 R 25,515. Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue. 61. Satellite Antenna The receiver in a parabolic television dish antenna is 4.5 feet from the vertex and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the vertex is at the origin.) y Receiver 4.5 ft x 333202_1002.qxd 12/8/05 9:00 AM Page 742 742 Chapter 10 Topics in Analytic Geometry Model It 62. Suspension Bridge Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top |
of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height of the suspension cables over the roadway at a distance of x meters from the center of the bridge. y Distance, x Height, y 0 250 400 500 1000 63. Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). 32 ft 0.4 ft Not drawn to scale Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle? 64. Highway Design Highway engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highway (see figure). Find an equation of the parabola. y 800 400 −400 −800 Interstate (1000, 800) 400 800 1200 1600 x (1000, −800) Street FIGURE FOR 64 65. Satellite Orbit A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by the satellite will have the minimum velocity necessary to escape Earth’s gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). 2, Circular t orbi y 4100 miles Parabolic path x Not drawn to scale (a) Find the escape velocity of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 4000 miles). 66. Path of a Softball The path of a softball is modeled by 12.5 y 7.125 x 6.252 where the coordinates x 0 was thrown. x y are measured in feet, with corresponding to the position from which the ball and (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory. feet per second |
at a height of Projectile Motion In Exercises 67 and 68, consider the path of a projectile projected horizontally with a velocity of feet, where the model for v the path is x2 v2 16 y s. s In this model (in which air resistance is disregarded), the height (in feet) of the projectile and distance (in feet) the projectile travels. is is the horizontal y x 333202_1002.qxd 12/8/05 9:00 AM Page 743 67. A ball is thrown from the top of a 75-foot tower with a (a) Find the area when p 2 and b 4. Section 10.2 Introduction to Conics: Parabolas 743 velocity of 32 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontally before striking the ground? 68. A cargo plane is flying at an altitude of 30,000 feet and a speed of 540 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground? Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 69 and 70, determine whether 69. It is possible for a parabola to intersect its directrix. 70. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical. 71. Exploration Consider the parabola x 2 4py. (a) Use a graphing utility to graph the parabola for p 1, Describe the effect on the p 2, graph when p 3, p p 4. and increases. (b) Give a geometric explanation of why the area p approaches 0 as approaches 0. x1, y1 73. Exploration Let the parabola the parabola at the point is x 2 4py. be the coordinates of a point on The equation of the line tangent to y y1 x1 2p x x1. What is the slope of the tangent line? 74. Writing In your own words, state the reflective property of a parabola. Skills Review 75. In Exercises 75–78, list the possible rational zeros of given by the Rational Zero Test. f x x3 2x2 2x 4 f x 2x3 4x2 3x 10 f x 2x5 x2 16 f x 3x |
3 12x 22 76. 77. 78. f (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the chord passing through the focus and parallel to the directrix (see figure). How can the length of this chord be determined directly from the standard form of the equation of the parabola? y Chord Focus 2 x = 4py x (d) Explain how the result of part (c) can be used as a sketching aid when graphing parabolas. 72. Geometry The area of the shaded region in the figure is A 8 3 p12 b32. y 2 x = 4py y = b x 79. Find a polynomial with real coefficients that has the zeros 3, 2 i, 2 i. 80. Find all the zeros of and f x 2x3 3x 2 50x 75 if one of the zeros is x 3 2. 81. Find all the zeros of the function gx 6x4 7x3 29x 2 28x 20 if two of the zeros are x ±2. 82. Use a graphing utility to graph the function given by hx) 2x4 x3 19x 2 9x 9. Use the graph to approximate the zeros of h. In Exercises 83–90, use the information to solve the triangle. Round your answers to two decimal places. 85. 84. 83. 86. A 35, a 10, b 7 B 54, b 18, c 11 A 40, B 51, c 3 B 26, C 104, a 19 a 7, b 10, c 16 a 58, b 28, c 75 A 65, b 5, c 12 89. 90. B 71, a 21, c 29 88. 87. 333202_1003.qxd 12/8/05 9:01 AM Page 744 744 Chapter 10 Topics in Analytic Geometry 10.3 Ellipses What you should learn • Write equations of ellipses in standard form and graph ellipses. • Use properties of ellipses to model and solve real-life problems. • Find eccentricities of ellipses. Why you should learn it Ellipses can be used to model and solve many types of real-life problems. For instance, in Exercise 59 on page 751, an ellipse is used to model the orbit |
of Halley’s comet. Harvard College Observatory/ SPL/Photo Researchers, Inch k (, c a 2 b2 + c 2 = 2a b2 + c 2 = a 2 FIGURE 10.21 Introduction The second type of conic is called an ellipse, and is defined as follows. Definition of Ellipse An ellipse is the set of all points distances from two distinct fixed points (foci) is constant. See Figure 10.18. in a plane, the sum of whose x, y (x, y) d 2 d 1 Focus Focus Major axis Vertex Center Minor axis Vertex d2 d1 FIGURE 10.18 is constant. FIGURE 10.19 The line through the foci intersects the ellipse at two points called vertices. The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the major axis at the center is the minor axis of the ellipse. See Figure 10.19. You can visualize the definition of an ellipse by imagining two thumbtacks placed at the foci, as shown in Figure 10.20. If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced by the pencil will be an ellipse. FIGURE 10.20 To derive the standard form of the equation of an ellipse, consider the ellipse foci, h ± a, k; vertices, h, k; in Figure 10.21 with the following points: center, h ± c, k. Note that the center is the midpoint of the segment joining the foci. 333202_1003.qxd 12/8/05 9:01 AM Page 745 Section 10.3 Ellipses 745 The sum of the distances from any point on the ellipse to the two foci is constant. Using a vertex point, this constant sum is a c a c 2a Length of major axis Consider the equation of the ellipse y k2 b2 x h2 a2 If you let tion can be rewritten as a b, then the equa- 1. x h2 y k2 a2 which is the standard form of the equation of a circle with r a (see Section 1.2). radius Geometrically, when for an ellipse, the major |
and minor axes are of equal length, and so the graph is a circle. a b or simply the length of the major axis. Now, if you let ellipse, the sum of the distances between That is, x, y x, y and the two foci must also be be any point on the 2a 2a. b2 a2 c 2, which implies that the Finally, in Figure 10.21, you can see that equation of the ellipse is b2x h2 a 2y k 2 a 2b 2 x h 2 a 2 y k2 b2 1. You would obtain a similar equation in the derivation by starting with a vertical major axis. Both results are summarized as follows. Standard Equation of an Ellipse The standard form of the equation of an ellipse, with center respectively, where major and minor axes of lengths and 2b, 2a h, k and 0 < b < a, is x h2 a 2 x h 2 b2 y k2 b2 y k2 a 2 1 1. Major axis is horizontal. Major axis is vertical. c The foci lie on the major axis, units from the center, with 0, 0, If the center is at the origin forms. c 2 a2 b2. the equation takes one of the following x 2 a2 y 2 b2 1 Major axis is horizontal. x 2 b2 y 2 a2 1 Major axis is vertical. Figure 10.22 shows both the horizontal and vertical orientations for an ellipse. y y (x − h)2 2a + (y − k)2 2b = 1 (x − h)2 2b + (y − k)2 2a = 1 (h, k) 2b (h, k) 2a 2a Major axis is horizontal. FIGURE 10.22 x 2b x Major axis is vertical. 333202_1003.qxd 12/8/05 9:01 AM Page 746 746 Chapter 10 Topics in Analytic Geometry y 4 3 b = 5 x (0, 1) (2, 1) (4, 1) −1 −1 −2 1 3 a = 3 FIGURE 10.23 ( x + 2 3) 22 + 2 ( 1) y − 12 = 1 (−5, 1) (−3, 2) (−1, 1) ( ) −3 − 3, 1 (−3, 1) ( )3 |
, 1 −3 + y 4 3 2 1 −5 − 4 −3 (−3, 0) −1 x −1 Example 1 Finding the Standard Equation of an Ellipse Find the standard form of the equation of the ellipse having foci at 4, 1 and a major axis of length 6, as shown in Figure 10.23. 0, 1 and Solution Because the foci occur at the distance from the center to one of the foci is know that c 2 a2 b2, Now, from b a2 c2 32 22 5. a 3. 4, 1, 0, 1 and you have the center of the ellipse is c 2. Because 2, 1) 2a 6, and you Because the major axis is horizontal, the standard equation is x 2 2 32 y 12 52 1. This equation simplifies to y 12 5 x 22 9 1. Now try Exercise 49. Example 2 Sketching an Ellipse Sketch the ellipse given by x 2 4y 2 6x 8y 9 0. Solution Begin by writing the original equation in standard form. In the fourth step, note that 9 and 4 are added to both sides of the equation when completing the squares. x 2 4y 2 6x 8y 9 0 Write original equation. x 2 6x 4y 2 8y 9 x 2 6x 4y 2 2y 9 Group terms. Factor 4 out of y-terms. x 2 6x 9 4y 2 2y 1 9 9 41 x 3 2 4y 1 2 4 x 32 4 x 32 22 y 12 1 y 12 12 1 1 Divide each side by 4. Write in standard form. Write in completed square form. x a 2 22, h, k 3, 1. Because From this standard form, it follows that the center is the endpoints of the major axis lie two the denominator of the -term is units to the right and left of the center. Similarly, because the denominator of the y the endpoints of the minor axis lie one unit up and down from -term is So, the the center. Now, from foci of the ellipse are The ellipse is shown in Figure 10.24. c2 a2 b2, 3 3, 1 c 22 12 3. 3 3, 1. b 2 12, you have and FIGURE 10.24 Now try Exercise 25. 333202_1003.qxd 12/8/05 9:01 AM |
Page 747 Section 10.3 Ellipses 747 Example 3 Analyzing an Ellipse Find the center, vertices, and foci of the ellipse 4x 2 y 2 8x 4y 8 0. Solution By completing the square, you can write the original equation in standard form. 4x 2 y 2 8x 4y 8 0 4x 2 8x y 2 4y 8 4x 2 2x y 2 4y 8 Write original equation. Group terms. Factor 4 out of terms. (x − 1)2 22 + (y + 2)2 42 y = 1 Vertex Focus 2 (1, 2)− Center Focus Vertex (1, 2) x 4 (1, 6)− −( 1, 2 + 2 3 2 ( −4 −2 ( − − 1, 2 2 3 ( FIGURE 10.25 4x 2 2x 1 y 2 4y 4 8 41 4 4x 1 2 y 2 2 16 x 1 2 4 x 1 2 22 y 22 16 y 22 42 1 1 Divide each side by 16. Write in standard form. Write in completed square form. The major axis is vertical, where h 1, k 2, a 4, b 2, and c a2 b2 16 4 12 23. So, you have the following. Center: 1, 2 Vertices: 1, 6 1, 2 Foci: 1, 2 23 1, 2 23 The graph of the ellipse is shown in Figure 10.25. Now try Exercise 29. Te c h n o l o g y You can use a graphing utility to graph an ellipse by graphing the upper and lower portions in the same viewing window. For instance, to graph the ellipse in Example 3, first solve for to get y 2 41 y1 x 12 4 and 2 41 y2 x 12 4. Use a viewing window in which the graph shown below. 6 ≤ x ≤ 9 and 7 ≤ y ≤ 3. You should obtain −6 3 −7 9 333202_1003.qxd 12/8/05 9:01 AM Page 748 748 Chapter 10 Topics in Analytic Geometry Application Ellipses have many practical and aesthetic uses. For instance, machine gears, supporting arches, and acoustic designs often involve elliptical shapes. The orbits of satellites and planets are also ellipses. Example 4 investigates the elliptical orbit of the moon about Earth. Example 4 An Application Involving an Ell |
iptical Orbit The moon travels about Earth in an elliptical orbit with Earth at one focus, as shown in Figure 10.26. The major and minor axes of the orbit have lengths of 768,800 kilometers and 767,640 kilometers, respectively. Find the greatest and smallest distances (the apogee and perigee), respectively from Earth’s center to the moon’s center. Solution Because 2a 768,800 2b 767,640, you have a 384,400 and and b 383,820 Moon 767,640 km Earth 768,800 km Perigee Apogee FIGURE 10.26 which implies that c a 2 b2 384,4002 383,8202 21,108. So, the greatest distance between the center of Earth and the center of the moon is Note in Example 4 and Figure 10.26 that Earth is not the center of the moon’s orbit. a c 384,400 21,108 405,508 kilometers and the smallest distance is a c 384,400 21,108 363,292 kilometers. Now try Exercise 59. Eccentricity One of the reasons it was difficult for early astronomers to detect that the orbits of the planets are ellipses is that the foci of the planetary orbits are relatively close to their centers, and so the orbits are nearly circular. To measure the ovalness of an ellipse, you can use the concept of eccentricity. Definition of Eccentricity The eccentricity of an ellipse is given by the ratio e e c a. Note that 0 < e < 1 for every ellipse. 333202_1003.qxd 12/8/05 9:01 AM Page 749 Section 10.3 Ellipses 749 To see how this ratio is used to describe the shape of an ellipse, note that because the foci of an ellipse are located along the major axis between the vertices and the center, it follows that 0 < c < a. For an ellipse that is nearly circular, the foci are close to the center and the ratio ca is small, as shown in Figure 10.27. On the other hand, for an elongated ellipse, the foci are close to the vertices, and the ratio is close to 1, as shown in Figure 10.28. ca y y Foci Foci c e is small is close to 1. FIGURE 10.27 FIGURE 10.28 a a The |
orbit of the moon has an eccentricity of e 0.0549, and the eccentricities A S A N The time it takes Saturn to orbit the sun is equal to 29.4 Earth years. Venus: Mercury: of the nine planetary orbits are as follows. e 0.2056 e 0.0068 e 0.0167 e 0.0934 e 0.0484 Mars: Jupiter: Earth: Saturn: Uranus: Neptune: Pluto: e 0.0542 e 0.0472 e 0.0086 e 0.2488 W RITING ABOUT MATHEMATICS Ellipses and Circles a. Show that the equation of an ellipse can be written as x h2 a2 y k2 a21 e2 1. b. For the equation in part (a), let utility to graph the ellipse for e 0.1. a 4, e 0.95, h 1, and e 0.75, k 2, e 0.5, and use a graphing e 0.25, e and Discuss the changes in the shape of the ellipse as approaches 0. c. Make a conjecture about the shape of the graph in part (b) when e 0. What is the equation of this ellipse? What is another name for an ellipse with an eccentricity of 0? 333202_1003.qxd 12/8/05 9:01 AM Page 750 750 Chapter 10 Topics in Analytic Geometry 10.3 Exercises VOCABULARY CHECK: Fill in the blanks. x, y 1. An ________ is the set of all points in a plane, the sum of whose distances from two distinct fixed points, called ________, is constant. 2. The chord joining the vertices of an ellipse is called the ________ ________, and its midpoint is the ________ of the ellipse. 3. The chord perpendicular to the major axis at the center of the ellipse is called the ________ ________ of the ellipse. 4. The concept of ________ is used to measure the ovalness of an ellipse. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–6, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d |
), (e), and (f).] (a) y (b) In Exercises 7–30, identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. x 2 4 −4 x 4 y 4 2 −4 y 6 2 −6 y 4 −4 x 4 6 x 4 2 −4 (c) (e) −4 −6 1. 3. 5. 6 25 4 x 2 2 16 x 2 2 9 (d) x 2 4 −4 y 4 −4 (f) x 2 −4 y 2 −2 −2 −6 1 1 y 1 2 1 y 22 4 1 2. 4 144 81 y 2 x2 9 9 y 2 x 2 28 64 x 42 12 x 52 94 x 32 254 y 32 16 1 y 12 1 y 12 254 1 y 2 x 2 16 25 y 2 x2 25 25 y 2 x 2 9 5 x 32 16 1 1 1 8. 10. 12. y 52 25 1 14. x2 49 y 12 49 1 16. 18. 1 x 22 y 42 14 9x 2 4y 2 36x 24y 36 0 9x 2 4y 2 54x 40y 37 0 x2 y2 2x 4y 31 0 x2 5y2 8x 30y 39 0 3x2 y2 18x 2y 8 0 6x2 2y2 18x 10y 2 0 x2 4y2 6x 20y 2 0 x2 y 2 4x 6y 3 0 9x 2 9y 2 18x 18y 14 0 16x 2 25y 2 32x 50y 16 0 9x 2 25y 2 36x 50y 60 0 16x2 16y 2 64x 32y 55 0 7. 9. 11. 13. 15. 17. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. In Exercises 31–34, use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for and obtain two equations.) y 31. 5x 2 3y 2 15 12x 2 20y 2 12x 40y 37 0 33. 34. 36x 2 9y 2 48x 36y 72 0 32. |
3x 2 4y 2 12 333202_1003.qxd 12/8/05 9:01 AM Page 751 In Exercises 35–42, find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 35. y 8 36. (0, 4) (2, 0) 8 4 (0, 4)− x − ( 2, 0) −8 −4 −8 y 4 −4 ( ) 0, 3 2 (2, 0) 4 ( 0, − ) 3 2 x (−2, 0) −4 37. Vertices: 38. Vertices: 39. Foci: 40. Foci: ±5, 0; ±2, 0; ±6, 0; 0, ±8; foci: ±2, 0 0, ±4 major axis of length 12 foci: major axis of length 8 41. Vertices: 4, 2 42. Major axis vertical; passes through the points passes through the point 0, ±5; 0, 4 2, 0 In Exercises 43–54, find the standard form of the equation of the ellipse with the given characteristics. 44. y 43. y (1, 3) 6 5 4 3 2 1 (2, 6) (3, 3) (2, 0) 1 432 5 6 x 45. y 46. − ( 2, 6) − ( 6, 3) 8 4 2 −6 −4 − ( 2, 0) (2, 3) x 2 4 4 3 2 1 −1 −2 −3 −4 y 1 −1 −1 −2 −3 −4 (4, 4) (7, 0) (1, 0) 6 2 4 5 3 (4, −4) x 8 x (2, 0) 2 1 (0, −1) 3 (2, −2) (4, −1) 47. Vertices: 0, 4, 4, 4; minor axis of length 2 48. Foci: 49. Foci: 50. Center: 51. Center: 52. Center: 0, 0, 4, 0; 0, 0, 0, 8; 2, 1; 0, 4; a 2c; 3, 2; a 3c; 0, 2, 4, 2; 53. Vertices: 2, 3, 2, 1 major axis of length 8 vertex: major axis of length 16 ; 2 |
, 1 2 vertices: minor axis of length 2 4, 4, 4, 4 foci: 1, 2, 5, 2 endpoints of the minor axis: Section 10.3 Ellipses 751 54. Vertices: 5, 0, 5, 12; 1, 6, 9, 6 endpoints of the minor axis: 55. Find an equation of the ellipse with vertices eccentricity e 3 5. 56. Find an equation of the ellipse with vertices eccentricity e 1 2. ±5, 0 and 0, ±8 and 57. Architecture A semielliptical arch over a tunnel for a one-way road through a mountain has a major axis of 50 feet and a height at the center of 10 feet. (a) Draw a rectangular coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points. (b) Find an equation of the semielliptical arch over the tunnel. and (c) You are driving a moving truck that has a width of 8 feet and a height of 9 feet. Will the moving truck clear the opening of the arch? 58. Architecture A fireplace arch is to be constructed in the shape of a semiellipse. The opening is to have a height of 2 feet at the center and a width of 6 feet along the base (see figure). The contractor draws the outline of the ellipse using tacks as described at the beginning of this section. Give the required positions of the tacks and the length of the string3 −2 −1 −2 Model It 59. Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately 0.967. The length of the major axis of the orbit is approximately 35.88 astronomical units. (An astronomical unit is about 93 million miles.) (a) Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the x -axis. (b) Use a graphing utility to graph the equation of the orbit. (c) Find the greatest (aphelion) and smallest (perihelion) distances from the sun’s center to the comet’s center. 333202_1003.qxd 12/8/05 9:01 AM Page 752 752 Chapter 10 Topics in Analy |
tic Geometry 60. Satellite Orbit The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in 1957). Its highest point above Earth’s surface was 947 kilometers, and its lowest point was 228 kilometers (see figure). The center of Earth was the focus of the elliptical orbit, and the radius of Earth is 6378 kilometers. Find the eccentricity of the orbit. Focus 228 km 947 km 61. Motion of a Pendulum The relation between the velocity (in radians per second) of a pendulum and its angular from the vertical can be modeled by a y 0 radian and 0, y displacement semiellipse. A 12-centimeter pendulum crests when the angular displacement is 0.2 radian. When the pendulum is at equilibrium the velocity is radians per second. 0.2 1.6 (a) Find an equation that models the motion of the pendulum. Place the center at the origin. (b) Graph the equation from part (a). (c) Which half of the ellipse models the motion of the pendulum? 62. Geometry A line segment through a focus of an ellipse with endpoints on the ellipse and perpendicular to the major axis is called a latus rectum of the ellipse. Therefore, an ellipse has two latera recta. Knowing the length of the latera recta is helpful in sketching an ellipse because it yields other points on the curve (see figure). Show that the length of each latus rectum is 2b2a. y Latera recta F1 F2 x Synthesis True or False? the statement is true or false. Justify your answer. In Exercises 67 and 68, determine whether 67. The graph of x2 4y4 4 0 is an ellipse. 68. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricity of the ellipse is large (close to 1). 69. Exploration Consider the ellipse x 2 a2 y 2 b2 1, a b 20. (a) The area of the ellipse is given by area of the ellipse as a function of A ab. a. Write the (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a), |
and make a conjecture about the shape of the ellipse with maximum area. 8 9 10 11 12 13 a A (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c). 70. Think About It At the beginning of this section it was noted that an ellipse can be drawn using two thumbtacks, a string of fixed length (greater than the distance between the two tacks), and a pencil. If the ends of the string are fastened at the tacks and the string is drawn taut with a pencil, the path traced by the pencil is an ellipse. (a) What is the length of the string in terms of a? (b) Explain why the path is an ellipse. Skills Review In Exercises 71–74, determine whether the sequence is arithmetic, geometric, or neither. 71. 80, 40, 20, 10, 5,... 5 2, 2,1 2,... 73. 3 2, 7 2, 1 72. 66, 55, 44, 33, 22,... 74. 1 4, 1 2, 1, 2, 4,... In Exercises 63– 66, sketch the graph of the ellipse, using latera recta (see Exercise 62). 1 y 2 x 2 16 9 5x 2 3y 2 15 63. 65. 1 y 2 x 2 1 4 9x 2 4y 2 36 64. 66. In Exercises 75–78, find the sum. 75. 77. 6 n0 10 n0 3n n 54 3 76. 6 n0 3n 78. 10 n1 n1 43 4 333202_1004.qxd 12/8/05 9:03 AM Page 753 10.4 Hyperbolas What you should learn • Write equations of hyperbolas in standard form. • Find asymptotes of and graph hyperbolas. • Use properties of hyperbolas to solve real-life problems. • Classify conics from their general equations. Why you should learn it Hyperbolas can be used to model and solve many types of real-life problems. For instance, in Exercise 42 on page 761, hyperbolas are used in long distance radio navigation for aircraft and ships. AP/Wide World Photos Section 10.4 Hyperbolas 753 Introduction The third type of conic is called a hyperb |
ola. The definition of a hyperbola is similar to that of an ellipse. The difference is that for an ellipse the sum of the distances between the foci and a point on the ellipse is fixed, whereas for a hyperbola the difference of the distances between the foci and a point on the hyperbola is fixed. Definition of Hyperbola A hyperbola is the set of all points distances from two distinct fixed points (foci) is a positive constant. See Figure 10.29. in a plane, the difference of whose x, y x y (, ) 2d 1d Focus Focus c a Branch Branch Vertex Center Vertex Transverse axis d 2 d− 1 is a positive constant. FIGURE 10.29 FIGURE 10.30 The graph of a hyperbola has two disconnected branches. The line through the two foci intersects the hyperbola at its two vertices. The line segment connecting the vertices is the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola. See Figure 10.30. The development of the standard form of the equation of a hyperbola is similar to that of an ellipse. Note in the definition below that and are related differently for hyperbolas than for ellipses. b, a, c Standard Equation of a Hyperbola The standard form of the equation of a hyperbola with center h, k is x h2 a 2 y k 2 a2 y k2 b2 x h2 b 2 1 1. Transverse axis is horizontal. Transverse axis is vertical. c 2 a2 b2. a The vertices are units from the center, and the foci are units from the center. Moreover, 0, 0, origin y 2 x 2 b2 a2 the equation takes one of the following forms. If the center of the hyperbola is at the Transverse axis is vertical. Transverse axis is horizontal. x 2 b2 y 2 a2 1 1 c 333202_1004.qxd 12/8/05 9:03 AM Page 754 754 Chapter 10 Topics in Analytic Geometry Figure 10.31 shows both the horizontal and vertical orientations for a hyperbola. 2 ) ( y − 2 ) k− 2 b = 1 ( x h Transverse axis is horizontal. FIGURE 10.31 2 ) ( x − 2 ) h− 2 |
b = 1 ( y k− ) Transverse axis is vertical. Example 1 Finding the Standard Equation of a Hyperbola Find the standard form of the equation of the hyperbola with foci 5, 2 and vertices 4, 2. 0, 2 and 1, 2 and When finding the standard form of the equation of any conic, it is helpful to sketch a graph of the conic with the given characteristics. Solution By the Midpoint Formula, the center of the hyperbola occurs at the point Furthermore, a 4 2 2, c 5 2 3 and it follows that and 2, 2. b c2 a2 32 22 9 4 5. So, the hyperbola has a horizontal transverse axis and the standard form of the equation is x 22 22 y 22 52 1. See Figure 10.32. This equation simplifies to y 22 5 x 22 4 1. (x − 2)2 2 2 − (y − 2)0, 2) (−1, 2) (4, 2) (2, 2) (5, 2) 1 2 3 4 −1 x FIGURE 10.32 Now try Exercise 27. 333202_1004.qxd 12/8/05 9:03 AM Page 755 Conjugate axis (h, k + b) Asy m ptote (h, k) (h, k − b) A s y m (h − a, k) (h + a, k) p t o t e FIGURE 10.33 Asymptotes of a Hyperbola Section 10.4 Hyperbolas 755 Each hyperbola has two asymptotes that intersect at the center of the hyperbola, as shown in Figure 10.33. The asymptotes pass through the vertices of a rectangle 2b by of dimensions is the conjugate joining and axis of the hyperbola. with its center at or h b, k and h b, k The line segment of length 2b, h, k b 2a h, k b h, k. Asymptotes of a Hyperbola The equations of the asymptotes of a hyperbola are. Transverse axis is horizontal. Transverse axis is vertical. Example 2 Using Asymptotes to Sketch a Hyperbola Sketch the hyperbola whose equation is 4x 2 y 2 16. Solution Divide each side of the original equation by 16, and rewrite the equation in standard |
form. y 2 42 Write in standard form. x 2 22 1 2, 0, 0, 4 a 2, 2, 0 0, 4. b 4, and the transverse axis is horiFrom this, you can conclude that and and the endpoints of the conzontal. So, the vertices occur at Using these four points, you are able to and jugate axis occur at sketch the rectangle shown in Figure 10.34. Now, from you have 25, 0 c 22 42 20 25. Finally, by drawing the asymptotes through the corners of this recand tangle, you can complete the sketch shown in Figure 10.35. Note that the asymptotes are So, the foci of the hyperbola are c2 a2 b2, 25, 0. y 2x. y 2x and y 8 6 (0, 4) − ( 2, 0) −6 −4 (2, 0) 4 6 x 2(− ) 5, 0 −6 −4 (0, 4)− −6 FIGURE 10.34 FIGURE 10.35 Now try Exercise 7. y 8 6 −6 2( ) 5, 0 4 2 x 22 6 − 2 y 42 x = 1 333202_1004.qxd 12/8/05 11:22 AM Page 756 756 Chapter 10 Topics in Analytic Geometry Example 3 Finding the Asymptotes of a Hyperbola Sketch the hyperbola given by of its asymptotes and the foci. 4x 2 3y 2 8x 16 0 and find the equations Solution 4x 2 3y 2 8x 16 0 4x2 8x 3y2 16 4x 2 2x 3y 2 16 4x 2 2x 1 3y 2 16 4 4x 12 3y 2 12 y2 4 x 12 32 x 12 3 y 2 22 1 1 Write original equation. Group terms. Factor 4 from x- terms. Add 4 to each side. Write in completed square form. Divide each side by 12. Write in standard form. 1, 0, has vertices From this equation you can conclude that the hyperbola has a vertical transverse 1, 2 and axis, centered at and has a conjugate 1 3, 0. To sketch the hyperbola, axis with endpoints draw a rectangle through these four points. The asymptotes are the lines passing through the corners of the rectangle. Using you |
can conclude that the equations of the asymptotes are 1 3, 0 1, 2, b 3, a 2 and and y 2 3 x 1 and y 2 3 x 1. Finally, you can determine the foci by using the equation have 1, 2 7. c 22 32 7, The hyperbola is shown in Figure 10.36. and the foci are c2 a2 b2. 1, 2 7 So, you and Now try Exercise 13. Te 7 −1, 2 + 4 3 (−1, 2) (−1, 0) 1 y2 22 − (x + 1)2 ( ) 3 2 −4 −3 −3 (−1, −2) ( ) −1, −2 − 7 FIGURE 10.36 You can use a graphing utility to graph a hyperbola by graphing the upper and lower portions in the same viewing window. For instance, to graph the hyperbola in Example 3, first solve for to get 21 y1 x 12 3 and 21 y2 y x 12 3. Use a viewing window in which You should obtain the graph shown below. Notice that the graphing utility does not draw the asymptotes. However, if you trace along the branches, you will see that the values of the hyperbola approach the asymptotes. and. −9 6 −6 9 333202_1004.qxd 12/8/05 9:03 AM Page 757 y = 2x − 8 Example 4 Using Asymptotes to Find the Standard Equation Section 10.4 Hyperbolas 757 Find the standard form of the equation of the hyperbola having vertices and and having asymptotes 3, 5 3, 1 y 2x 8 and y 2x 4 y 2 (3, 1) −2 2 4 6 x −2 − 4 −6 (3, −5) FIGURE 10.37 as shown in Figure 10.37. Solution By the Midpoint Formula, the center of the hyperbola is a 3. hyperbola has a vertical transverse axis with you can determine the slopes of the asymptotes to be 3, 2. Furthermore, the From the original equations, y = −2x + 4 m1 2 a b a 3 and, because 2 a b and m2 2 a b you can conclude 2 3 b b 3 2. So, the standard form of the equation is y 22 32 1. x 32 |
2 3 2 Now try Exercise 35. As with ellipses, the eccentricity of a hyperbola is e c a Eccentricity c > a, it follows that and because If the eccentricity is large, the branches of the hyperbola are nearly flat, as shown in Figure 10.38. If the eccentricity is close to 1, the branches of the hyperbola are more narrow, as shown in Figure 10.39. e > 1. y y e is large. e is close to 1. Vertex Focus x Focus Vertex FIGURE 10.38 FIGURE 10.39 333202_1004.qxd 12/8/05 9:03 AM Page 758 758 Chapter 10 Topics in Analytic Geometry Applications The following application was developed during World War II. It shows how the properties of hyperbolas can be used in radar and other detection systems. Example 5 An Application Involving Hyperbolas Two microphones, 1 mile apart, record an explosion. Microphone A receives the sound 2 seconds before microphone B. Where did the explosion occur? (Assume sound travels at 1100 feet per second.) y 3000 2000 0 0 2 2 B A x 2000 Solution Assuming sound travels at 1100 feet per second, you know that the explosion took place 2200 feet farther from B than from A, as shown in Figure 10.40. The locus of all points that are 2200 feet closer to A than to B is one branch of the hyperbola x2 a2 where y2 b2 1 2200 c a− c a− and 2 = 5280 c 2200 + 2( c a− ) = 5280 c 5280 2 2640 a 2200 2 1100. FIGURE 10.40 Hyperbolic orbit Vertex Sun p Elliptical orbit b2 c 2 a 2 26402 11002 5,759,600, So, the explosion occurred somewhere on the right branch of the hyperbola and you can conclude that x 2 1,210,000 y 2 5,759,600 1. Now try Exercise 41. Another interesting application of conic sections involves the orbits of comets in our solar system. Of the 610 comets identified prior to 1970, 245 have elliptical orbits, 295 have parabolic orbits, and 70 have hyperbolic orbits. The center of the sun is a focus of each of these orbits, and each orbit has a vertex at the point where the comet is closest to the sun, as shown in |
Figure 10.41. Undoubtedly, there have been many comets with parabolic or hyperbolic orbits that were not identified. We only get to see such comets once. Comets with elliptical orbits, such as Halley’s comet, are the only ones that remain in our solar system. is the distance between the vertex and the focus (in meters), and is the velocity of the comet at the vertex in (meters per second), then the type of orbit is determined as follows. If p v Parabolic orbit 1. Ellipse: 2. Parabola: 3. Hyperbola: v < 2GMp v 2GMp v > 2GMp FIGURE 10.41 In each of these relations, G 6.67 1011 gravitational constant). M 1.989 1030 kilograms (the mass of the sun) and cubic meter per kilogram-second squared (the universal 333202_1004.qxd 12/8/05 9:03 AM Page 759 Section 10.4 Hyperbolas 759 General Equations of Conics Classifying a Conic from Its General Equation Ax 2 Cy 2 Dx Ey F 0 The graph of is one of the following. 1. Circle: 2. Parabola: A C AC 0 A 0 or C 0, but not both. 3. Ellipse: AC > 0 A and have like signs. C 4. Hyperbola: AC < 0 A and have unlike signs. C The test above is valid if the graph is a conic. The test does not apply to equations such as x 2 y 2 1, whose graph is not a conic. Example 6 Classifying Conics from General Equations Classify the graph of each equation. a. b. c. d. 4x 2 9x y 5 0 4x 2 y 2 8x 6y 4 0 2x 2 4y 2 4x 12y 0 2x2 2y2 8x 12y 2 0 Solution a. For the equation 4x 2 9x y 5 0, you have AC 40 0. Parabola So, the graph is a parabola. b. For the equation 4x 2 y 2 8x 6y 4 0, you have AC 41 < 0. Hyperbola So, the graph is a hyperbola. c. For the equation 2x 2 4y 2 4x 12y 0, you have AC 24 > 0. Ellipse So, |
the graph is an ellipse. d. For the equation A C 2. 2x2 2y2 8x 12y 2 0, you have Circle So, the graph is a circle. Now try Exercise 49. W RITING ABOUT MATHEMATICS Sketching Conics Sketch each of the conics described in Example 6. Write a paragraph describing the procedures that allow you to sketch the conics efficiently Historical Note Caroline Herschel (1750–1848) was the first woman to be credited with detecting a new comet. During her long life, this English astronomer discovered a total of eight new comets. 333202_1004.qxd 12/8/05 9:03 AM Page 760 760 Chapter 10 Topics in Analytic Geometry 10.4 Exercises VOCABULARY CHECK: Fill in the blanks. 1. A ________ is the set of all points x, y in a plane, the difference of whose distances from two distinct fixed points, called ________, is a positive constant. 2. The graph of a hyperbola has two disconnected parts called ________. 3. The line segment connecting the vertices of a hyperbola is called the ________ ________, and the midpoint of the line segment is the ________ of the hyperbola. 4. Each hyperbola has two ________ that intersect at the center of the hyperbola. 5. The general form of the equation of a conic is given by ________. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.Eduspace.com. In Exercises 1–4, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) y 8 4 −8 4 8 −4 −8 y 8 −8 −4 4 8 −8 1 y 2 4 1 1. 3. x 2 y 2 25 9 x 1 2 16 (b) (d) x x y 8 x x −8 −4 4 8 −8 y 8 4 −4 4 8 −4 −8 2. 4. x 2 y 2 9 25 x 12 16 1 y 22 9 1 In Exercises 5–16, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph using the asymptotes |
as an aid. 5. x 2 y 2 1 7. 9. x 2 y 2 81 25 x 12 4 1 y 22 1 1 6. 8. x 2 9 x 2 36 y 2 25 y 2 4 1 1 1 1 x 32 144 y 62 19 y 12 14 y 22 25 x 22 14 x 32 116 9x 2 y 2 36x 6y 18 0 x 2 9y 2 36y 72 0 x 2 9y 2 2x 54y 80 0 16y 2 x 2 2x 64y 63 0 1 10. 11. 12. 13. 14. 15. 16. In Exercises 17–20, find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes. 17. 18. 19. 20. 2x 2 3y 2 6 6y 2 3x 2 18 9y 2 x 2 2x 54y 62 0 9x 2 y 2 54x 10y 55 0 In Exercises 21–26, find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. 21. Vertices: 22. Vertices: 23. Vertices: 24. Vertices: 0, ±2; ±4, 0; ±1, 0; 0, ±3; foci: 0, ±4 ±6, 0 asymptotes: foci: asymptotes: y ±5x y ±3x 25. Foci: 26. Foci: 0, ±8; ±10, 0; asymptotes: asymptotes: y ±4x y ± 3 4x In Exercises 27–38, find the standard form of the equation of the hyperbola with the given characteristics. 27. Vertices: 28. Vertices: 2, 0, 6, 0; 2, 3, 2, 3; foci: 0, 0, 8, 0 foci: 2, 6, 2, 6 333202_1004.qxd 12/8/05 9:03 AM Page 761 Section 10.4 Hyperbolas 761 29. Vertices: 30. Vertices: 31. Vertices: 4, 1, 4, 9; 2, 1, 2, 1); 2, 3, 2, 3; passes through the point 2, 1, 2, 1; |
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