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C-10 Appendix C PROJECT Specific or General? Whatever Works! Sometimes in math and science, when we’re stuck on a problem, we can make progress by turning from the specific to the general. This approach sometimes surprises students because of a mistaken belief that the general or more abstract case is always tougher than the specific. The specific equation that you’ll work to solve in this project is (a) Use a graphing utility to graph the equation y 24 14 x x. Then, by zooming in on the x-intercept, approximate the root of equation (1) to the nearest hundredth. (b) Now let’s try to find the exact value for the root. By using the process of isolating the radical and then squaring, show that equation (1) eventually becomes (c) At this point we are stuck; the equation obtained in part (b) can’t be solved by using any of the elementary factoring techniques from intermediate algebra. Here is where we shift our focus to a slightly more general problem. Instead of equation (1), we consider 2k 1k x x 0 (2) where k is a constant. Use the process of isolating the radical and then squaring to show that equation (2) eventually becomes x 4 2kx 2 x k 2 k 0 (3) (d) Although equation (3) is a fourth-degree equation in x, it is only a quadratic equation in k. That is, equation (3) can be written k 2 (2x 2 1)k (x 4 x) 0 24 14 x x 0 x 4 8x 2 x 12 0 Verify that this is equivalent to equation (3). Use the quadratic formula to solve this last equation for k in terms of x. You should obtain (after simplifying) two solutions: k x 2 x 1 or k x 2 1 (e) Finally now, in both of the equations obtained in part (d), replace k with 4 [that’s what it is in equation (1)], and solve each equation for x. (f) Three of the four values that you obtain in part (e) are extraneous solutions for equation (1). Use your approximation in part (a) and a calculator to decide which value is the actual root of equation (1). (1)
Appendix C C-11 MINI PROJECT An Inequality for the Garden Part of a landscape architect’s plans call for a rectangular garden with a perimeter of 150 ft. The architect wants to find the width w and length l that maximize the area A of the garden. Two preliminary sketches are shown in the figure that follows. (In both cases, note that the perimeter is 150 ft, as required.) Of the two sketches, the one on the right yields the larger area. In this project you’ll use the inequality given in Exercise 40(b)* to find exactly which combination of w and l yields the largest possible area. (a) As background, first review or rework Exercise 40(b). If you are working in a group, assign one person to do this and then present the result to the others. 25 ft 30 ft 50 ft 45 ft A=25 ft 50 ft=1250 ft@ A=30 ft 45 ft=1350 ft@ (b) Letting w denote the width of the rectangular garden, and assuming that the perimeter is 150 ft, show or explain why the area A is given by A w(75 w). (c) Use the inequality in Exercise 40(b) to show that A 56254 1406.25. In words: Given a perimeter of 150 ft, no matter how we pick w and l, the area can never exceed 1406.25 ft 2 . Hint: Apply the inequality to the product w(75 w). (d) From part (c) we know that the area can’t exceed 1406.25 ft 2 , so now the job is to find a combination of w and l yielding that area. To accomplish this, take the equation obtained in part (b), replace A with 1406.25, and then use a graphing utility to solve for w. (e) As a check on the work with the graphing utility (and to stay in shape with the algebra), use the quadratic formula to solve the same equation. (f) Now that you know w, find l and describe the dimensions of the required rectangle. *Exercise Set 2.3, Precalculus: A Problems-Oriented Approach, 7th edition
Page 1 and 2: APPENDIX C Title and Page Number Pr
Page 3 and 4: Appendix C C-3 MINI PROJECT Discuss
Page 5 and 6: Appendix C C-5 MINI PROJECT Thinkin
Page 7 and 8: Appendix C C-7 1. equilateral trian
Page 9: Appendix C C-9 MINI PROJECT Flying
Page 13 and 14: Appendix C C-13 (a) The two data po
Page 15 and 16: Appendix C C-15 and http://www.svob
Page 17 and 18: Appendix C C-17 For Figure D we can
Page 19 and 20: Appendix C C-19 MINI PROJECT A Grap
Page 21 and 22: Appendix C C-21 MINI PROJECT Who Ar
Page 23 and 24: Appendix C C-23 MINI PROJECT 1 How
Page 25 and 26: Appendix C C-25 either the quadrati
Page 27 and 28: Appendix C C-27 PROJECT The Least-S
Page 29 and 30: Appendix C C-29 Then, for any line
Page 31 and 32: Appendix C C-31 PROJECT Finding Som
Page 33 and 34: Appendix C C-33 3. When a person co
Page 35 and 36: Appendix C C-35 2. Use part (b) of
Page 37 and 38: Appendix C C-37 PROJECT Coffee Temp
Page 39 and 40: Appendix C C-39 MINI PROJECT More C
Page 41 and 42: Appendix C C-41 The calculations co
Page 43 and 44: Appendix C C-43 PROJECT Transits of
Page 45 and 46: Appendix C C-45 E A ¨ V ¨ Bª d S
Page 47 and 48: Appendix C C-47 PROJECT A Linear Ap
Page 49 and 50: Appendix C C-49 Continuing with the
Page 51 and 52: Appendix C C-51 PROJECT Constructin
Page 53 and 54: Appendix C C-53 1 y Figure A A grap
Page 55 and 56: Appendix C C-55 PROJECT Fourier Ser
Page 57 and 58: Appendix C C-57 sine and cosine fun
Page 59 and 60: Appendix C C-59 PROJECT The Motion
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Appendix C C-61 PROJECT The Design
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Appendix C C-63 denote the angles o
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Appendix C C-65 inverse trigonometr
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Appendix C C-67 PROJECT Inverse Sec
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Appendix C C-69 Alternatively, usin
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Appendix C C-71 PROJECT Snell’s L
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Appendix C C-73 vertically downward
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0 0 0 Appendix C C-75 PROJECT Vecto
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Appendix C C-77 Exercise 8 We outli
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Appendix C C-79 y y=2x+5 Îy=2 A (1
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 Appendi
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Appendix C C-83 PROJECT Parameteriz
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Appendix C C-85 It is worth noting
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Appendix C C-87 6. Show that the li
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Appendix C C-89 2. An altitude of a
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Appendix C C-91 PROJECT The Leontie
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Appendix C C-93 n x units of that
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Appendix C C-95 him/herself through
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Appendix C C-97 PROJECT The Leontie
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Appendix C C-99 As in the solution
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Appendix C C-101 MINI PROJECT 2 Con
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Appendix C C-103 PROJECT Using Hype
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Appendix C C-105 PROJECT Two Method
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Appendix C C-107 (b) Use Method 1 t
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Appendix C C-109 MINI PROJECT An Un
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Appendix C C-111 PROJECT More on Su
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Appendix C C-113 SOLUTION 25 25 a (
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Appendix C C-115 4. Simplify the fo
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