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per minute 23. (a) 9 25. (ab)β(e) Answers will vary. (b)β(e) Answers will vary. Chapter 4 Section 4.1 (page 290) Vocabulary Check (page 290) f1x lnx x2 4 2 ln c2 ln c1 ln 1 1 k2 2 252,6061.0310t 400.88t2 1464.6t 291,782 1 k1 t 13. y1 y2 11. c 15. (a) (b) 2. angle 5. acute; obtuse 1. Trigonometry 4. radian 6. complementary; supplementary 8. linear 9. angular 10. 3. coterminal 7. degree 2r2 A 1 1. 2 radians 3. 7. (a) Quadrant I 9. (a) Quadrant IV 11. (a) Quadrant III 3 radians (b) Quadrant III 5. 1 radian (b) Quadrant III (b) Quadrant II 333200_04_AN.qxd 12/9/05 1:32 PM Page A132 A132 13. (a) 15. (a) Answers to Odd-Numbered Exercises and Tests (b) 37. (a) (b) Ο 5 4 y y Ο 11 b) x x y y 405Β° 480Β° x x 405, 315 39. (a) 600, 120 41. (a) 43. (a) Complement: 324, 396 180, 540 Supplement: (b) (b) 72; (b) Complement: none; Supplement: 11 (b) Complement: none; Supplement: 45. (a) Complement: ; Supplement: 162 65 101 30 β3 47. (a) 51. (a) 55. 2.007 63. 69. 73. (a) 75. (a) 77. (a) 6 79. 5 50 85. 29 25.714 114.592 85.308 240 36 2 30 radians radians 6 270 (b) 5 6 (b) 210 3.776 57. 65. 49. (a) (b) 9 53. (a) 420 59. 9.285 67. 4 3 (b) 61. 66 0.014 337.500 71. (a) 54.75 756.000 128.5 (b) (b) 81. (b |
) (b) 330.007 145 48 3 34 48 32 83. radians 7 15 inches 47.12 inches 87. 8 3 radian 2 9 89. 3 meters 91. square inches 8.38 square inches 2 1.14 0.071 radian 4.04 93. 12.27 square feet 97. 101. (a) 728.3 revolutions per minute (b) 4576 radians per minute 95. 591.3 miles 5 12 radian 99. 103. (a) 10,400 radians per minute 32,672.56 radians per minute 17. (a) 19. (a) 13 6 8 3, 11 6, 4 3 (b), 7 17 6 6, 23 25 12 12 (b) 21. (a) Complement: ; 6 Supplement: 2 3 (b) Complement: none; Supplement: 4 23. (a) Complement: Supplement: 1 0.57; 2 1 2.14 (b) Complement: none; Supplement: 210 60 165 29. 27. 25. 31. (a) Quadrant II 33. (a) Quadrant III 35. (a) (b) Quadrant IV (b) Quadrant I (b) y y 30Β° x 150Β° x 94253 400, 1000 2400, 6000 (b) 105. (a) (b) 107. feet per minute 9869.84 feet per minute radians per minute centimeters per minute 140Β° 35 109. False. A measurement of A 476.39 square meters 1496.62 square meters 4 radians corresponds to two complete revolutions from the initial to the terminal side of an angle. 333200_04_AN.qxd 12/9/05 1:32 PM Page A133 Answers to Odd-Numbered Exercises and Tests A133 111. False. The terminal side of the angle lies on the -axis. 113. Increases. The linear velocity is proportional to the radius. 115. The arc length is increasing. If is constant, the length of x 117. 121. β2 the arc is proportional to the radius 2 2 210 119. s r. y y = x5 x β 2)5 123. β3 β2 y 6 5 4 3 1 β1 β2 β3 y = x5 1 2 3 x y = 2 β x5 Section 4.2 (page 299) Vocabulary Check (page 299) 1. unit circle 3. |
period 2. periodic 4. odd; even csc 17 15 sec 17 8 cot 8 15 csc 13 5 sec 13 12 cot 12 5 3, 1 2 2 7. 9. 1 2, 3 2 15. 19. 1 sin 2 6 cos 3 2 6 tan 3 6 3 sin 11 1 6 2 11 3 6 2 11 3 6 3 cos tan sin 15 17 cos 8 17 tan 15 8 sin 5 13 cos 12 13 tan 5 12 2 2 2 2 0, 1, sin cos 4 4 2 2 2 2 1. 3. 5. 11. 13. 17. 21. 2 2 2 2 1 tan 4 sin7 4 cos7 4 1 tan7 4 1 sin3 2 0 cos3 2 tan3 2 is undefined. 23. 25. 27. sin cos tan 4 1 sin 2 cos 0 2 tan 2 sin4 3 2 3 1 cos4 2 3 3 tan4 3 is undefined. sec csc 2 2 3 4 3 4 3 1 cot 4 1 csc 2 sec is undefined. 2 cot 0 2 23 csc4 3 3 2 sec4 3 cot4 3 3 3 8 3 2 3 cos cos 1 2 31. 0 35. 33. 29. sin 5 sin 0 cos cos15 2 2 sin sin9 7 4 4 1 3 37. (a) (b) 3 4 41. (a) 5 0.1288 47. 1 53. (a) 55. (a) 0.25 or 2.89 57. (a) (b) (b) 4 5 49. 1.3940 0.4 2 2 1 5 39. (a) 43. 0.7071 51. (b) 5 45. 1.0378 1.4486 (b) 1.82 or 4.46 t y 0 1 4 0.25 0.0138 1 2 0.1501 3 4 0.0249 1 0.0883 t 5.5 (b) 59. False. sint sin t (c) The displacement decreases. means that the function is odd, not that the sine of a negative angle is a negative number. (b) cos t1 65. y -axis symmetry cos t1 61. (a) (c) f 1x 2 3 sin t1 x 1 sin t1 f 1x x2 4, x β₯ 0 69. 63. 67. y 8 6 4 2 β6 β4 β2 β2 β4 β6 |
β8 2 4 6 8 10 x β6 β5 β2 y 4 3 2 1 β1 β1 β2 β3 β 333200_04_AN.qxd 12/9/05 1:32 PM Page A134 A134 Answers to Odd-Numbered Exercises and Tests Section 4.3 (page 308) Vocabulary Check (page 308) (b) iv 1. (a) v 2. opposite; adjacent; hypotenuse 3. elevation; depression (c) vi (d) iii (e) i (f) ii 1. sin cos tan 3. sin cos tan 5. sin cos tan 3 5 4 5 3 4 9 41 40 41 9 40 1 3 22 3 2 4 csc sec cot csc sec cot 5 3 5 4 4 3 41 9 41 40 40 9 csc 3 sec 32 4 cot 22 The triangles are similar, and corresponding sides are proportional. 3 5 4 5 3 4 7. sin cos tan The triangles are similar, and corresponding sides are proportional. 5 3 5 4 4 3 csc sec cot 9. 4 3 ΞΈ 7 cos 7 4 tan 37 7 csc 4 3 sec 47 7 7 3 cot 11. sin 3 2 1 2 cos tan 3 csc cot 23 3 3 3 sin 213 13 2 cos 313 13 csc sec 13 2 13 3 tan 2 3 60; 3 2 3 3 23. 30; (d) 3 2 13 2 3 (c) (d) 1 2 2 3 (c) (d) 2 4 43. (a) 0.1736 1 3 (b) 0.1736 13 3 15. 17. 25. ΞΈ ; 6 1 2 45; 19. 60; 3 21. 27. (a) 3 (b) 4 213 13 29. (a) 31. (a) 3 (b) 313 13 (c) (b) 22 3 33β 41. Answers will vary. 45. (a) 0.2815 47. (a) 1.3499 49. (a) 5.0273 51. (a) 1.8527 (b) 3.5523 (b) 1.3432 (b) 0.1989 (b) 0.9817 53. (a) 30 55. (a) 60 57. (a) 60 6 3 3 (b) 30 (b) 45 (b) 45 6 4 4 59. 303 61. 323 3 |
r cot 1. 6. 2. csc 3. y x 4. r x 5. cos 7. reference 1. (a) sin cos tan csc sec cot 3. (a) sin cos b) sin cos tan csc sec cot (b) sin cos 15 17 8 17 15 8 17 15 17 8 8 15 17 17 417 17 tan csc sec 3 3 2 23 3 3 cot 24 25 7 25 24 7 5. sin cos tan csc sec cot 25 24 25 7 7 24 tan 1 4 csc 17 sec 17 4 cot 4 tan 3 cot 3 3 23. sin 0 cos 1 tan 0 25. sin 2 2 cos 2 2 tan 27. sin 1 25 5 5 5 cos tan 2 is undefined. csc sec 1 cot is undefined. csc 2 sec 2 cot 1 csc 5 2 sec 5 cot 1 2 333200_04_AN.qxd 12/9/05 1:32 PM Page A136 A136 Answers to Odd-Numbered Exercises and Tests 29. 0 37. 23 31. Undefined 33. 1 39. 65 y 35. Undefined 79. 0.4142 203Β° β²ΞΈ β245Β° 43. 3.5 y 3.5 β²ΞΈ 47. sin 51. sin 750 750 750 cos tan cos tan 3 4 3 4 3 4 3 11 4 11 4 11 4 55. sin cos tan 2 2 2 2 1 81. (a) 83. (a) 85. (a) 87. (a) (b) (b) 30 60 210 7 6 135 3 4 150 5 6, 150 5 6 6, 120 2 3 3, 225 5 4 N 22.099 sin0.522t 2.219 55.008 F 36.641 sin0.502t 1.831 25.610 N 34.6, F 1.4 45 (b) 4, 330 11 6, 315 7 4, 330 11 6 (b) February: March: May: June: August: September: November: N 41.6, F 13.9 N 63.4, F 48.6 N 72.5, F 59.5 N 75.5, F 55.6 N 68.6, F 41.7 N 46.8, F 6.5 (c) Answers will vary. 89. (a) 2 centimeters 1.98 (c) centimeters (b) 0. |
14 centimeter x x 95. As 91. 0.79 ampere 93. False. In each of the four quadrants, the signs of the secant function and cosine function will be the same, because these functions are reciprocals of each other. 0 y 0 cm and sin y12 decreases from 1 to 0. Thus, 90 without bound. When x y decreases from 12 cm to increases from 0 cm to 12 cm. Therefore, cos x12 increases from 0 to 1 and tan yx and increases, the tangent is undefined. -intercepts: increases from 90, to x 1, 0, 4, 0 y -intercept: Domain: all real numbers 0, 4 x 2, 0 x -intercept: 0, 8 y -intercept: Domain: all real numbers x 97. 99. (1, 0) 2 4 6 8 x (0, β4) (β4, 0) β6 β8 8 6 4 2 β2 β2 β4 β8 y 12 10 (0, 8) (β2, 0) x β8 β6 β4 2 4 6 8 β4 x x y 3 y β²ΞΈ 41. Ο 2 3 β²ΞΈ 45. sin 225 cos 225 2 2 2 2 tan 225 1 49. sin150 1 2 3 2 cos150 3 tan150 3 1 sin 2 6 cos 3 2 6 tan 3 3 6 1 sin3 2 0 cos3 2 tan3 2 4 5 0.3420 13 2 69. 61. 53. 57. 59. is undefined. 67. 73. 4.6373 75. 0.3640 77. 63. 8 5 1.4826 65. 0.1736 71. 3.2361 0.6052 333200_04_AN.qxd 12/12/05 11:19 AM Page A137 Answers to Odd-Numbered Exercises and Tests A137 29. y 101 (β 0, ( (7, 0) 6 8 x 7, 0 x -intercept: 0, 7 y -intercept: 4 Vertical asymptote: x 2 Horizontal asymptote: y 0 27. g Domain: all real numbers x 2 except x 31. 35. 39. 43. 103. y 5 4 3 2 1 ) 2) 0 105. y 12 9 6 (β1, 0) (1, 0) β 12 β 9 β 6 β 3 3 6 9 12 x x 0 |
, 1 y -intercept: Horizontal asymptote: 2 y 0 Domain: all real numbers x Β±1, 0 x -intercepts: x 0 Vertical asymptote: Domain: all real numbers except x 0 x Section 4.5 (page 328) Vocabulary Check (page 328) 1. cycle 2. amplitude 3. 4. phase shift 5. vertical shift 2 b 1. Period: Amplitude: 3 7. Period: 2 3. Period: 4 Amplitude: 9. Period: 5 5. Period: 6 Amplitude: 1 2 5 2 Amplitude: 3 Amplitude: 3 11. Period: 3 Amplitude: g g 1 2 f is a shift of is a reflection of 13. Period: 1 1 Amplitude: 4 15. units to the right. 17. xin the axis. 19. The period of f is twice the period of 21. f is a shift of three units upward. 23. The graph of has twice the amplitude of the graph of g 25. The graph of g is a horizontal shift of the graph of g. f g f f. units to the right1 y 4 3 2 1 x Ο3 2 g f Ο3 33. 37. 41. 45. x x x x β Ο3 2 β Ο 2 Ο 2 Ο3 2 β4 y 2 Οβ 2 Ο2 Ο4 β 1 β 2 y 3 2 β1 2 3 β2 β1 y 3 β3 y 4 3 1 2 3 x Ο2 1 4 β 3 y 2 1 ββ Ο β2 β3 333200_04_AN.qxd 12/9/05 1:32 PM Page A138 A138 47. 51. β 4 β 6 y 2.2 1.8 Answers to Odd-Numbered Exercises and Tests 493 β2 β1 β1 1 2 3 x 53. x Ο Ο 2 y 4 2 β8 4 β0.1 0 y 55. x 0.1 0.2 57 59. β 3 β6 6 Ο x Ο4 β4 61. 0.12 β20 20 3 β0.12 63. a 2, d 1 65. a 4, d 4 67. a 3, b 2, c 0 69. a 2, b 1, c 4 71. β2 2 β 11 6 73. (a) 6 seconds (b) 10 cycles per |
minute (c) v 1.00 0.75 0.50 0.25 β0.25 β1.00 2 4 8 10 t 75. (a) Ct 56.55 26.95 cos 6 t 3.67 (b) 100 0 0 12 The model is a good fit. 100 (c) 0 0 12 The model is a good fit. 77.90; (d) Tallahassee: Chicago: The constant term gives the annual average temperature. 56.55 (e) 12; yes; one full period is one year. (f) Chicago; amplitude; the greater the amplitude, the greater the variability in temperature. 1 440 (b) 440 cycles per second 77. (a) second 79. (a) 365; answers will vary. (b) 30.3 gallons; the constant term (c) 60 124 < t < 252 81. False. The graph of f x sin x graph of the two graphs look identical. cos x sinx 83. True. Because f x sinx 2 translates the exactly one period to the left so that reflection in the -axis of y sinx x, 2 is a y cos x. 2 2 0 0 365 333200_04_AN.qxd 12/9/05 1:32 PM Page A139 Answers to Odd-Numbered Exercises and Tests A139 7. 11. 15. 19. 23. 85. y 2 1 f = g β Ο3 2 Ο 2 Ο3 2 x β2 Conjecture: sin x cosx 2 87. (a) β2 2 β 2 2 The graphs appear to coincide from 2 to. 2 (b) β2 2 β 2 2 The graphs appear to coincide from 2 to. 2 (c) x7 7!, x 6 6! β2 2 β 2 2 β2 2 β2 2 The interval of accuracy increased. 91. x 2 89. 1 2 log10 3 ln t lnt 1 3x y4 93. log10 xy 95. ln 97. Answers will vary. Section 4.6 (page 339) Vocabulary Check (page 339) 1. vertical 5. 4. 2 7. 2. reciprocal 6. x n 3. damping, 1 1, 1. e, 5. f, 4 2 2. c, 6. b, 4 3. a, 1 4. d. 13. 17. 212 β1 1 2 x x β3 |
β 2Ο x β Ο x Ο 25. y 4 3 2 1 β4 4 x β Ο3 2 Ο 2 x y y y β3 β2 β1 β1 Οβ2 3 2 1 6 4 2 333200_04_AN.qxd 12/9/05 1:32 PM Page A140 Answers to Odd-Numbered Exercises and Tests 29. y 2 1 4 β4 3 β3 x Ο2 2 2 5 4 41. 7 4, 3, 4 4, A140 27. y 4 3 2 1 βΟ β1 Ο 2Ο 3Ο x 31. 35. 39. 43. 47. β.6 β6 6 β 0. 33. 5 β 2 37. 3 2 β 2 45 49. Even (b Ο3 4 x Ο 51. (a) y 3 2 1 β1 53. (c) β3 approaches 0 and f approaches cosecant is the reciprocal of the sine. g because the 2 β 2 3 The expressions are equivalent except that when y1 is undefined. sin x 0, 55. β2 4 β4 2 The expressions are equivalent. f β 0 as x β 0. g β 0 as x β 0. 58. a, 60. c, f β 0 as x β 0. g β 0 as x β 0. 57. d, 59. b, 61. y 3 2 1 β1 β2 β3 63. y 3 2 1 2 3 x Οβ Ο β1 x β3 β2 β1 The functions are equal. The functions are equal. 65. 1 67. 6 β8 8 β9 9 β1 x β, gx β 0. As 6 69. β6 x β, f x β 0. As 71. 2 0 β2 As x β 0, y β. 8 β6 6 β1 x β 0, gx β 1. As 73. β 2 β2 75. x β 0, f x As d 7 cot x d oscillates between 1 and 1 14 10 6 2 β2 β6 β10 β14 Ο 4 Ο 2 Ο 3 4 x Ο Angle of elevation 333200_04_AN.qxd 12/9/05 1:32 PM Page A141 Answers to Odd-Numbered Exercises and Tests A141 77. (a) 50,000 R C 0 0 100 17. β |
1.5 1 β1 f g the coordinate of csc x is the 43. 0.3 45. 0.1 47. 0 49. 3 5 51. 5 5 (b) As the predator population increases, the number of prey decreases. When the number of prey is small, the number of predators decreases. C : 24 months H: 24 months; 12 months; 12 months R : L: (c) 79. (a) (c) 1 month (b) Summer; winter 81. True. For a given value of ysin x. reciprocal of the coordinate of f from the left, y2 from the right, 83. As approaches 2 approaches x, x f approaches. approaches x. As 85. (a) 2 β2 β3 0.7391 3 (b) 1, 0.5403, 0.8576, 0.6543, 0.7935, 0.7014, 0.7640, 0.7221, 0.7504, 0.7314,... ; 0.7391 87. β 3 2 6 β 6 3 2 The graphs appear to coincide on the interval 1.1 β€ x β€ 1.1. ln 54 2 2 e73 3 ln 2 0.693 1.684 1031 1.994 91. 89. 93. 95. Β± e3.2 1 Β±4.851 97. 2 Section 4.7 (page 349) Vocabulary Check (page 349) 1. 2. 3. y sin1 x; 1 β€ x β€ 1 y arccos x; 0 β€ y β€ y tan1 x. 11. 6 2 3 3. 3 13. 3 5. 6 7. 5 6 9. 3 15. 0 1.5 1.25 23. 31. 0.85 37. arctan 25. 0.32 33. 1.29 x 4 0.85 19. 1.29 27. 1.99 35., 3 21. 29. 0.74 3 3, 1 39. arcsin x 2 5 41. arccos x 3 2x 53. 12 13 55. 34 5 57. 5 3 59. 61. 1 4x 2 63. 1 x 2 65 67. 69. β3 2 β2 3 71. 73. 75. y Β±1, x > 0; Asymptotes: 9 x2 81 x 1 x2 2x 10 9 x2 81, x < 0 |
y Ο 77. β1 1 2 3 x y Ο2 Ο β Ο β2 β1 1 2 g is a The graph of horizontal shift one unit to the right of f. 79. y Ο 81. y Ο β4 β2 2 4 x β 333200_04_AN.qxd 12/9/05 1:32 PM Page A142 A142 Answers to Odd-Numbered Exercises and Tests 83. 2 85. 103. Domain: Range:, 1 1, 2, 0 0, 2 β2 4 β y Ο 2 β2 β1 1 2 x β Ο 2 87. β1 β4 0 4 β2 32 sin2t 89. 4 β2 6 β6 1 5 2 The graph implies that the identity is true. 91. (a) arcsin 5 s 93. (a) 1.5 (b) 0.13, 0.25 0 β 0.5 6 (b) 2 feet (c) 95. (a) 26.0 As 0; x (b) 24.4 feet increases, 97. (a) arctan x 20 (b) 14.0, 31.0 approaches 0. is not in the range of the arctangent. 99. False. 5 4 101. Domain:, Range: 0 105. (a) 107. (a) (b) 4 f f 1 2 2 (c) 1.25 (d) 2.03 f 1 f β β β2 2 β2 (b) The domains and ranges of the functions are restrictdiffer because of f f 1 f 1 f ed. The graphs of the domains and ranges of and 111. 117.391 and f f 1. 109. 1279.284 113. 7 115. sec 47 7 7 3 cot 7 cos 4 tan 37 7 csc 4 3 3 4 ΞΈ 11 6 11 5 sec 6 5 cot 511 11 sin tan csc 611 11 6 5 ΞΈ 11 117. Eight people 119. (a) $21,253.63 (c) $21,285.66 (b) $21,275.17 (d) $21,286.01 Section 4.8 (page 359) Vocabulary Check (page 359) 1. elevation; depression 3. harmonic motion 2. bearing 333200_04_AN.qxd 12/9/05 1:32 PM Page A143 1. |
7. a 3.64 c 10.64 B 70 a 49.48 A 72.08 B 17.92 13. 19.99 inches 19. (a) 5. c 11.66 A 30.96 B 59.04 11. 2.56 inches 3. 9. a 8.26 c 25.38 A 19 a 91.34 b 420.70 B 7745 15. 107.2 feet 17. 19.7 feet h x y 47Β° 40β² 50 ft 35Β° h 50tan 4740 tan 35 (b) (c) 19.9 feet (b) tan 121 2 171 3 (c) 35.8 21. 2236.8 feet 23. (a) 1 12 ft 2 ΞΈ 1 17 ft 3 2.06 27. 0.73 mile 25. 29. 554 miles north; 709 miles east 31. (a) 58.18 nautical miles west; 104.95 nautical miles south distance 130.9 nautical miles (b) 68.82 meters 37. 1933.3 feet W; (b) S 36.7 N 58 E N 56.31 W 3.23 miles or 17,054 feet 78.7 y 3 r 35.3 49. 33. (a) 35. 39. 41. 47. 43. a 12.2, b 7 45. 29.4 inches 51. d 4 sint 53. 59. (c) 528 32 d 3 cos4t 3 (d) 1 16 (d) 1 120 (b) 4 (c) 4 (b) 60 (c) 0 (b) 8 Ο 8 Ο 4 Ο3 8 Ο 2 t 55. (a) 4 1 16 57. (a) 61. (a) y 1 β 1 Answers to Odd-Numbered Exercises and Tests A143 63. (a) Base 1 Base 2 Altitude Area 8 8 8 8 8 8 8 8 16 cos 30 8 sin 30 8 16 cos 40 8 sin 40 8 16 cos 50 8 sin 50 8 16 cos 60 8 sin 60 8 16 cos 70 8 sin 70 8 16 cos 80 8 sin 80 8 16 cos 90 8 sin 90 59.7 72.7 80.5 83.1 80.7 74.0 64.0 (b) Base 1 Base 2 Altitude Area 8 8 8 8 8 8 8 16 cos 56 8 sin 56 8 16 cos 58 8 sin 58 8 16 cos 59 8 sin 59 |
82.73 83.04 83.11 8 16 cos 60 8 sin 60 83.14 8 16 cos 61 8 sin 61 8 16 cos 62 8 sin 62 83.11 83.04 (c) (d) 83.14 square feet A 641 cos sin 100 0 0 90 83.1 square feet when The answers are the same. 60 65. False. The tower is leaning, so it is not perfectly vertical and does not form a right angle with the ground. C H A P T E R 4 67. No. 69. N 24 E y 4x 6 y 7 6 5 3 2 1 means 24 degrees east of north. 71. y 4 5x 22 5 y 7 6 4 3 2 1 β4 β3 β2 β1 β1 1 2 3 4 x β2 β1 β1 1 2 3 4 5 x 333200_04_AN.qxd_pg A144 1/9/06 8:57 AM Page A144 A144 Answers to Odd-Numbered Exercises and Tests Review Exercises (page 365) 1. 0.5 radian 3. (a) y 5. (a) y 41. sin 441 41 cos 541 41 43. sin 3 2 cos 1 2 Ο 11 4 (b) Quadrant II, 5 4 3 4 (c) 7. (a) y 70Β° x x x Ο β 4 3 (b) Quadrant II, 10 3 2 3 (c) 9. (a) y x 59. β110Β° (b) Quadrant III 250, 470 (c) 128.571,, 13. 15. 27. 25. 23. 29. sin csc cos 2 0.589 19. 478.17 inches 430, 290 200.535 662 3 400 3 2 1 2 3 2 (b) Quadrant I (c) 11. 8.378 17. 21. (a) radians per minute (b) inches per minute Area 339.28 square inches 1 1 2 2 7 6 7 6 7 tan 6 sin2 3 cos2 3 tan2 3 11 sin 4 sin17 6 75.3130 3 3 3 2 1 2 3 3 sin 4 sin 7 6 39. 3.2361 csc2 3 sec2 3 cot2 3 sec cot 37. 31. 35. 33. 23 3 23 3 2 3 3 15 15 55. 71.3 meters csc tan 4 5 41 4 41 5 sec |
tan 3 csc 23 3 sec 2 cot 5 4 cot (b) (b) 22 3 15 4 51. 0.5621 (c) (c) (d) 32 4 415 15 53. 3.6722 45. (a) 3 47. (a) 1 4 49. 0.6494 57. 3 3 2 4 (d) csc 5 4 sec 5 3 cot 3 4 sin 4 5 cos 3 5 tan 4 3 sin 15241 241 cos 4241 241 csc sec 241 15 241 4 cot 4 15 82 9 csc sec 82 cot 1 9 17 4 csc sec 17 cot 1 4 61. 63. tan 15 4 sin 982 82 82 82 cos tan 9 sin 417 17 17 17 cos tan 4 65. sin 11 6 67. cos 5 6 11 5 tan csc 611 11 cot 511 11 55 8 cos tan 355 55 csc 8 3 sec 855 55 55 3 cot 333200_04_AN.qxd 12/9/05 1:33 PM Page A145 69. sin 21 5 21 tan 2 csc 521 21 sec 5 2 cot 221 21 71. 84 y 73. 5 y 264Β° β²ΞΈ β²ΞΈ x x β Ο 6 5 1 3 ; tan 2 3 1 cos7 2 3 ; 75. 77. 79. sin 3 sin7 3 tan7 3 3 ; cos 2 3 3 ; 2 3 2 2 ; sin 495 cos 495 2 2 ; tan 495 1 81. sin240 3 2 ; cos240 1 2 ; tan240 3 0.7568 85. 0.0584 83. 89. y 2 1 β Ο3 2 Ο 2 x β 2 87. 3.2361 91. y 6 4 2 β2 β6 x Ο6 Answers to Odd-Numbered Exercises and Tests A145 93. 951 β2 β3 β4 Ο t 97. (a) 99. y 2 sin 528x (b) 264 cycles per second 101β Ο 105β β1 β2 β3 β4 Ο 6 x β Ο3 2 Ο 2 β3 β4 β9 9 β6 x β, f x β As 111. 0.41 113. 6 0.46 115. 6 β1.5 119. 1.24 121. 0.98 125. 1.5 β4 β 2 β 2 103. 107 |
333200_05_AN.qxd 12/9/05 1:50 PM Page A147 Answers to Odd-Numbered Exercises and Tests A147 (b) 3705 feet (c) 3. (a) 4767 feet w 2183 tan 63 w 3705 feet, 3000 5. (a) (b) 3 β1 β2 2 β2 3 β1 2 Even Even h 51 50 sin8t 7. 2 9. (a) 2 E P I 7300 7380 β2 2 (b) 7348 I 7377 E P β2 (c) P7369 0.631 E7369 0.901 I7369 0.945 (b) 11. (a) 3.35, 7.35 0.65 (c) Yes. There is a difference of nine periods between the values. 40.5 (b) 1.75 feet x 1.71 feet; y 3.46 feet 13. (a) (c) (d) As you move closer to the rock, must get smaller and and will decrease along with d 2 smaller. The angles d the distance y, 1 so will decrease. Chapter 5 Section 5.1 (page 379) Vocabulary Check (page 379) 1. 5. 9. tan u cot2 u cos u 2. 6. 10. cos u sec2 u tan u 3. cot u 7. cos u 4. csc u 8. csc u 1. 5. 9. sin x 3 2 cos x 1 2 tan x 3 csc x 23 3 sec x 2 cot x 3 3 sin x 5 13 cos x 12 13 tan x 5 12 sec x 13 12 csc x 13 5 cot x 12 5 sin x 1 3 cos x 22 3 2 4 tan x csc x 3 sec x 32 4 cot x 22 3. sin 2 2 2 2 tan 1 cos sec 2 csc 2 cot 1 7. sin cos 2 3 tan 5 3 5 2 sec 3 2 csc 35 5 cot 25 5 25 5 5 5 cos 11. sin tan 2 csc 5 2 sec 5 cot 1 2 13. sin 1 cos 0 tan cot 0 csc 1 sec is undefined. csc is undefined. 16. a 22. c 17. b 23. f cos2 29. tan x 37. cos u sin u sec x 1 51. cot2 xcsc x 1 4 cot2 x 61 |
. 1 cos y 39. 45. sec4 x 57. 2 csc2 x 15. d 21. b 27. 35. 1 43. 49. 55. 59. 65. 18. f 24. a 31. cos x 1 sin y 19. e 25. e 33. 41. sin2 x sec sin2 x tan2 x 47. sin2 x cos2 x 53. sin2 x 1 2 sin x cos x 63. 2 sec x 67. 3sec x tan x C H A P T E R 5 20. c 26. d 333200_05_AN.qxd 12/9/05 1:50 PM Page A148 A148 69. 1 0 0 x y1 y2 x y1 y2 y1 71. x y1 y2 x y1 y2 12 Answers to Odd-Numbered Exercises and Tests 109. Not an identity because tan k sin k cos k 1 sin 115. x 25 119. 5x2 8x 28 x2 4x 4 123. 2 0.2 0.4 0.6 0.8 1.0 113. Answers will vary. x2 6x 8 x 5x 8 117. 111. An identity because sin 1 0.1987 0.3894 0.5646 0.7174 0.8415 121. 0.1987 0.3894 0.5646 0.7174 0.8415 1.2 1.4 0.9320 0.9854 0.9320 0.9854 y2 y 2 1 β1 β2 β3 β4 x Ο2 Ο3 2 0.2 0.4 0.6 0.8 1.0 Section 5.2 (page 387) 1.2230 1.5085 1.8958 2.4650 3.4082 1.2230 1.5085 1.8958 2.4650 3.4082 1.2 1.4 5.3319 11.6814 5.3319 11.6814 Vocabulary Check (page 387) 1. identity 4. cot u 8. sec u 2. conditional equation cos2 u 6. sin u 7. tan u 3. csc u 5. 1β37. Answers will vary. 39. (a) (b) 0 1 y2 y1 csc x 5 sec 73. 81. 75. 2 tan x 83. 77. 3 sin 3 tan 3 cos 3; |
sin 0; cos 1 79. 85. 4 sin 22; sin 2 2 ; cos 2 2 87. 0 β€ β€ lncot x 91. 95. (a) 0 β€ < 89. lncsc t sec t 93. 2, 3 2 < < 2 (b) 97. (a) 1.6360 0.6360 1 csc2 132 cot 2 132 1.8107 0.8107 1 cot2 2 csc2 2 7 7 cos90 80 sin 80 0.9848 cos 0.8 sin 0.8 0.7174 2 tan 99. 101. True. For example,, 0 103. 1, 1 107. Not an identity because cos Β± 1 sin2 sinx sin x. 105. (b) β 5 5 β5 5 (c) Answers will vary. 41. (a) β2 5 β1 y2 y1 2 Identity (b) (c) Answers will vary. 43. (a) (b) Not an identity 5 β1 β2 2 Identity (c) Answers will vary. 333200_05_AN.qxd 12/9/05 1:50 PM Page A149 45. (a) (b) y2 y1 β2 3 β3 2 (c) Answers will vary. Not an identity 47 and 49. Answers will vary. 55. Answers will vary. 57. False. An identity is an equation that is true for all real 51. 1 53. 2 values of. 59. The equation is not an identity because sin Β±1 cos2. 7 4 Possible answer: 2 3 26i 3 Β± 21 61. 65. 8 4i 63. 1 Β± 5 67. Section 5.3 (page 396) Vocabulary Check (page 396) 1. general 2. quadratic 3. extraneous 1β5. Answers will vary. 7. 2 n, 2n 2 3 6 4 3 5 6 11. n, n 9. 3 2n, 2 3 2n 13. 17. 21. 25. 31. 37. 23. n 2 8 0, n, 2n 12 2 6n, 2 6n 11 6 3,, 6, 39. 15. 3 n, n 2 3 19. 0 11 6 29.,, 3 27. No solution 5 3 5 6 2 4n, 4n 7 2 45. 2.678, 5.820 41. 1 4n 49. 0.860, 3 |
.426 43. 47. 1.047, 5.236 51. 0, 2.678, 3.142, 5.820 53. 0.983, 1.768, 4.124, 4.910 55. 0.3398, 0.8481, 2.2935, 2.8018 57. 1.9357, 2.7767, 5.0773, 5.9183 59., 4 5 4, arctan 5, arctan 5 61., 3 5 3 Answers to Odd-Numbered Exercises and Tests A149 4 5 4 0.7854 3.9270 63. (a) 3 (b) 2 0 β3 Maximum: Minimum: 0.7854, 1.4142 3.9270, 1.4142 65. 1 67. (a) All real numbers except x x 0 y -axis symmetry; Horizontal asymptote: y 1 (d) Infinitely many solutions (b) (c) Oscillates (e) Yes, 0.6366 69. 0.04 second, 0.43 second, 0.83 second 71. February, March, and April 75. (a) Between (b) 5 times: t 8 seconds and t 16, 48, 80, 112, 144 73. t 24 36.9, 53.1 seconds seconds 0.6 < x < 1.1 77. (a) 2 (b) 0 2 β2 A 1.12 79. True. The first equation has a smaller period than the second equation, so it will have more solutions in the interval 0, 2. 81. 1 83. C 24 a 54.8 b 50.1 87. 85. cos 390 sin 390 1 2 3 2 3 3 tan 390 sin1845 cos1845 2 2 2 2 tan1845 1 C H A P T E R 5 Vocabulary Check (page 404) 1. 2. 4. 5. sin u cos v cos u sin v cos u cos v sin u sin v sin u cos v cos u sin v cos u cos v sin u sin v 3. 6. tan u tan v 1 tan u tan v tan u tan v 1 tan u tan v 1. (a) 2 6 4 (b) 1 2 2 33. 2 35. 6 n, n 89. 1.36 91. Answers will vary. Section 5.4 (page 404) 333200_05_AN |
.qxd 12/9/05 1:50 PM Page A150 A150 Answers to Odd-Numbered Exercises and Tests 3. (a) 5. (a) 2 6 4 (b) 2 1 2 1 2 (b) 3 1 2 7. sin 105 cos 105 2 4 2 4 3 1 1 3 tan 105 2 3 9. sin 195 2 4 1 3 cos 195 2 4 3 1 tan 195 2 3 11. sin cos tan 13. sin cos 3 1 2 4 2 1 3 4 11 12 11 12 11 12 17 12 17 12 17 12 tan 2 3 31. 39. 49. 16 65 5 3 3 2 33. 3 2 35. 1 37. 63 65 63 16 41. 51. 1 43. 53. 0 65 56 45. 3 5 47. 44 117 55β63. Answers will vary. 65. sin x 67. cos 69. 2 71. 5, 4 7 4 sin2t 0.6435 (c) 1 cycle per second feet sinu Β± v sin u cos v Β± cos u sin v cos x cos 2 sin x sin 2 sin x 2 81β 83. Answers will vary. 2 sin 13 sin 3 0.3948 85. (a) 4 (b) 2 cos (b) 4 13 cos3 1.1760 73., 4 75. (a) 7 4 y 5 12 (b) 5 12 77. False. 79. False. cosx 87. (a) 89. 95. 2 cos β2 91. Proof 93. 15 3 β3 2 3 1 sin 285 2 4 2 3 1 cos 285 4 tan 285 2 3 sin165 3 1 cos165 1 3 2 4 2 4 tan165 2 3 2 4 1 3 sin cos 13 12 13 12 13 tan 12 sin13 12 cos13 12 tan13 12 cos 40 tan 239 25. 3 1 27. sin 1.8 29. tan 3x 15. 17. 19. 21. 23. sin2 sin2 4 f 1x x 15 5 97. 1 4 99. Because 4x 3 101. f is not one-to-one, 6x 3 103. Section 5.5 (page 415) f 1 does not exist. Vocabulary Check (page 415) 1. 3. 4. 6. 2. cos2 u 2 sin u cos u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u Β±1 cos u tan2 u 5. 2 1 cos u |
sin u sin u 1 cos u 1 2 1 2 7. 8. cosu v cosu v sinu v sin u v cosu v 2 sinu v 9. 2 2 sinu v 10. 2 sinu v 2 2 333200_05_AN.qxd 12/9/05 1:50 PM Page A151 Answers to Odd-Numbered Exercises and Tests A151 3 4 cos 2x cos 4x 1. 11. 15. 17 17, 5, 12, 12, 5, 6 6 2 3 sin 2x 3. 15 17 17 13, 12 12 3 7,, 2 6 21. 11 6 4 cos 2x 9. 0, 5. 8 15 13. 0, 17 8 4 3 7. 2, 3 17.,, 3 2 0, 2 19. 23. sin cos tan 27. sin 2u 24 25 2u 7 25 2u 24 7 1 8 29. cos 25. sin cos tan 2u 24 25 2u 7 25 2u 24 7 2u 421 25 2u 17 25 2u 421 17 1 cos 4x 1 8 1 cos 2x cos 4x cos 2x cos 4x 1 16 417 17 17 tan 37. 39. 31. 33. 35. 41. 1 4 2 3 2 3 sin 75 1 2 cos 75 1 2 tan 75 2 3 112 30 1 2 112 30 1 2 112 30 1 2 2 2 2 2 43. sin cos tan 45. sin 2 2 cos 2 2 2 2 47. sin cos tan cos tan 178 2 1 89 889 89 889 8 89 5 178 49. 51. sin 1 2 1 2 8 8 2 1 tan sin tan cos tan 2 sin 3x cos 526 26 26 26 5 310 10 10 10 3 53. sin 55. 57. tan 4x 3,, 5 3 59. 61. 3,, 5 3 2 0 β2 sin 0 3sin sin 10 sin 2 2 1 2 2 2 0 2 β2 65. 5cos 60 cos 90 69. 5 2 cos 8 cos 2 cos 2y cos 2x 1 2 2 cos 4 sin 77. 79. 2 cos sin 81. sin 2 sin 2 1 2 73. 2 cos 4x cos 2x 2 sin sin 63. 67. 71. 75. 83. 87. 3 1 2,, 0, 4 2 2 0 β2 89., 6 5 6 2 0 β2 25 169 β2 91. 111. 115 85 93. 4 |
13 3 β3 95β109. Answers will vary. 113. 2 β2 3 β3 2 Ο Ο 2 x 333200_05_AN.qxd 12/9/05 1:50 PM Page A152 A152 Answers to Odd-Numbered Exercises and Tests 23.85 Review Exercises (page 420) 117. 121. (a) 2x1 x2 119. (b) 0.4482 (c) 760 miles per hour; 3420 miles per hour (d) 2 sin1 1 M u < 0, 123. False. For sin 2u sin2u 2 sinu cosu 2sin u cos u 2 sin u cos u. 125. (a) 4 (b) 0 0 2, 3 Maximum: 3 cos 4x 1 (b) 4 1 2 sin2 x cos2 x 2 cos4 x 2 cos2 x 1 127. (a) (d) (c) (e) No. There is often more than one way to rewrite a 2 sin2 2 x 1 1 trigonometric expression. 129. (a) (β1, 4) y 6 5 4 3 2 1 (5, 2b) 131. (a) Distance 210 y (c) Midpoint: 2, 1 2 β 1 1 2 x (b) Distance 2 3 133. (a) Complement: 13 35; 2 3, 3 (c) Midpoint: 125 supplement: 2 3. cos x 9. 1. 7. sec x tan x 3 4 csc x 5 3 sec x 5 4 cot x 4 3 5. cos x cot x 2 2 tan x 1 csc x 2 sec x 2 cot x 1 sin2 x 13. 1 sec x 2 sin x 15. 11. 19. 21. 23β31. Answers will vary. cot 2 tan2 17. cot2 x 2n, 2n 35. n n, n 39. 0, 4 3 41. 0 11 45. 0, 9, 8 11, 8 13, 8 15 8 47. 0, arctan4, arctan4 2, arctan 3, arctan 3 sin 285 cos 285 3 1 2 4 2 3 1 4 tan 285 2 3 33. 37. 43. 49. 51. 53. 55. 61. 3 1 3 1 sin cos 2 4 2 4 25 12 25 12 25 12 sin 15 57. 57 36 1 52 tan 2 |
2 sec2, 1 β€ sin x β€ 1 for all 1.8431, 2.1758, 3.9903, 8.8935, 9.8820 1 cot2 csc2 y2 115. y1 x 1 (b) 113. 117. Chapter Test (page 423) 2. 1 3. 1 4. csc sec 1. sin 313 13 cos 213 13 13 3 13 2 csc sec cot 2 3 21. 2.938, 2.663, 1.170 5, tan 2u 4 sin 2u 4 23. 24. Day 123 to day 223 t 0.26 minute 25. 0.58 minute 0.89 minute 1.20 minutes 1.52 minutes 1.83 minutes Problem Solving (page 427) 1. (a) cos Β± 1 sin2 tan Β± cot Β± sec Β± sin 1 sin2 1 sin2 sin 1 1 sin2 csc 1 sin sin Β±1 cos2 1 cos2 cos 1 1 cos2 csc Β± tan Β± sec 1 cos cot Β± cos 1 cos2 3. Answers will vary. 5 333200_06_AN.qxd 12/9/05 1:51 PM Page A154 A154 Answers to Odd-Numbered Exercises and Tests 7. sin 2 2 2 2 1 cos 1 cos sin 2 1 cos cos tan 9. (a) 20 0 0 365 t 91, t 274; (b) (c) Seward; The amplitudes: 6.4 and 1.9 (d) 365.2 days Spring Equinox and Fall Equinox β€ x β€ 5 6 (b) 2 3 β€ x β€ 4 3 6 11. (a) (c) (d) 13. (a sinu sin u cos v cos w cos u sin v cos w sin u sin v sin w cos u cos v sin w (b) tanu v w tan u tan v tan w tan u tan v tan w 1 tan u tan v tan u tan w tan v tan w 15. (a) 15 (b) 233.3 times per second 0 0 1 Chapter 6 Section 6.1 (page 436) Vocabulary Check (page 436) 1. oblique 2. b sin B 3. 1 2 ac sin B 1. 3. 5. 7. 9. 11. 13. 15. 17. C 105, b 28.28, c 38.64 C 120, b 4.75, c 7 |
.17 B 21.55, C 122.45, c 11.49 B 60.9, b 19.32, c 6.36 B 42 4, a 22.05, b 14.88 A 10 11, C 154 19, c 11.03 A 25.57, B 9.43, a 10.53 B 18 13, C 51 32, c 40.06 C 83, a 0.62, b 0.51 B 48.74, C 21.26, c 48.23 19. 21. No solution 23. Two solutions: B 72.21, C 49.79, c 10.27 B 107.79, C 14.21, c 3.30 25. (a) (c) 27. (a) b β€ 5, b 5 sin 36 b > 5 sin 36 b β€ 10.8, b 10.8 sin 10 (b) 5 < b < 5 sin 36 (b) 10.8 < b < 10.8 sin 10 (c) b > 29. 10.4 16.1 37. 41. (a) 10.8 sin 10 31. 1675.2 39. 77 meters 17.5Β° 18.8Β° x y 9000 ft 33. 3204.5 35. 15.3 meters (b) 22.6 miles (c) 21.4 miles (d) 7.3 miles z 43. 3.2 miles 45. True. If an angle of a triangle is obtuse greater than 90, then the other two angles must be acute and therefore less than The triangle is oblique. 90. arcsin0.5 sin 47. (a) (b) (c) (d) (e) 1 0 0 Domain: 0 < < Range: 0 < < 6 c 18 sin arcsin0.5 sin sin 27 0 0 c c Domain: Range: 0 < < 9 < c < 27 0.4 0.8 1.2 1.6 0.1960 0.3669 0.4848 0.5234 25.95 23.07 19.19 15.33 2.0 2.4 2.8 0.4720 0.3445 0.1683 12.29 10.31 9.27 increases from 0 to As decreases, and decreases from 27 to 9. c, increases and then 333200_06_AN.qxd 12/9/05 1:51 PM Page A155 49. cos |
x 51. sin2 x Section 6.2 (page 443) Vocabulary Check 1. Cosines 3. Heronβs Area Formula 2. (page 443) b2 a2 c2 2ac cos B 1. 3. 5. 7. 9. 11. 13. 15. A 23.07, B 34.05, C 122.88 B 23.79, C 126.21, a 18.59 A 31.99, B 42.39, C 105.63 A 92.94, B 43.53, C 43.53 B 13.45, C 31.55, a 12.16 A 14145, C 2740, b 11.87 A 27 10, C 27 10, b 56.94 A 33.80, B 103.20, c 0.54 a c b d 135.1 111.8 102.8 17. 5 19. 10 21. 15 23. 16.25 29. 8 14 16.96 12.07 20 25 25. 10.4 5.69 13.86 20 27. 52.11 45 68.2 77.2 N 37.1 E, S 63.1 E W N S E 3000 m C 1700 m B 3700 m A N 58.4 W 31. 373.3 meters 37. (a) 41. 24.2 miles 43. 45. d (inches) PQ 9.4, QS 5, RS 12.8 33. (b) 72.3 S 81.5 W 35. 43.3 miles 39. 63.7 feet 9 10 12 13 14 Answers to Odd-Numbered Exercises and Tests A155 59. 2 61. 3 63. 3 65. 1 1 4x2 69. cos 1 67. 1 x 2 71. sec 1 csc is undefined. 3 tan 3 sec 23 3 csc 2 73. 2 sin 7 12 sin 4 Section 6.3 (page 456) Vocabulary Check (page 456) 2. initial; terminal 4. vector 1. directed line segment 3. magnitude 5. standard position 7. multiplication; addition 8. resultant 9. linear combination; horizontal; vertical 6. unit vector slopev and have the same magnitude and direction, so they are slopeu 1 4 v 1. 3. 7. 11. 15. u v 17, u equal. v 3, 2; v 13 v 0, 5; v 5 v 8, 6; v 10 y |
9. x 5. v 3, 2; v 13 v 16, 7; v 305 13. v 9, 12; v 15 y 17degrees) 60.9 69.5 88.0 98.2 109.6 s (inches) 20.88 20.28 18.99 18.28 17.48 d (inches) 15 16 (degrees) 122.9 139.8 s (inches) 16.55 15.37 47. 46,837.5 square feet 51. False. For s 49. $83,336.37 to be the average of the lengths of the three s sides of the triangle, would be equal to a b c3. v vβ 19. y u + 2v 2v 53. False. The three side lengths do not form a triangle. 55. (a) 570.60 57. Answers will vary. (b) 5910 (c) 177 u x 333200_06_AN.qxd 12/9/05 1:51 PM Page A156 A156 Answers to Odd-Numbered Exercises and Tests 21. (a) 3, 4 (b) 1β β+ u 2 3 4 5 (c) 1, 7 y 2 2u β β 10 β3v vβ 3u 2 23. (a) 5, 3 (b) 5c) 4i 11j y 12 10 8 vβ 3u 2 β3v β 8 β 6 β 4 β 2 β 2 2u 2 4 6 x 27. (a) 2i j (b) 2i j u x 3 β v u vβ β 1 y 1 β1 β2 βc) 4i 3j y 1 u v 2u β1 β2 β3 β4 β3v 2u β 2v 2 2 2, 2 i 25 5 j 39. 10 10 j 310 33. 10 52 2, i 52 2 43. 7i 4j 45. 3i 8j 29. 1, 0 31. 5 5 37. j 1829, 29 v 3, 3 2 4529 29 (c) 10 12 10 8 6 4 2 β 12 β 10 β 3v 2 x 25. (a) 3i 2j (b) i 4j 3 β2 β1 x 3 u v+ y 5 4 u vβ β 35. 41. 47. y 1 β1 β2 1 2 3 x u 3 2 u 333200_06_AN. |
i sin 2.62 7cos 0 i sin 0 23. Imaginary axis 253 β2 3 3+ i β3 β i 1 2 3 4 Real axis Imaginary axis Real axis β1 β2 β3 β 4 23cos 6 i sin 6 10 cos 3.46 i sin 3.46 27. y Imaginar axis 29. 5 4 3 2 1 5 + 2i Imaginary axis Real axis β10 β8 β6 β4 β2 β2 β4 β6 β8 β10 29cos 0.38 i sin 0.38 139cos 3.97 i sin 3.97 31. Imaginar y axis 33. Imaginary axis β 3 2 3 Real axis Real axis 1 1 β1 β2 3 4 33 4 i 32cos 7 4 i sin 7 4 2cos 6 i sin 6 15. Imaginary axis 17. Imaginary axis β 4 β3 β2 β1 Real axis β 4 β 2 2 4 Real axis 35. β2 β3 β 4 β2( 1 + 3i) β 2 β4 β6 β8 β5i 4cos 4 3 i sin 4 3 5cos 3 2 i sin 3 2 3 2 33 2 i β 15 2 8 + 15 2 8 i β 4 β3 β2 β1 152 8 152 8 i Imaginary axis 37. Imaginar y axis 3 2 1 β1 Real axis 8i 2 4 6 8 10 Real axis 10 8 6 4 2 β 2 β 2 8i 9. 10 cos 5.96 i sin 5.96 β 1 β 1 1 2 3 4 5 Real axis β 8 β 5 3i 333200_06_AN.qxd 12/9/05 1:51 PM Page A159 39. Imaginary axis 2 1 β 1 β 2 2.8408 + 0.9643i 1 2 3 4 Real axis 2.8408 0.9643i 4.6985 1.7101i Imaginary axis 41. 45. 43. 2.9044 0.7511i 2 z 2 = i 2 z3 = (β1 + i) 2 2 z = (1 + i) 2 β 2 z 4 = β1 1 Real axis β 1 Answers to Odd-Numbered Exercises and Tests A159 67. Imaginary axis 69. Imaginary axi s 3 1 β1 β 1 β 3 4 2 1 3 Real axis β4 β2 2 4 Real axis β 2 β 4 73. i 71. 77. 1253 |
2 4 4i 125 2 608.0 144.7i 813 81 2 2 89. (a) 85. 81. i 32i 75. 1283 128i 79. 1 83. 597 122i 87. 32i 5 cos 60 i sin 60 5 cos 240 i sin 240 The absolute value of each is 1. 12cos 10 9 0.27cos 150 i sin 150 i sin 49. 3 3 47. 51. cos 200 i sin 200 (b) Imaginary axis 3 1 cos 30 i sin 30 β3 β1 1 3 Real axis 53. 55. cos 59. (a) (b) 61. (a) (b) (c) 57. 7 4 i sin i sin 2 3 2 4cos 302 i sin 302 i sin 3 2cos 22cos 4 4 4 cos 0 i sin 0 4 (c) 4 3 3 2cos 2cos 4 2 2 7 2 2i 22cos 4 2i 2i 2 2i 2 2 2i i sin i sin i sin 7 4 4 7 4 63. (a) 5cos 0.93 i sin 0.93 2cos 5 3 i sin 5 3 cos 1.97 i sin 1.97 0.982 2.299i (c) 91. (a) 15 2 i β 3 i, i sin i sin 5 2 2cos 2cos 2cos 15 2 2 9 8 9 14 9 5 2 2 9 8 9 14 9 i sin (b) Imaginary axis 3 1 5 2 0.982 2.299i 5cos 0 i sin 0 (b) (c) 65. (a) (b) (c) 5 13 10 13 15 13 13 cos 0.98 i sin 0.98 β 3 β 1 β 1 1 3 Real axis cos 5.30 i sin 5.30 0.769 1.154i i 0.769 1.154i β 3 (c) 1.5321 1.2856i, 1.8794 0.6840i, 0.3473 1.9696i C H A P T E R 6 333200_06_AN.qxd 12/9/05 1:51 PM Page A160 Answers to Odd-Numbered Exercises and Tests (b) Imaginary axis 3 4 7 4 i sin i sin 3 5cos 4 7 5cos 4 52 52 2 2 52 52 2 2 i i β6 β2 6 4 2 β2 β4 |
β6 2 4 6 Real axis A160 93. (a) (c) 95. (a) 5cos 5cos 5cos 4 9 10 9 16 9 i sin i sin i sin 4 9 10 9 16 9 (c) 101. (a) 99. (a) cos 0 i sin 0 cos cos cos cos 2 5 4 5 6 5 8 5 i sin i sin i sin i sin 2 5 4 5 6 5 8 5 β2 1, 0.3090 0.9511i, 0.8090 0.5878i, 0.8090 0.5878i, 0.3090 0.9511i 5cos 3 5cos i sin 5 5cos 3 5 3 i sin i sin 3 (b) Imaginary axis 2 β2 Real axis 2 (b) Imaginary axis Real axis 4 6 (c) 97. (a) 0.8682 4.9240i, 4.6985 1.7101i, 3.8302 3.2140i 2cos 0 i sin 0 2cos 2 2cos i sin 2cos 3 2 i sin i sin 2 3 2 (b) Imaginary axis 3 1 (b) Imaginary axis 6 4 2 β6 β2 2 4 6 Real axis β 4 β 6 (c) 103. (a) i, 5, 5 2 53 5 2 2 22cos 22cos 22cos 22cos 22cos 3 20 11 20 19 20 27 20 7 4 i sin i sin i sin i sin i 53 2 3 20 11 20 19 20 27 20 7 4 i sin β 3 β 1 1 3 Real axis (b) Imaginary axis β 1 β 3 (c) 2, 2i, 2, 2i 1 β 2 β 1 1 2 Real axis β 2 (c) 2.5201 1.2841i, 0.4425 2.7936i, 2.7936 0.4425i, 1.2841 2.5201i, 2 2i 333200_06_AN.qxd 12/9/05 1:51 PM Page A161 105. 107. 109. 111. cos cos cos cos i sin i sin i sin i sin i sin i sin 11 11 8 8 15 15 8 8 3cos 5 3 3cos 5 3cos i sin 7 3cos 5 9 3cos 5 2cos 3 8 2cos 7 8 2cos 11 8 2cos 15 8 7 5 9 5 3 8 7 8 11 |
8 15 8 i sin i sin i sin i sin i sin i sin 62cos 62cos 62cos 7 12 5 4 23 12 i sin i sin i sin 7 12 5 4 23 12 Imaginary axis 1 2 β 1 2 Imaginary axis 4 Real axis β 4 β 2 2 4 Real axis β 4 Imaginary axis β 3 β 1 3 1 β 3 Imaginary axis 2 Real axis 3 β2 Real axis 2 β 2 113. True, by the definition of the absolute value of a complex number. cos 1 and/or 2 0. r2 i sin 1 2 0 if cos 2 i sin 2 115. True. r1r2 z1z2 0 r1 and only if 117. Answers will vary. 119. (a) 121. Answers will vary. 123. (a) (b) r 2 2cos 30 i sin 30 2cos 150 i sin 150 2cos 270 i sin 270 B 68, b 19.80, c 21.36 B 60, a 65.01, c 130.02 B 47 45, a 7.53, b 8.29 16; 4 1 135. 133. 5 125. 127. 129. 131. 16; 2 Answers to Odd-Numbered Exercises and Tests A161 Review Exercises (page 482) 15. 33.5 19. 31.01 feet C 74, b 13.19, c 13.41 A 26, a 24.89, c 56.23 C 66, a 2.53, b 9.11 B 108, a 11.76, c 21.49 A 20.41, C 9.59, a 20.92 B 39.48, C 65.52, c 48.24 1. 3. 5. 7. 9. 11. 13. 7.9 17. 31.1 meters A 29.69, B 52.41, C 97.90 21. A 29.92, B 86.18, C 63.90 23. A 35, C 35, b 6.55 25. A 45.76, B 91.24, c 21.42 27. 4.3 29. 31. 615.1 meters 37. 39. 45. (a) (d) 47. (a) (d) 49. (a) (d) 51. (a) 53. feet 33. 9.80 u v 61, slopeu 7, 5 7, 7 41. 4, 3 11, 3 1 |
12 Imaginary axis i sin i sin (b Real axis 4 (c) 113. (a) (b) i, 32 2 i, 0.7765 2.898i, 32 2 32 2.898 0.7765i, 32 2 2 0.7765 2.898i, 2.898 0.7765i 2cos 0 i sin 0 2cos 2 3 2cos 4 3 Imaginary axis 2 3 4 3 i sin i sin 3 β3 β1 1 3 Real axis β3 115. 117. i i sin i sin (c) 3cos 3cos 3cos 3cos 2, 1 3 i, 1 3 i 32 2 4 3 32 4 2 5 32 4 2 7 32 4 2 4 3 4 5 4 7 4 32 2 32 2 32 2 32 2 i sin i sin i i i Imaginary axis 4 2 β4 β2 2 4 Real axis β2 β4 i sin 2cos 2cos 2cos 2 7 6 11 6 2i 2 7 3 i 6 11 3 i 6 i sin i sin Imaginary axis 3 1 β1 β3 β3 Real axis 3 333200_06_AN.qxd 12/12/05 11:21 AM Page A163 119. True. sin 90 is defined in the Law of Sines. so u v v, x2 8i 0 v vu. are x 2 2i and b2 a2 c2 2ac cos B, the direction is the same and the magnitude is k the result is a vector in the opposite direction times as great. (b) 64 121. True. By definition, 123. False. The solutions to 125. x 2 2i. a2 b2 c2 2bc cos A, c2 a2 b2 2ab cos C 127. A 129. If C and k > 0, times as great. k < 0, If and the magnitude is k 4cos 60 i sin 60 4cos 180 i sin 180 4cos 300 i sin 300 131. (a) 133. z1z2 4; z1 z2 cos2 i sin2 cos 2 i sin 2 Chapter Test (page 486) C 88, b 27.81, c 29.98 A 43, b 25.75, c 14.45 1. 2. 3. Two solutions: B 29.12, C 126.88, c 22.03 B 150.88, C 5.12, c 2.46 |
. 32.6; 543.9 kilometers per hour 45. 425 foot-pounds Problem Solving (page 493) 1. 2.01 feet 3. (a) A 75 mi 135Β° 15Β° y B 75Β° 30Β° x 60Β° Lost party 9. 6.7 10. 3 4 A164 4. 8 Ο2 βb) Station A: 27.45 miles; Station B: 53.03 miles (c) 11.03 miles; 5. (a) (i) 2 (iv) 1 (b) (i) 1 (iv) 1 (c) (i) 5 2 (iv) 1 (d) (i) 25 (iv) 1 5 S 21.7 E (ii) (v) 1 (ii) (v) 1 32 (ii) 13 (v) 1 (ii) (v) 1 52 w 1 2 7. 9. (a) u v F1 F2 P Q The amount of work done by of work done by F2. F1 (iii) 1 (vi) 1 (iii) (vi) 1 13 (iii) 85 2 (vi) 1 (iii) (vi) 1r 52 is equal to the amount (b) F1 F2 60Β° 30Β° P Q is The amount of work done by as the amount of work done by F1. F2 3 times as great 2 3 2 tan,, 3, 2 16 20. 63 sin 5 3 21. 4 3 11. 1 4x2 12. 1 13. 14β16. Answers will vary. 17., 5 2 23., 18. 22. 11 6 6 5 5 19. sin 25 5 2 cos 6x cos 2x B 26.39, C 123.61, c 15.0 B 52.48, C 97.52, a 5.04 B 60, a 5.77, c 11.55 A 26.38, B 62.72, C 90.90 24. 25. 26. 27. 28. 29. 36.4 square inches 31. 3i 5j 32. 2 2, 30. 85.2 square inches 2 2 5 33. i sin 3 4 36. 123 12i 34. 35. 37. 5, 1 21 13 3 4 1 1, 5; 13 22cos cos 0 i sin sin i sin cos cos 333200_07_AN.qxd 12/12/05 11:23 AM Page |
A165 Chapter 7 Section 7.1 (page 503) Vocabulary Check (page 503) 1. system of equations 3. solving 5. point of intersection 4. substitution 2. solution 6. break-even (d) Yes (d) No 0, 0, 2, 4 (c) No (c) No (b) No (b) Yes 7. 1. (a) No 3. (a) No 2, 2 2, 6, 1, 3 5. 0, 5, 4, 3 11. 9. 0, 1, 1, 1, 3, 1 13. 20 1, 1 3, 40 21. 19. 3 2, 4, 0, 0 27. No solution 25. 5 2, 3 33. 35. 31. 4, 1 37. 43. 2, 2, 4, 0 39. No solution 5, 5 45. 2 2 6 2, 3 1 15. 17. 23. No solution 29. 1, 4, 4, 7 4, 3 41. 4, 3, 4, 3 5 β6 6 β 2 10 0, 1 47. β2 16 β3 4, 2 β24 24 β16 0, 13, Β±12, 5 1, 2 51. 0.287, 1.751 1 2, 2, 4, 1 49. 55. 59. 63. (a) 781 units 65. (a) 8 weeks 4 2, 0, 29 57. 10, 21 1, 0, 0, 1, 1, 0 10 53. No solution 61. 192 units (b) 3708 units (b) 1 2 3 4 360 24x 336 312 228 264 24 18x 42 60 78 96 5 6 7 8 360 24x 240 216 192 168 24 18x 114 132 150 168 67. More than $11,666.67 Answers to Odd-Numbered Exercises and Tests A165 69. (a) (b) x 0.06x 27,000 y 0.085y 25,000 2,000 0 12,000 10,000 Decreases; Interest is fixed. (c) $5000 71. (a) Solar: Wind: 150 (b) 0.1429t2 4.46t 96.8 16.371t 102.7 8 0 13 10.3, 66.01. (c) Point of intersection: Consumption of solar and wind energy are equal at this point in time in the year 2000. t 10.3 |
, 135.47 (d) (e) The results are the same, but due to the given parameters, t 135.47 is not of significance. (f) Answers will vary. 6 meters 9 meters 75. 8 kilometers 12 kilometers 73. 77. 79. False. To solve a system of equations by substitution, you can solve for either variable in one of the two equations and then back-substitute. 9 inches 12 inches 81. 1. Solve one of the equations for one variable in terms of the other. 2. Substitute the expression found in Step 1 into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step 2. 4. Back-substitute the value obtained in Step 3 into the expression obtained in Step 1 to find the value of the other variable. 5. Check that the solution satisfies each of the original C H A P T E R 7 equations. y 2x y 0 (b) 2x 7y 45 0 30x 17y 18 0 83. (a) 85. 89. 91. Domain: All real numbers except (c) y 3 0 87. y x 2 Horizontal asymptote: Vertical asymptote4 93. Domain: All real numbers except x y 1 Horizontal asymptote: Vertical asymptotes: x Β±4 333200_07_AN.qxd 12/12/05 11:24 AM Page A166 A166 Answers to Odd-Numbered Exercises and Tests Section 7.2 (page 515) Vocabulary Check (page 515) 51. (a) x 0.2x y 10 0.5y 3 (b) 12 1. elimination 3. consistent; inconsistent 2. equivalent 4. equilibrium point 1. 2, 1 3. 1, 1 (c) 20% solution: 50% solution: 62 3 liters 31 3 liters y 4 3 2 3x + 2y = 1 x + y = 0 β4 β3 β2 β1 2 3 4 β2 β3 β4 7. a, 3 2a 5 2 3x β 2y = 5 y 4 3 2 1 β3 β2 β1 2 3 4 5 β2 β6x + 4y = β10 β6 18 β4 Decreases 55. 400 adult, 1035 student y 0.97x 2.1 y 2x 4 53. $6000 57. 61. 63. (a) |
65. False. Two lines that coincide have infinitely many points (b) 41.4 bushels per acre y 0.32x 4.1 y 14x 19 59. of intersection. 69. 67. No. Two lines will intersect only once or will coincide, and if they coincide the system will have infinitely many solutions. 39,600, 398. It is necessary to change the scale on the axes to see the point of intersection. k 4 x β€ 22 3 x β€ 19 16 71. 73. 75. β 22 3 x 19 16 β 9 β 8 β7 β6 β 5 β1 0 1 2 3 77. 2 < x < 18 β2 β 3 0 3 6 9 12 15 18 x 795 β 4 β3 β 2 β 81. ln 6x 83. log9 87. Answers will vary. 12 x 85. No solution Section 7.3 (page 527) Vocabulary Check (page 527) 1. row-echelon 3. Gaussian 5. nonsquare 2. ordered triple 4. row operation 6. position 2 β1 1 2 4 5 6 2x + y = 5 β3 β4 5. No solution β2x + 2y = 5 y 4 1 β4 β2 β 1 2 3 4 x x 9. 1 3, 2 3 β2 β4 y 4 3 β4 β3 β2 β1 β2 β3 β4 x β y = 2 3x β 6y = 5 x 2 3 4 9x + 3y = 1 12 7, 18 7 31 17. 13. 27. 5 6 2 29. 4, 1 3, 4 21. 90 31, 67 a, 1 6a 5, 2 5 2, 3 15. 11. 4 18 5, 3 19. No solution 5 23. Infinitely many solutions: 35, 43 25. 35 31. b; one solution; consistent 32. a; infinitely many solutions; consistent 33. c; one solution; consistent 34. d; no solutions; inconsistent 6, 3 35. 37. 43. 550 miles per hour, 50 miles per hour 45. 49. Cheeseburger: 310 calories; fries: 230 calories 2,000,000, 100 80, 10 2, 1 4, 1 41. 39. 47. (c) No (c) Yes 3, 10, 2 1. (a) No 3. (a) No 5. 11. 1, 2, 4 2y |
y x 2x (b) No (b) No 7. 3z 5 2z 9 3z 0 (d) Yes (d) No 2, 2, 2 1 9. 43 6, 25 6 15. First step in putting the system in row-echelon form 1, 2, 3 13. 19. No solution 23. 4, 8, 5 1 21. 3a 10, 5a 7, a 2a, 21a 2, 8a 2 25. 29. 3 2a 1 a 3, a 1, a 3a 1, a 5, 2, 0 2, 1, 3 2, 2 17. 27. 333200_07_AN.qxd 12/9/05 1:53 PM Page A167 Answers to Odd-Numbered Exercises and Tests A167 75. x 5 y 5 5 77. x Β± 2 2 y 1 2 1 or x 0 y 0 0 79. False. Equation 2 does not have a leading coefficient of 1. 81. No. Answers will vary. 83. x 2y z 0 x y 3z 1 3x y z 9 x 2y 4z x 4y 8z x 6y 4z 5 13 7 x 2z 0 2y z 0 x y z 5 x 2y 4z 9 y 2z 3 x 4z 4 12 85. 91. 11 i 89. 80,000 7 2i 95. 7 2 22 3i 87. 6.375 93. 97. (a) (b) 4, 0, 3 y 25 20 15 C H A P T E R 7 β β 10 β 15 β 20 4, 3 2, 3 99. (a) (b) y 30 20 10 β 5 β 3 β2 1 2 4 x x β30 β40 β50 β60 0 2 4 5 4.996 4.938 4 1 101. x y 2 5 y 12 10 31. 37. 41. 43. 1, 1, 1, 1 33. No solution 9a, 35a, 67a 39. s 16t 2 32t 500 y 1 2x 2 2x 45. 35. 0, 0, 0 s 16t 2 144 5 β4 β3 x2 y 2 4x 0 47. β3 3 β3 8 6 y x2 6x 8 10 β2 β6 49. x2 y 2 6x 8y 0 10 β2 6 β12 51. 6 touchdowns, 6 extra |
-point kicks, and 1 field goal 53. $300,000 at 8% $400,000 at 9% $75,000 at 10% 250,000 1 2s 125,000 1 2s 125,000 s s in growth stocks in certificates of deposit in municipal bonds in blue-chip stocks 55. 59. Vanilla 2 lb Hazelnut 4 lb French Roast 4 lb X 4 lb 57. Brand Y 9 lb Brand Z 9 lb Brand Television 30 ads Radio 10 ads Newspaper 20 ads 61. 63. (a) Not possible (b) No gallons of 10%, 6 gallons of 15%, 6 gallons of 25% (c) 4 gallons of 10%, No gallons of 15%, 8 gallons of 25% 1, I1 y x2 x I3 69. y 0.0075x2 1.3x 20 24 x2 3 10x 41 y 5 2, 1 I2 6 65. 67. 71. (a) (b) 100 (c) 75 0 x y 175 100 120 140 75 68 55 The values are the same. (d) 24.25% (e) 156 females Touchdowns 8; Two-point conversions 1; Field goals 2; Extra-point kicks 5 73. 333200_07_AN.qxd 12/9/05 1:53 PM Page A168 A168 103. Answers to Odd-Numbered Exercises and Tests x y 2 1 5.793 4.671 0 4 1 2 3.598 3.358 53. (a) (b) 2 3 x 4 x y x 12 xx 6 β4 2 8 10 x β 6 2 8 10 x βc) The vertical asymptotes are the same. 55. (a) 5 3 x 3 x 3 y 24x 3 x2 b4 4 6 8 x β4 2 4 6 8 x β4 β6 β8 y = 5 x + 3 β4 β6 β8 y = 3 x β 3 (c) The vertical asymptotes are the same. 2000 7 4x (b) Ymax Ymin (c) 1000 2000 11 7x 7 4x 2000 2000 11 7x, 0 < x β€ 1 (d) Maximum: Minimum: 400F 266.7F Ymax 0 β100 Ymin 1 59. False. The partial fraction decomposition is 40, 40 105. 107. Answers will vary. Section 7.4 (page 539) Vocabulary Check (page 539) 1 |
. partial fraction decomposition 3. linear; quadratic; irreducible 2. improper 4. basic equation 1. b A x 5. 2. c B x 14 3. d 4. a B x2 A x 7. B x 52 C x 53 A x 5 A x 1 x Bx C x2 1 1 x 1 Dx E x2 12 1 x 19. 2 C x 10 A 11. x 1 x 1 1 2 Bx C x2 10 1 x 1 15. 2x x2 2 1 2x 1 x2 x 1 39. 31. 1 1 x 1 x 2 4x 4 x2 1 23. x 1 x2 x2 x 2x 1 x 2 x2 2 1 2x 1 1 x 1 2x 7 17 2x 1 2 x2 2x x2 22 3 2x 1 1 x2 2 x 1 4 x 12 2 x 47. 51. 9. 13. 17. 21. 25. 29. 33. 35. 37. 41. 43. 45. 49. 1 x 13 2 1 x2 2x 1 2 x 1 3 x 4 A x 10 1 1 a x 2a B x 10 1 a x C x 102. 1 1 a y 63. 1 a y 61. 1 x 2 27. 3 x 3 9 x 32 57. (a) 333200_07_AN.qxd 12/9/05 1:53 PM Page A169 65. y 67. 8 6 4 2 β2 β 2 β 4 2 4 8 10 x β3 β2 y 5 4 3 β1 β1 β2 β3 1 2 4 5 x 69. y 5 β20 β 15 β 10 5 10 15 20 x Section 7.5 (page 548) Vocabulary Check (page 548) 1. solution 4. solution 2. graph 5. consumer surplus 3. linear 33 1 3 4 5 7. 5. 92 β1 1 2 3 4 β 2 β2 y 115 β4 β2 6 4 3 2 1 β2 No solution x 2 3 Answers to Odd-Numbered Exercises and Tests A169 15. 2 0 β 2 β6 β9 19. 23. 6 6 9 4 β4 3 β9 27. y β€ 1 2 x 2 13. 17. 21. 25. y 3 2 β3 β2 β1 1 2 3 x β2 β3 2 β2 6 0 4 β2 1 3 4 β8 |
β3 β5 (b) No (b) No (c) Yes (c) Yes (d) Yes (d) Yes (0, 1) (1, 0) 1 2 x 37. (β1, 4) (β1, 0) β 4 β 3 41. (β 2, 0 29. 31. (a) No 33. (a) Yes 35. y 3 2 β1 y (β 1, 0) β2 39. 4 1 β1 β2 β2 β1 2 3 4 x β3 β1 1 3 4 β2 β3 ( 5, 0 ( x 1 2 3 4 ( 10 333200_07_AN.qxd 12/9/05 1:53 PM Page A170 A170 43. y 3 2 1 Answers to Odd-Numbered Exercises and Tests 45. y 4 2 (4, 2) β1 1 2 3 4 5 x β4 β2 x 2 4 (1, β1) β 2 β 3 47. y 4 3 2 1 (4, 4) (β 1, β1) 1 2 3 4 5 51. 55. 59. β 3 β 4 β6 5 β x2 y2 β€ 16 x β₯ 0 y β₯ 0 β2 β4 7 β 1 49. β5 53. 5 β2 β1 x 6 7 7 57. 61 63b) Consumer surplus: $1600 Producer surplus: $400 (b) Consumer surplus: $40,000,000 Producer surplus: $20,000,000 65. (a) p 50 40 30 20 10 67. (a) p 160 140 120 100 80 Consumer Surplus Producer Surplus p = 50 β 0.5x p = 0.125x (80, 10) x 10 20 30 40 50 60 70 80 Consumer Surplus Producer Surplus p = 140 β 0.00002x (2,000,000, 100) p = 80 + 0.00001x 1,000,000 2,000,000 x 69 β€ 12 β€ 15 β₯ 0 β₯ 0 71. x x y y y β€ β₯ β₯ β₯ 20,000 2x 5,000 5,000 73. 55x x 70y β€ β₯ y β₯ 7500 50 40 y 12 10 6 4 2 2 y 15,000 10,000 y 120 100 80 60 40 20 4 6 8 10 x 10,000 15,000 x 20 40 60 80 100 120 x 75. (a) 10y 10y 20y |
20x 15x 10x x y β₯ 300 β₯ 150 β₯ 200 β₯ 0 β₯ 0 (b) y 30 x 30 (c) Answers will vary. y 19.17t 46.61 77. (a) (b) 225 8 0 14 (c) Total retail sales h 2 a b $821.3 billion 79. True. The figure is a rectangle with a length of 9 units and a width of 11 units. 81. The graph is a half-line on the real number line; on the rectangular coordinate system, the graph is a half-plane. 333200_07_AN.qxd 12/9/05 1:53 PM Page A171 Answers to Odd-Numbered Exercises and Tests A171 83. (a) y2 x2 β₯ y > x > 10 x 0 (b) β6 17. y 10 (0, 8) 6 4 β4 19. y 10 (0, 8) (c) The line is an asymptote to the boundary. The larger the circles, the closer the radii can be and the constraint will still be satisfied. 87. c 88. a 28x 17y 13 0 91. 86. b 5x 3y 8 0 x y 1.8 0 85. d 89. 93. 95. (a) 2.17t 22.5 0.241t2 7.23t 3.4 271.05t y1 y2 y3 (b) 60 y3 y1 y2 5 30 18 (c) The quadratic model is the best fit for the data. (d) $48.66 Section 7.6 (page 558) Vocabulary Check (page 558) 1. optimization 3. objective 5. vertex 2. linear programming 4. constraints; feasible solutions 1. Minimum at Maximum at 5. Minimum at Maximum at 9. Minimum at Maximum at 11. Minimum at 0, 0: 0 5, 0: 20 0, 0: 0 3, 4: 17 0, 0: 0 60, 20: 740 0, 0: 0 3. Minimum at Maximum at 7. Minimum at Maximum at 0 40 0 0, 0: 0, 5: 0, 0: 4, 0: 20 Maximum at any point on the line segment connecting 60, 20 30, 45: 2100 and 13. y 4 3 1 β 1 (0, 2) (0, 0) 2 3 4 5 (5, 0) x 15 |
. y 4 3 1 β1 (0, 2) (0, 0) 2 3 4 5 (5, 0) x Minimum at Maximum at 0, 0: 0 5, 0: 30 Minimum at Maximum at 0, 0: 0 0, 2: 48 4 2 (5, 3) 4 2 (5, 3) 2 4 6 8 x (10, 0) 2 4 6 8 x (10, 0) 5, 3: 35 Minimum at No maximum 21. 15 10, 0: 20 Minimum at No maximum 23. 15 20 β5 50 20 β5 50 24, 8: 104 40, 0: 160 Minimum at Maximum at 25. Maximum at 29. Maximum at 3, 6: 12 0, 5: 25 36, 0: 36 24, 8: 56 Minimum at Maximum at 27. Maximum at 31. Maximum at 0, 10: 10 22 : 271 3, 19 6 6 33. y (0, 3) ( 20 19, 45 19 ( 2 1 (0, 0) 1 (2, 0 The maximum, 5, occurs at any point on the line segment. connecting y 20 19, 45 2, 0 and 19 35. (0, 7) 10 6 4 2 (0, 0) 2 4 6 (7, 0) x The constraint x β€ 10 is extraneous. Maximum at 0, 7: 14 2x y β€ 4 The constraint is extraneous. Maximum at 0, 1: 4 37. y 3 2 (0, 0) (0, 1) (1, 0) x 3 4 333200_07_AN.qxd 12/12/05 11:28 AM Page A172 A172 Answers to Odd-Numbered Exercises and Tests 39. 750 units of model A 1000 units of model B Optimal profit: $83,750 43. Three bags of brand X Six bags of brand Y Optimal cost: $195 47. $62,500 to type A $187,500 to type B Optimal return: $23,750 41. 216 units of $300 model 0 units of $250 model Optimal profit: $8640 45. 0 tax returns 12 audits Optimal revenue: $30,000 49. 53. 57. 61. 49. True. The objective function has a maximum value at any point on the line segment connecting the two vertices. 51. (a) t β₯ 9 (b) 3 4 β€ t β€ 9 53. z x 5y 55. |
z 4x y 57. 9 2x 3, x 0 x2 2x 13 xx 2 4 ln 38 14.550 4, 3, 7 59. 63. 67., x Β±3 61. ln 3 1.099 65. 65. 1 3e127 1.851 Review Exercises (page 563) 1. 7. 11. 13. 1, 1 3. 0, 0, 2, 8, 2, 8 1.41, 0.66, 0.25, 0.625 9. 1.41, 10.66 5, 4 5. 4, 2 β6 2 β6 6 17. 23. 0, 2 96 meters 144 15. 3847 units meters 8 0, 0 0.5, 0.8 5 a 14 21. 27. d, one solution, consistent 28. c, infinite solutions, consistent 29. b, no solution, inconsistent 30. a, one solution, consistent 25. 5, a,, 33. 500,000 159 7 7 24 5 22, 8 5 a 4, a 3, a x2 y2 4x 4y 1 0 y 3x2 14.3x 117.6 41. 37. 5 31. 35. 39. 43. 45. (a) (b) 2, 4, 5 3a 4, 2a 5, a y 2x2 x 5 130 (c) 195.2; yes. 0 6 80 The model is a good fit. 47. $16,000 at 7% $13,000 at 9% $11,000 at 11% 51. A x C B x2 1 25 x 5 B A x x 20 x2 1 55. 8x 5 3x x2 1 9 8x 3 x x2 12 59. 63 10 10 8 6 4 2 β 2 β2 β4 y 100 60 40 20 (0, 80) (40, 60) β 2 (2, 15) (2, 9) (6, 3) 4 y 67. y 16 12 8 4 x 71. (15, 15) ( 15, β 3 2 ( x 12 8 6 4 2 (0, 0) β2 1600 (6, 4) (4, 0) x 2 4 6 8 β400 1600 β400 (0, 0) (60, 0) 20 40 80 100 69. y 6 5 4 3 2 (2, 3) 19. 2, 3 5 (β1, 0 |
) β 4 β3 1 2 3 4 x 73. 20x 12x x β 2 30y β€ 8y β€ β₯ y β₯ 24,000 12,400 0 0 75. (a) p 175 150 125 100 75 50 Consumer Surplus Producer Surplus p = 160 β 0.0001x (300,000, 130) p = 70 + 0.0002 x 100,000 200,000 300,000 x (b) Consumer surplus: $4,500,000 Producer surplus: $9,000,000 333200_07_AN.qxd 12/9/05 1:53 PM Page A173 79. y 27 24 21 18 15 12 9 6 3 (0, 25) (5, 15) (15, 0) x 3 6 9 12 15 18 21 24 27 Minimum at No maximum 15, 0: 26.25 77. y 15 12 9 6 3 (0, 10) (5, 8) (0, 0) (7, 0) x 3 6 9 12 15 Minimum at Maximum at y 81. 0, 0: 0 5, 8: 47 6 5 4 3 2 1 (0, 4) (0, 0) (3, 3) (5, 0) x 1 2 3 4 5 6 Minimum at Maximum at 0 0, 0: 3, 3: 48 Answers to Odd-Numbered Exercises and Tests A173 6. y (1, 12) (0.034, 8.619) 16 12 4 x 9. 12. β1 1, 12, 1, 5 1 1 2 3 0.034, 8.619 2, 1 8. 11. 13. 15. y 4 3 2 1 (0, 0) 10. No solution 2, 3, 1 3 2 x2 2 x 2 x 3x x2 2 14. 16. y 6 3 (1, 4) (1, 2) β 12 β9 β6 β3 6 9 12 x β 2 β1 1 3 4 x (β 4, β16) β18 18. Maximum at Minimum at 12, 0: 240 0, 0: 0 ( 1, 15 ( 83. 72 haircuts 85. Three bags of brand X 0 permanents Optimal revenue: $1800 Two bags of brand Y Optimal cost: $105 87. False. To represent a region covered by an isosceles trape- β 2 17. zoid, the last two inequality signs should be 91. β€ |
. 89. 93. 2 14 95. 3x y 7 6x 3y 1 2x 2y 3z x 2y z x 4y z 97. An inconsistent system of linear equations has no solution. 99. Answers will vary. 7 4 1 β5 β3 β 2 β1 32 5 x β2 β5 ( ( 7, β3 (1, β3) Chapter Test 3, 4 2. 8, 4, 2, 2 1. 3. 4. (page 567) 0, 1, 1, 0, 2, 1 5. y y 8 6 4 2 β2 β4 3, 2 (3, 2) 2 4 6 10 x (β3, 0) β 9 β 6 12 9 6 3 β 3 β 6 (2, 5) x 6 9 19. 8%: $20,000 8.5%: $30,000 20. y 1 2 x2 x 6 21. 0 units of model I, 5300 units of model II Optimal profit: $212,000 Problem Solving (page 569) 1. (β 10, 0) y 12 8 a c (6, 8) b β8 β4 4 8 (10, 0) x β4 β8 β12 3, 0, 2, 5 a 85, b 45, c 20 852 452 202 Therefore, the triangle is a right triangle. 333200_08_AN.qxd 12/12/05 11:30 AM Page A174 A174 Answers to Odd-Numbered Exercises and Tests 5. (a) One (b) Two (c) Four ad bc 3. 7. 10.1 feet high; 11. (a) 13. (a) 5a 16 6 252.7 feet long 2 3, 4 (b), a 5 5a 16, a, 6 13a 40 11a 36 14 14 a 3, a 3, a a 0.15a 193a t β€ β₯ 772t β₯ 32 1.9 11,000, (b) (c) t 30 25 15. 9. $12.00, 1 a 1 4a 1, a (d) Infinitely many 17. (a) x x 0 y < y β€ 200 β₯ 35 β€ 130 (b) 20 10 5 β5 β5 y 250 200 150 100 50 5 10 15 20 25 30 a (70, 130) (35,130) (c) No, because the total cholesterol |
is greater than 200 milligrams per deciliter. 50 100 150 250 x (d) LDL: 140 milligrams per deciliter HDL: 50 milligrams per deciliter Total: 190 milligrams per deciliter 50, 120; 170 50 3.4 < 4; (e) answers will vary. Chapter 8 Section 8.1 (page 582) Vocabulary Check (page 582) 2. square 1. matrix 4. row; column 7. row-equivalent 9. Gauss-Jordan elimination 3. main diagonal 5. augmented 6. coefficient 8. reduced row-echelon form 1. 7. 1 2 4 1 11. 15. 7 19 2x 3 3 5 0 3. 5. 3 1 5 12 9. 2 2 1 5 2 1 8 5z 2z 13. 13 10 12 7 2 y 3y 6x 2 0 6 10 3 1 2 4 0 x 2y 7 2x 3y 4 17. 9x 2x x 3x 12y 18y 7y 3z 5z 8z 2z 19. 21 20 1 6 4 2w 0 10 4 10 1 0 0 1 1 3 4 2 5 20 23. Add 5 times Row 2 to Row 1. 25. Interchange Row 1 and Row 2. 1 6 5 4 Add 4 times new Row 1 to Row 3. (b) (d 10 10 2 1 0 3 2 0 27. (a) (c) (e 10 1 3 10 0 0 1 0 1 2 0 The matrix is in reduced row-echelon form. 29. Reduced row-echelon form 31. Not in row-echelon form 33. 37. 41. 45. 47. 53. 59. 65. 67. 71. 75. 81. 1 6 0 1 3 0 35 2y y 2 16 12 5 0 1 39. 43. x y 2z y z z 4 2 2 51. 3, 2 57. Inconsistent 4, 3, 6 69. Inconsistent 63. 61. 49. 55. 7, 3, 4 4, 10, 4 1, 4 8, 0, 2 3, 4 5, 6 4, 3, 2 2a 1, 3a 2, a 4 5b 4a, 2 3b 3a, b, a 0, 2 4a, a 1, 0, 4, 2 2a, a, a, 0 73. 77. Yes; |
,, 1, 3 3 1 2 79. No 333200_08_AN.qxd 12/9/05 2:40 PM Page A175 83. 4x2 x 12x 1 85. $150,000 at 7% $750,000 at 8% $600,000 at 10% 1 x 1 3 2 x 12 x 1 y x2 2x 5 87. 89. (a) (b) y 0.004x2 0.367x 5 18 0 0 120 (c) 13 feet, 104 feet (e) The results are similar. (d) 13.418 feet, 103.793 feet (c) (b) 91. (a) 500, 0, x5 s t, 600 s, x4 s, x7 t 600, x4 t, x3 s, x2 500 t, x6 0, x2 0, x7 0, x2 1000, x6 2 4 x1 x5 x1 x6 x1 x5 93. False. It is a 95. False. Gaussian elimination reduces a matrix until a rowechelon form is obtained; Gauss-Jordan elimination reduces a matrix until a reduced row-echelon form is obtained. 0, x3 0 500, x3 0, x7 matrix. 600, x4 500 500, 97. (a) There exists a row with all zeros except for the entry in the last column. (b) There are fewer rows with nonzero entries than there are variables and no rows as in (a). 99. They are the same. 101. 1032 β1 β1 1 2 3 4 x 105 Answers to Odd-Numbered Exercises and Tests A175 Section 8.2 (page 597) Vocabulary Check (page 597) 1. equal 5. (a) iii 6. (a) ii 2. scalars (b) iv (b) iv (c) i (c) i 3. zero; O (d) v (d) iii 4. identity (e) ii 3. 1 3 x 2, y 3 0 9 (c) (b) 3 6 3 3 (b) 5 3 4 5 1 5 (c) 18 6 9 3 12 15 1. x 4, y 22 5. (a) (d) 7. (a) (d) 9. (a) 2 7 1 19 3 9 15 11 16 8 11 3 2 1 4 |
6 3 4 9 (c) (b) 1 3 6 3 4 5 11. (a), (b), and (d) not possible 0 11 3 6 1 24 1 6 0 0 2 12 (d 11 18 0 12 3 7 1 8 9 9 0 15. 19. 3.739 13.249 0.362 4.252 9.713 (c) 8 15 10 59 1.581 3 3 1 2 13 2 0 11 2 4 16 46 10 26 3 13. 17. 21. 25. 29. 24 12 17.143 11.571 23. 4 32 12 12 2.143 10.286 9 0 10 6 1 17 27. Not possible 31. 3 0 0 0 4 0 0 0 10 Order: 3 2 Order 333200_08_AN.qxd 12/9/05 2:40 PM Page A176 A176 Answers to Odd-Numbered Exercises and Tests 33 35. 41 42 10 7 5 25 7 25 45 65. 37. 41. (a) 43. (a) 45. (a) 47. 51. (a) 53. (a) 55. (a) 57. (a) Order: 151 516 47 0 6 0 10 7 5 4 (c) (c) (b) (b) (b) 13 4 8 8 1 10 14 9 12 14 16 2 48 387 87 (c) Not possible 39. Not possible 6 12 8 6 6 8 2 31 0 10 2 14 10 0 3 3 25 279 20 15 12 10 0 7 8 1 4 8 49. 16 3 x1 4 1 (b) x2 0 1 4 3 x1 36 x2 1 x1 9 2 3 1 3 5 5 x1 20 5 1 2 30 60 84 120 A 125 100 The entries represent the numbers of bushels of each crop that are shipped to each outlet. $6.00 B $3.50 The entries represent the profits per bushel of each crop. BA $1037.50 The entries represent the profits from both crops at each of the three outlets1012.50 2 1 5 8 16 1 2 3 0 6 17 1 2 75 125 100 175 3 2 $1400 x2 x3 x2 x3 (b) (b) (b) 59. 84 42 61. (a) (b) (c) 63. $15,770 $26,500 $21,260 $18,300 $29,250 $24, |
150 The entries represent the wholesale and retail values of the inventories at the three outlets. 0.314 0.461 0.315 0.435 0.308 0.392 P3 0.300 P4 0.250 P5 0.225 P6 0.213 P7 0.206 P8 0.203 0.308 0.486 0.311 0.477 0.305 0.492 0.175 0.433 0.392 0.188 0.377 0.435 0.194 0.345 0.461 0.197 0.326 0.477 0.198 0.316 0.486 0.199 0.309 0.492 0.175 0.217 0.608 0.188 0.248 0.565 0.194 0.267 0.539 0.197 0.280 0.523 0.198 0.288 0.514 0.199 0.292 0.508 Approaches the matrix 0.2 0.3 0.5 0.2 0.3 0.5 0.2 0.3 0.5 67. (a) Sales $ Profit 115 161 188 447 624.5 731.2 (b) $464 The entries represent the total sales and profits for each type of milk. 69. (a) 2 0.5 3 (b) 120 lb 150 lb 473.5 588.5 The entries represent the total calories burned. 71. True. The sum of two matrices of different orders is undefined. 73. Not possible 79. 2 3 81. 75. Not possible 3 3 AC BC 2 2 77. 2 2 83. AB is a diagonal matrix whose entries are the products of the corresponding entries of A and B. 85. 91. 4 8, 3 7, 1 2 5 Β± 37 4 3, 1 87. 0, 93. 89. 4, Β± 15 3 i Section 8.3 (page 608) Vocabulary Check (page 608) 1. square 2. inverse 3. nonsingular; singular 4. A1B 333200_08_AN.qxd 12/9/05 2:40 PM Page A177 1β9. AB I 0 1 3 and BA I 3 2 13. 2 1 15. 1 2 1 1 1 2 0 11. 17. Does not exist 1 2 3 1 1 2 19. Does not exist 0 1 4 1 4 1 3 4 7 20 23. 0 0 1 5 21. 25. 29 |
. 33. 37.5 1 4.5 1 0 10 10.5 3.5 1 1.81 5 2.72 1 3 1 0.90 5 3.63 27. 175 95 14 37 20 3 13 7 1 31. 12 4 8 5 2 4 9 4 6 35. Does not exist 39. 3 19 2 19 2 19 5 19 15 59 70 59 3, 8, 11 55. No solution 7, 3, 2 16 59 4 59 41. Does not exist 43. 45. 51. 57. 61. 5, 0 2, 1, 0, 0 4, 8 5 16, 19 16a 13 5, 0, 2, 3 49. 47. 8, 6 53. 59. 16a 11 2, 2 1, 3, 2 16, a 63. 65. 67. $7000 in AAA-rated bonds $1000 in A-rated bonds $2000 in B-rated bonds 69. $9000 in AAA-rated bonds $1000 in A-rated bonds $2000 in B-rated bonds (b) I1 I2 I3 71. (a) amperes amperes amperes 3 8 5 2 3 5 73. True. If B is the inverse of A, then 75. Answers will vary. 77. x β₯ 5 or x β€ 9 I1 I2 I3 amperes amperes amperes AB I BA. β 10 β 9 β 8 β 7 β6 β5 β4 x 79. 2 ln 315 ln 3 10.472 81. 26.5 90.510 83. Answers will vary. Answers to Odd-Numbered Exercises and Tests A177 Section 8.4 (page 616) Vocabulary Check (page 616) 1. determinant 3. cofactor 2. minor 4. expanding by cofactors 5. 27 17. 11. 4.842 3 3 3 7. 0 0.002 2, M21 2, C21 2, M21 2, C21 4, M13 9. 6 19. 4, M22 4, C22 1, M22 1, C22 1, M21 4, M32 10, M33 1, C21 4, C13 4, C32 12, M13 7, M31 12, C13 4, C32 10, C33 11, M21 4, M32 11, C21 42, C33 75 33. (a) 96 (b) |
1. 5 13. 72 23. (a) (b) 25. (a) (b) 27. (a) M11 C11 M11 C11 M11 M23 C11 C23 M11 M22 C11 C23 75 31. (a) 35. (a) 170 58 43. 51. 412 3. 5 11 15. 6 5, M12 5, C12 4, M12 4, C12 3, M12 4, M31 3, C12 4, C31 30, M12 26, M23 30, C12 7, C31 (b) (b) 170 30 126 45. 53. 29. (a) (b) 61. (a) 3 (b) 2 63. (a) 8 (b) 0 (c) 65. (a) 21 (b) 19 67. (a) 2 (b) 6 (c) 39. 0 168 37. 0 47. 55. 0 2 (c) 0 4 1 57. 0 3 4 1 (c, 4 1 ln x 9 21. 0 2, 2, 3 2, M22 8 2, C22 8 36, 42, M33 36, C22 12 (b) 96 41. 9 12 26, 49. 0 336 (d) 6 59. 410 (d) 0 4 3 9 3 3 0 (d) 399 (d) 12 69β73. Answers will vary. 79. 83. 85. True. If an entire row is zero, then each cofactor in the 8uv 1 75. 81. 77. e5x 1, 4 expansion is multiplied by zero. 87. Answers will vary. 89. A square matrix is a square array of numbers. The deter- minant of a square matrix is a real number. 91. (a) Columns 2 and 3 of A were interchanged. A 115 B A (b) Rows 1 and 3 of were interchanged. A 40 B 93. (a) Multiply Row 1 by 5. (b) Multiply Column 2 by 4 and Column 3 by 3. 95. All real numbers x C H A P T E R 8 333200_08_AN.qxd 12/9/05 2:40 PM Page A178 A178 Answers to Odd-Numbered Exercises and Tests 97. All real numbers such that 99. All real numbers t such that 101. y 12 4 103. β |
8 β 4 4 8 12 x 105. Does not exist Section 8.5 (page 628) Vocabulary Check (page 628) 1. Cramerβs Rule 2. collinear x1 3. A Β± 1 2 y1 y2 y3 5. uncoded; coded x2 x3 1 1 1 4. cryptogram 67. y 6 4 2 (0, 5) (0, 0) (6, 4) ( 20 3 (, 0 x 2 4 6 Minimum at Maximum at 0, 0: 0 6, 4: 52 Review Exercises 1. 3 1 3. 1 1 5. (page 632) 10 3 4 5 1 9. 0 0 15 22 3 1 1 2 1 0 13. x 5y 4z 1 y 2z 3 z 4 9 10 3 7z 4x 9x y 2y 4y 5x x 2y 3z 9 y 2z 2 z 0 2z 10 1 17. 2, 2a 1, a 27. 5, 7 23. 2, 3, 1 33. 5, 2, 6 40, 5, 4 19. 1, 0, 4, 3 29. 2, 6, 10, 3 7. 11. 15. 21. 25. 31. 5 2 5. 7 0, 1 32 7, 30 11. 19. 33 8 2, 1 2 21. 3. Not possible 2, 1, 1 17. 14 y 0 or 9. 15. 7 y 16 5 y 11 2, 2 1, 3, 2 1, 2, 1 1. 7. 13. 23. 28 y 3 27. 31. Collinear y 3 37. 2x 3y 8 0 43. 45. Uncoded: 25. or 29. 250 square miles 33. Not collinear 3x 5y 0 39. 35. Collinear 41. x 3y 5 0 Encoded: 20 18 15, 21 2 12, 5 0 9, 14 0 18, 9 22 5, 18 0 3, 9 20 25 52 49 49 10 27 94 12 22 54 1 1 121 41 55 3 34 7 0 13 27 9 47. 49. 6 35 69 11 20 17 6 16 58 46 79 67 5 41 87 91 207 257 11 5 41 40 80 84 76 177 227 51. HAPPY NEW YEAR 53. CLASS IS CANCELED 55. SEND PLANES 59. False. The denominator is the determinant of the coeff |
i- 57. MEET ME TONIGHT RON cient matrix. 61. False. If the determinant of the coefficient matrix is zero, the system has either no solution or infinitely many solutions. 6, 4 1, 0, 3 63. 65. 5, 2, 0 10, 12 2a 3 2, 3, 3 x 12, y 7 1 8 13 15 8 8 20 12 5 20 7 14 42 3 31 (c) 28 44 16 8 8 (c) 35. (a) 37. (a) 39. 17 13 17 2 41. 43. 48 15 18 51 3 33 4 3 10 3 3 100 12 84 2 3 11 3 0 220 4 212 47. 51. 49. 30 51 53. 14 14 36 (b) (d) (b) (d) 54 2 4 45. x 1, y 11 12 5 3 9 7 28 29 39 5 5 11 9 1 10 38 13 38 122 5 71 4 24 32 14 7 17 4 70 2 10 12 4 17 2 8 40 48 333200_08_AN.qxd 12/12/05 11:30 AM Page A179 Answers to Odd-Numbered Exercises and Tests A179 5.5 2 14.5 2.5 3.5 1 9.5 1.5 5. 44 20 14 4 8 22 41 66 55. 61. 65. 22 80 66 57. 24 36 59. 17 36 8 12 76 38 1 12 114 95 133 76 63. 19 42 $274,150 $303,150 The merchandise shipped to warehouse 1 is worth $274,150 and the merchandise shipped to warehouse 2 is worth $303,150. AB I and BA I 73. 6, 1, 1 97. 550 83. 36, 11 89. 42 2 95. 1, M22 1, C22 2 21, 22, M31 5, 67β69. 4 5 71. 2 1 2 1 1 0 75. 79. 85. 91. 99. (a) (b) 101. (a 15 2.5 77. 12 5 6 5 2 13 3 1 7 1 20 3 1 1 6 10 2, 1, 2 2 1 2 5 6 1 3 81. 87. 4 6, 1 3, 1 M11 C11 M11 M21 M32 C11 C21 C31 93. 4, M12 4, C12 30, M12 20, M22 2, M33 30, C12 20, C |
22 5, C32 105. 279 113. 10 x 2y 4 0 (b) 1, 1, 2 7, M21 7, C21 12, M13 19, M23 19 12, C13 19, C23 19 2, C33 107. 4, 7 115. Collinear 21, 22, 103. 130 111. 16 117. 121. Uncoded: Encoded: 109. 1, 4, 5 12 15 2 12, 5 21 6 0 20 21 99 119. 15, 15 68 30 11 2x 6y 13 0 15, 0 21 23 0 20 0, 42 60 53 8 45 102 69 123. SEE YOU FRIDAY 125. False. The matrix must be square. 127. The matrix must be square and its determinant nonzero. 129. No. The first two matrices describe a system of equations with one solution. The third matrix describes a system with infinitely many solutions. Β±210 3 131. Chapter Test (page 637) 1, 3, 1 2 14 5 8 2 2 4 2. 3. 4. (a 15 12 7 4 4 0 (d) (b) (c) 12 12 14 12. 2 5 4 13, 22 7. 3, 5 11. 14. Uncoded: Encoded: 3 6 5 8. 12. 196 2, 4, 6 9. 29 10. 43 13. 7 11 14 15, 3 11 0, 15 14 0, 23 15 15, 4 0 0 115 41 59 14 3 11 29 15 14 128 53 60 15. 75 liters of 60% solution 25 liters of 20% solution 1. (a) Problem Solving AT 1 1 AAT 1 1 y (page 639) 2 3 3 2 4 2 2 4 AT 4 3 2 1 T β4 β3 β2 β1 1 2 3 4 x AAT β2 β3 β4 A represents a counterclockwise rotation. (b) AAT is rotated clockwise to obtain AT. AT is then rotated clockwise 90 90 to obtain T. 333200_09_AN.qxd 12/9/05 2:41 PM Page A180 A180 Answers to Odd-Numbered Exercises and Tests 3. (a) Yes 5. (a) Gold Cable Company: 28,750 subscribers (d) No (b) No (c) No 27. 10 29. 18 Galaxy Cable Company: 35,750 subscribers Nonsubscribers: 35 |
,500 Answers will vary. (b) Gold Cable Company: 30,813 subscribers Galaxy Cable Company: 39,675 subscribers Nonsubscribers: 29,513 Answers will vary. (c) Gold Cable Company: 31,947 subscribers Galaxy Cable Company: 42,329 subscribers Nonsubscribers: 25,724 Answers will vary. (d) Cable companies are increasing the number of subscribers, while the nonsubscribers are decreasing. x 6 7. 13. Sulfur: 32 atomic mass units 9β11. Answers will vary. Nitrogen: 14 atomic mass units Fluorine: 19 atomic mass units 2 0 1 5 1 BT 3 BTAT 2 3 AT 1 1 2 ABT 2 4 A1 1 1 0 15. 17. (a) 1 2 1 1 (b) JOHN RETURN TO BASE A 0 19. Chapter 9 Section 9.1 (page 649) Vocabulary Check (page 649) 1. infinite sequence 4. recursively 6. summation notation 8. series 9. 5. factorial n th partial sum 2. terms 3. finite 7. index; upper; lower 3. 2, 4, 8, 16, 32 3, 3 7. 0, 1, 0, 1 1. 4, 7, 10, 13, 16 5. 9. 2, 4, 8, 16, 32 3, 12 47, 15 13, 24 11, 9 11. 37 1 1 1 332, 232, 8 3, 2 3, 2 3, 2 3, 2 1,, 3 2 15. 19. 17. 1 532 21. 0, 0, 6, 24, 60 3, 2, 5 2, 0 1 4 1, 2, 7 5 13., 1 9 23., 53 17 5 3, 9, 27, 1, 1 25 16 73 (b) 109. (a) (b) (c) 161 81, 485 243 25. 44 239 31. 0 0 2 0 0 10 10 33. c 34. b 35. d 0 β10 10 an 3n 2 36. a 37. 1nn 1 n 2 39. an 43. an 49. an n 2 1 n 1 2n 1 1 1 n 41. an 45. an 1 n2 47. an 1n1 51. 28, 24, 20, 16, 12 53. 3, 4, 6, 10, 18 55. 6, 8, 10, 12, 14 |
an 57. 81, 27, 9, 3, 1 243 3n 1 24 67. 90 61. 65. 1 6 1 2 1,,,, 1 30 1 120 1 2n2n 1 9 81. 88 5 71. 79. 89. 95. i1 9 20 i1 2 3 103. 107. (a) an 2n 4 9 9 2 2 27 8,, 59. 1, 3, 63. 1, 1 2 n 1 69., 1 24, 1 720, 1 40,320 73. 35 75. 40 77. 30 85. 81 87. 47 60 93. 1i13i 6 i1 75 16 99. 101. 3 2 A3 A6 $5306.04, $5630.81, 7 9 i1 i1 97. 91. 8 2i 1 2i1 83. 30 3 2 i 8 5 1 3i 1i1 i 2 105. $5100.00, A2 $5412.16, A5 $5743.43, A8 $11,040.20 60.57n 182 1.61n2 26.8n 9.5 A1 A4 A7 A40 bn cn $5202.00, $5520.40, $5858.30 n an bn cn 8 9 10 11 12 13 311 357 419 481 548 608 303 363 424 484 545 605 308 362 420 480 544 611 The quadratic model is a better fit. (d) The quadratic model; 995 333200_09_AN.qxd 12/12/05 11:31 AM Page A181 Answers to Odd-Numbered Exercises and Tests A181 111. (a) a0 a3 a6 a9 a12 $3102.9, a1 $4425.3, a4 $5091.8, a7 $5550.9, a10 $6251.5, a13 $3644.3, a2 $4698.2, a5 $5245.7, a8 $5735.5, a11 $6616.3 $4079.6, $4914.8, $5393.2, $5963.5, 7000 0 0 14 (b) The federal debt is increasing. 113. True by the Properties of Sums 115. 1, 1, 2, 3, |
5, 8, 13, 21, 34, 55, 89, 144 21 13, 34 21, 55 34, 8 5,,,,, x, 125. 121. 123. 129. (a) 5 3 1, 2, 2, 3, 117. $500.95 x4 x3 x2 24 6, 2 x4 x2, x6, 720 24 2 f 1x x 3 4 8 1 2 6 18 9 7 10 3 7 4 4 2 7 42 23 194 4 1 135. 131. (a) 133. 26 4 1 (c) (c) 89 13 8, 55 119. Answers will vary. x5 120 x8 40,320, x10, x β₯ 0 127. (b) (d) 5 3,628,800 h1x x2 1 26 12 4 24 1 21 2 21 10 16 12 3 25 11 9 31 47 22 10 13 (b) (d) 10 3 8 42 31 25 4 1 3 16 45 48 Section 9.2 (page 659) Vocabulary Check (page 659) 1. arithmetic; common 3. sum of a finite arithmetic sequence an 2. dn c d 2 1. Arithmetic sequence, 3. Not an arithmetic sequence 5. Arithmetic sequence, 7. Not an arithmetic sequence 9. Not an arithmetic sequence 11. 8, 11, 14, 17, 20 d 1 4 13. 7, 3, Arithmetic sequence, 5, Arithmetic sequence, 1, 9 d 3 d 4 15. 17. 3 3 1, 1 3 5 1, 4,1, 1, 1, Not an arithmetic sequence 2,3, Not an arithmetic sequence an an an 25. 21. 3 3n 2 2 xn x 10 3 n 5 8n 108 19. an 5 2n 13 23. an 2 3n 103 29. 27. an 2.6, 3.0, 3.4, 3.8, 4.2 31. 5, 11, 17, 23, 29 33. 2, 2, 6, 10, 14 35. 2, 6, 10, 14, 18 37. d 4; 4n 11 39. 15, 19, 23, 27, 31; 41. 200, 190, 180, 170, 160; 1 1 5 43. 8, 2, 8 ; 45. 59 53. an d 10; 1 8 n 3 4 50. d 1 4, 47. 18.6 10n 210 an |
d) The slope of the line and the common difference of the arithmetic sequence are equal. 103. 4 105. Slope: y- 1 2; intercept: 0 107. Slope: undefined; y- intercept No y 8 6 4 2 β2 β2 β 4 β 6 β8 2 4 6 8 10 12 14 x 109. x 1, y 5, z 1 111. Answers will vary. Section 9.3 (page 669) Vocabulary Check (page 669) β4 2. an a1r n1 4. geometric series 1. geometric; common 3. Sn 5. S a11 r n 1 r a1 1 r r 3 1. Geometric sequence, 3. Not a geometric sequence r 1 5. Geometric sequence, 2 r 2 7. Geometric sequence, 9. Not a geometric sequence 11. 2, 6, 18, 54, 162 200, 1 20, 1 2, 1 5, 1 15. x4 x2 x3 x 2 128 8 32 21. 64, 32, 16, 8, 4; 1, 1 17. 13. 19. r 1 2; an 2, 2000,,, 8, 1 4, 1 2, 1 16 1, e, e2, e3, e4 128 1 n 2 51. 0 β16 24 0 0 10 0 10 β15 10 57. 43 63. 592.647 1365 59. 32 65. 2092.596 71. 3.750 7 21 4 n1 85. 5 3 93. 4 11 77. 87. 95. 0.14n1 6 n1 30 7 22 55. 171 53. 511 61. 29,921.311 67. 8 5 69. 6.400 53n1 75. n1 73. 7 79. 2 89. 32 97. n1 16 81. 3 91. Undefined 83. 2 3 20 β15 10 Horizontal asymptote: Corresponds to the sum of the series y 12 99. (a) an 1190.881.006n (b) The population is growing at a rate of 0.6% per year. (c) 1342.2 million. This value is close to the prediction. (d) 2007 101. (a) $3714.87 (d) $3728.32 (b) $3722.16 (e) $3729.52 (c) $3725.85 105. Answers |
will vary. (b) $26,263.88 (b) $118,788.73 113. $1600 103. $7011.89 107. (a) $26,198.27 109. (a) $118,590.12 111. Answers will vary. $2181.82 115. 119. $3,623,993.23 121. False. A sequence is geometric if the ratios of consecutive 117. 126 square inches terms are the same. 123. Given a real number between r n approaches zero. increases, n r 1 and 1, as the exponent 333200_09_AN.qxd 12/9/05 2:41 PM Page A183 Answers to Odd-Numbered Exercises and Tests A183 139. Answers will vary. β12β10 β8 β6 β4 (d) y 10 8 6 4 2 β4 β6 x 2 4 (0, 0) x2 2x 127. x3x 83x 8 3x2 6x 1 131. 125. 129. 133. 137., x 3 3x x 3 5x2 9x 30 x 2x 2 3x 12x 5, x 0, 1 2 2x 1 3 135. Section 9.4 (page 681) Vocabulary Check (page 681) 1. mathematical induction 3. arithmetic 4. second 2. first 1. 5 k 1k 2 3. Sn 37. 5β33. Answers will vary. n 10 10 9 10 45. 979 43. 91 41. 120 51. 0, 3, 6, 9, 12, 15 39. k 12k 22 4 Sn 35. n2n 1 n 2n 1 Sn 47. 70 49. 3402 First differences: 3, 3, 3, 3, 3 Second differences: 0, 0, 0, 0 Linear 53. 3, 1, 2, 6, 11, 17 First differences: Second differences: Quadratic 2, 3, 4, 5, 6 1, 1, 1, 1 55. 2, 4, 16, 256, 65,536, 4,294,967,296 First differences: 2, 12, 240, 65,280, 4,294,901,760 Second differences: 10, 228, 65,040, 4,294,836,480 Neither an 2 n 2 n 3 1 an n2 n 3 57. 59. 61. (a) 2.2, 2. |
4, 2.2, 2.3, 0.9 (b) A linear model can be used. 2.2n 102.7 2.08n 103.9 142.3; an an (c) (d) Part b: Part c: an These are very similar. an 141.34 P7 63. True. may be false. 65. True. If the second differences are all zero, then the first differences are all the same and the sequence is arithmetic. 4x4 4x2 1 64x3 240x2 300x 125 69. 67. 71. (a) Domain: all real numbers except 0, 0 x x 3 (b) Intercept: (c) Vertical asymptote: Horizontal asymptote: x 3 y 1 73. (a) Domain: all real numbers except t t 0 t -intercept: (b) (c) Vertical asymptote: 7, 0 t 0 Horizontal asymptote: y y 1 (d) 4 2 β8 β6 β4 β2 2 6 8 t (7, 0) β4 β6 β8 Section 9.5 (page 688) Vocabulary Check (page 688) 1. binomial coefficients 2. Binomial Theorem; Pascalβs Triangle 3. ; nCr n r 4. expanding a binomial C H A P T E R 9 3. 1 13. 35 5. 15,504 15. 7. 210 x 4 4x3 6x 2 4x 1 9. 4950 1. 10 11. 56 17. 19. 21. 23. a4 24a3 216a2 864a 1296 y3 12y2 48y 64 x5 5x 4y 10x 3y2 10x 2y 3 5xy 4 y5 r 6 18r 5s 135r 4s2 540r 3s3 1215r 2s4 1458rs5 729s6 25. 243a5 1620a4b 4320a3b2 5760a2b3 3840ab4 1024b5 27. 29. 31. 8x3 12x2y 6xy2 y3 x8 4x6y2 6x 4y 4 4x2y6 y8 1 x 5 2x4 24x3 113x2 246x 207 10y2 x 3 10y3 x 2 5y4 x 5y x 4 y5 33 |
. 35. 32t 5 80t 4s 80t 3s2 40t 2s3 10ts4 s5 333200_09_AN.qxd 12/12/05 11:32 AM Page A184 A184 Answers to Odd-Numbered Exercises and Tests 41. 360 x3y2 37. 39. 45. 49. 180 55. 57. x5 10x4y 40x3y2 80x2y3 80xy4 32y5 120x7y3 1,259,712 x2y7 32,476,950,000x4y8 51. 43. 47. 1,732,104 326,592 53. 210 x2 12x32 54x 108x12 81 x 2 3x 43y 13 3x 23y 23 y 59. 3x 2 3xh h 2, h 0 61. 1 x h x h 0, 4 63. 69. 1.172 73. g β 8 65. 2035 828i 71. 510,568.785 67. 1 4 f β 4 4 93. 954 β2 2 4 6 x β2 β3 β2 β1 1 2 3 β1 x gx x 32 4 5 5 6 97. Section 9.6 (page 698) gx x 2 1 f. Vocabulary Check (page 698) 1. Fundamental Counting Principle 2. permutation is shifted four units to the left of g gx x3 12x2 44x 48 77. 0.171 f t 0.0025t 3 0.015t 2 0.88t 7.7 75. 0.273 79. (a) (b) 24 (c) (d) 0 0 13 gt 0.0025t 3 0.06t 2 1.33t 17.5 60 0 0 g f 13 f t: gt: 33.26 gallons; (e) (f) The trend is for the per capita consumption of bottled water to increase. This may be due to the increasing concern with contaminants in tap water. 33.26 gallons; yes 81. True. The coefficients from the Binomial Theorem can be used to find the numbers in Pascalβs Triangle. 83. False. The coefficient of the x10 -term is 1,732,104 and the coefficient of the x14 -term is 192,456. 85. 1 8 28 56 70 56 28 8 1 1 9 36 84 126 |
126 84 36 9 1 1 10 45 120 210 252 210 120 45 87. The signs of the terms in the expansion of between positive and negative. 89β91. Answers will vary. 1 10 x yn n! 3. nPr n r! 5. combinations 4. distinguishable permutations 7. 8 5. 3 9. 30 11. 30 (c) 180 (d) 600 27. 120 3. 5 15. 175,760,000 (b) 648 21. (a) 40,320 1. 6 13. 64 17. (a) 900 19. 64,000 25. 336 31. 1,860,480 39. 11,880 45. ABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, CABD, CADB, DABC, DACB, BCAD, BDAC, CBAD, CDAB, DBAC, DCAB, BCDA, BDCA, CBDA, CDBA, DBCA, DCBA (b) 384 n 6 or 35. 15,504 33. 970,200 43. 2520 37. 120 41. 420 n 5 23. 24 29. 47. 1,816,214,400 51. AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, 49. 5,586,853,480 DF, EF 53. 324,632 57. (a) 3744 61. 5 65. (a) 146,107,962 63. 20 55. (a) 35 (b) 24 (b) 63 59. 292,600 (c) 203 (b) If the jackpot is won, there is only one winning number. alternate (c) There are 28,989,675 possible winning numbers in the state lottery, which is considerably less than the possible number of winning Powerball numbers. 67. False. It is an example of a combination. 69. They are equal. 71β73. Proof 75. No. For some calculators the number is too great. 77. (a) 35 (c) 83 4 79. (a) (c) 0 (b) 8 (b) 0 81. 8.30 83. 35 333200_09_AN.qxd 12/9/05 2:41 PM Page A185 Section 9.7 (page 709) Vocabulary Check ( |
page 709) 1. experiment; outcomes 3. probability 5. mutually exclusive 7. complement 4. impossible; certain 6. independent (b) i 8. (a) iii 2. sample space (c) iv (d) ii Answers to Odd-Numbered Exercises and Tests A185 71. y 12 10 8 4 2 73. y 2 β8 β6 β4 β2 4 6 8 x β4 β2 2 4 6 8 12 x Review Exercises (page 715) β8 β12 β14 11 17. 12 29. 0.86 1. 8, 5, 4, 4 n an 7. 17. 6050 21n 205 24 15. 7 2, 16 5 3. 72, 36, 12, 3, 3 5 5. an k1 19. 9. 120 20 1 2k $10,067, $10,269, $10,476, $10,687 $22,196.40 A2 A5 A8 25. (a) (b) A1 A4 A7 A10 A120 11. 1 13. 30 21. 5 9 $10,134, $10,338, $10,546, 23. 2 99 A3 A6 A9 $10,201, $10,407, $10,616, 31. 4, 7, 10, 13, 16 d 2 d 1 2 12n 5 35. an 7n 107 39. an 45. 25,250 (b) $192,500 53. 4, 1, 1 4, 1 16, 1 64 9, 6, 4, 8 ; 3.052 105 or n1 3ny 2y 43. 88 r 2 r 2 3, 16 27. Arithmetic sequence, 29. Arithmetic sequence, 33. 25, 28, 31, 34, 37 37. an 41. 80 47. (a) $43,000 49. Geometric sequence, 51. Geometric sequence, 3, 16 8 55. 9, 6, 4, 161 57. an 1001.05n1; 252.695 59. an 15 61. 127 16 71. 8 69. 5486.45 120,0000.7t 77. (a) 79β81. Answers will vary. 1 3 n 85. Sn 91. 5, 10, 15, 20, 25 First differences: 5, 5, 5, 5 Second differences: 0, 0, 0 Linear 83. 87. 465 65 |
. 31 73. 5 2 63. 10 9 at 2 5 67. 24.85 75. 12 (b) $20,168.40 Sn n2n 7 89. 4648 93. 16, 15, 14, 13, 12 First differences: Second differences: 0, 0, 0 Linear 1, 1, 1, 1 95. 15 103. 105. 107. 115. 56 97. 56 99. 35 101. 28 x 4 16x3 96x2 256x 256 a5 15a 4b 90a3b2 270a2b3 405ab4 243b5 41 840i 117. 119. (a) 43% (b) 82% 111. 10,000 109. 11 113. 720 1 9 1 5 1. (b) 27. 15. 3 13 2 5 1 12 3 4 11. 23. H, 1, H, 2, H, 3, H, 4, H, 5, H, 6, T, 1, T, 2, T, 3, T, 4, T, 5, T, 6 ABC, ACB, BAC, BCA, CAB, CBA 3. AB, AC, AD, AE, BC, 5. 3 7. 8 1 19. 3 18 31. 35 35. (a) 243 112 209 7 9. 8 21. 33. (a) 58% (b) 95.6% (c) 0.4% 16 25 274 627 BD, BE, CD, CE, DE 3 13. 26 25. 0.3 (b) PTaylor wins 1 2 PMoore wins PJenkins wins 1 4 49 (c) 323 45. (a) 54 12 (c) 55 55 (b) 0.9998 (c) (c) 41. (a) 43. (a) 47. (a) 51. (a) 0.9702 53. (a) 55. (a) 5 1 (b) 13 2 49. 0.4746 (c) 0.0002 (b) (b) (b) 21 1292 1 120 14 55 225 646 1 24 1 50 97 209 37. (a) (b) (b) 39. (c) (c) (e) 15 16 1 1444 1 8 9 19 1 16 1 38 10 19 729 6859 (c) 4 13 (d) (f) The probabilities are slightly better in European roulette. |
57. True. Two events are independent if the occurrence of one has no effect on the occurrence of the other. 59. (a) As you consider successive people with distinct birthdays, the probabilities must decrease to take into account the birth dates already used. Because the birth dates of people are independent events, multiply the respective probabilities of distinct birthdays. 365 363 365 364 365 (c) Answers will vary. is the probability that the birthdays are not distinct, Qn which is equivalent to at least two people having the same birthday. 365 362 (b) (d) 365 (e) n 10 15 20 23 30 40 50 0.88 0.75 0.59 0.49 0.29 0.11 0.03 0.12 0.25 0.41 0.51 0.71 0.89 0.97 Pn Qn (f) 23 61. No real solution 10 69. 67. 11 2 63. 0, 1 Β± 13 2 65. 4 333200_09_AN.qxd 12/9/05 2:41 PM Page A186 A186 Answers to Odd-Numbered Exercises and Tests 1 216 121. 125. True. 3 123. 4 n 2n 1n! n 2! n! n! 127. True by Properties of Sums 129. False. When equals 0 or 1, then the results are the same. 131. In the sequence in part (a), the odd-numbered terms are negative, whereas in the sequence in part (b), the evennumbered terms are negative. n 2n 1 r 2 3 1 3 0 4 3 10. 12. 15. 2 1 3 3 2 12 3 1 2 4 13. 2 2 9 9 7 6 0 16. 84 17. 11. 2, 3, 1 14. 175 95 14 6 3 6 2 37 20 3 13 7 1 133. Each term of the sequence is defined in terms of preceding 18. Gym shoes: $198.36 million terms. 135. d 136. a 139. 240, 440, 810, 1490, 2740 137. b 138. c (page 719), 1 17 4. 6. 86,100 an 2. an n 2 n! 0.8n 1.4 7. 189,, 1. 1 5 Chapter Test, 1 11 1 1 14 8 3. 50, 61, 72; 140 5. 5, 10, 20, 40, 80 8. 4 10. 12. |
(a) 72 14. 26,000 18. 25% 9. Answers will vary. x4 8x3y 24x2y2 32xy3 16y4 (b) 328,440 15. 720 13. (a) 330 17. 1 15 16. 108,864 11. (b) 720,720 3.908 1010 Cumulative Test for Chapters 7β9 (page 720),, 22. 19. 20., 1 11 3, 4, 2 1 13 Jogging shoes: $358.48 million Walking shoes: $167.17 million 5, 4 1, 1 1 9 7 5 25. (a) 65.4 24. 920 26. 3, 6, 12, 24, 48 29. 32. 70 36. 720 z4 12z3 54z2 108z 81 34. 453,600 33. 120 37. (b) 13 9 27. 23. an an 1 4 21. 9 n 1! n 3 3.2n 1.4 28. Answers will vary. 30. 210 35. 151,200 31. 600 Problem Solving (page 725) 1. 1, 1.5, 1.416, 1.414215686, 1.414213562, 1.414213562,... 2. xn 3. (a) approaches 8 n (b) If n if is odd, is even, an an and 2, 4. y 2 1 β3 β2 β1 1 3 4 6 7 x (c) 0 0 10 n an 1 2 10 101 1000 10,001 4 2 4 2 β 2 β3 β 4 β 5 β6 β8 1. 3. 5. 7. 1, 2, 3 4, 2, 3 2, 3 4 2, 1 2. 1, 2, 1 4. 6 12 10 (0, 5) 8 6 4 2 (4, 4) (6, 0) x 4 8 10 12 2 (0, 0) Maximum at Minimum at 4, 4: z 20 0, 0: z 0 8. $0.75 mixture: 120 pounds; $1.25 mixture: 80 pounds 9. 3 x2 2x 4 y 1 (d) It is not possible to find the value of an as approaches n infinity. 2n 1 5. (a) 3, 5, 7, 9, 11, 13, 15, 17; an (b) To obtain the arithmetic |
sequence, find the differences of consecutive terms of the sequence of perfect cubes. Then find the differences of consecutive terms of this sequence. (c) 12, 18, 24, 30, 36, 42, 48; (d) To obtain the arithmetic sequence, find the third sequence obtained by taking differences of consecutive terms in consecutive sequences. an 6n 6 (e) 60, 84, 108, 132, 156, 180; an 24n 36 7. sn an n1 1 2 3 4 9. Answers will vary. 11. (a) Answers will vary. $0.71 15. (a) 13. sn 2 1 3 (b) 17,710 (b) 2.53, 24 turns 333200_10a_AN.qxd 12/9/05 2:42 PM Page A187 Chapter 10 Section 10.1 (page 732) Vocabulary Check (page 732) 2. 1. inclination m1 1 m1m2 m2 3. tan Ax1 4. C By1 A2 B2 3. 1 5. 3 7. 3.2236 1. 9. 3 3 3 4 radians, 135 11. 13. 0.6435 radian, 17. 2.1112 radians, 21. 2.1112 radians, 25. 0.1974 radian, 29. 0.9273 radian, 33. 36.9 121.0 121.0 11.3 53.1 radian, 45 4 15. 1.0517 radians, 60.3 19. 1.2490 radians, 23. 1.1071 radians, 71.6 63.4 27. 1.4289 radians, 31. 0.8187 radian, 81.9 46.9 2, 1 β 4, 4: slope 3 2 4, 4 β 6, 2: slope 1 6, 2 β 2, 1: slope 1 4 2, 1: 42.3; 4, 4: 78.7; 6, 2: 59.0 4, 1 β 3, 2: slope 3 7 3, 2 β 1, 0: slope 1 1, 0 β 4, 1: slope 1 5 4, 1: 11.9; 3, 2: 21.8; 1, 0: 146.3 35. 37. 0 39. 7 5 41. 7 45. (a 43. 47. (a) 837 37 1.3 |
152 2 β1 β1 β2 3537 74 53. (c) 35 8 (b) 4 (c) 8 (b) 31.0 51. 0.1003, 1054 feet 22 33.69; 56.31 49. 55. 57. True. The inclination of a line is related to its slope by m tan., then the angle is in the second quadrant, where the tangent function is negative. If the angle is greater than but less than 2 59. (a) d 4 m2 1 Answers to Odd-Numbered Exercises and Tests A187 (b) d (c) m 0 6 5 2 1 β4 β3 β2 β1 1 2 3 4 β2 (d) The graph has a horizond 0. tal asymptote at As the slope becomes the distance larger, between the origin and y mx 4, the line becomes smaller and approaches 0. xy- intercepts: intercept: 5 Β± 5, 0 0, 20 m 63. 61. xy- intercept: intercept: 65. x- intercepts: 7, 0 0, 49 7 Β± 53 2 0, 1, 0 3 yintercept: f x 3x 1 3, 49 Vertex: f x 6x 1 1 12, 289 Vertex: 1 2 49 3 3 2 289 24 12 24 67. 71. 73. 69. 75. f x 5x 17 Vertex: 17 5 2 324 5 5, 324 5 y 6 5 2 1 β1 β1 β 12 9 6 3 β3 β3 3 6 9 12 x β4 β3 β2 Section 10.2 (page 740) Vocabulary Check (page 740) 1. conic 4. axis 2. locus 5. vertex 3. parabola; directrix; focus 7. tangent 6. focal chord x 1. A circle is formed when a plane intersects the top or bottom half of a double-napped cone and is perpendicular to the axis of the cone. 3. A parabola is formed when a plane intersects the top or bottom half of a double-napped cone, is parallel to the side of the cone, and does not intersect the vertex. 7. d 8. f 6. b 5. e 11. Vertex: Focus: Directrix: 0, 0 0, 1 2 y 1 2 9. a 13. Vertex: Focus: Directrix: 10. |
c 0, 0 3 21 β6 β5 β4 β3 β3 β4 333200_10a_AN.qxd 12/9/05 2:42 PM Page A188 A188 Answers to Odd-Numbered Exercises and Tests 15. Vertex: Focus: Directrix: 0, 0 0, 3 2 y 3 2 17. Vertex: Focus: Directrix: 1, 2 13 β2 β1 1 2 3 4 5 x β3 β 19. Vertex: Focus: Directrix: 2, 2 2, 3 y 1 1, 1 1, 2 21. Vertex: Focus: Directrix 321 β2 2 4 x 23. Vertex: Focus: Directrix: 2, 3 4, 3 x 0 β 10 β 8 β 6 β 4 27. Vertex: Focus: Directrix: 2 1 4, 1 0, 1 2 x 1 2 25. Vertex: Focus: Directrix: 2, 1 2, 1 2 x 2 β14 x 10 4 β12 y 2 β2 β 4 β6 β8 4 45. 49. 53. 55. 59. y 22 8x 5 y 22 8x 51. 47. x2 8 y 4 y 6x 1 3 10 β5 25 β10 2, 4 4x y 8 0; 2, 0 15,000 0 0 225 57. 61. 4x y 2 0; 1 y 1 18 x2 2, 0 x 106 units y 1 640 x2 (b) 8 feet 17,5002 miles per hour x2 16,400 y 4100 x2 64 y 75 63. (a) 65. (a) (b) 67. (a) 69. False. If the graph crossed the directrix, there would exist 24,750 miles per hour (b) 69.3 feet points closer to the directrix than the focus. 71. (a) p = 3 p = 2 21 p = 1 p = 4 β18 18 β3 p increases, the graph becomes wider. As 0, 1, 0, 2, 0, 3, 0, 4 (b) (d) Easy way to determine two additional points on the (c) 4, 8, 12, 16; 4p graph 75. Β±1, Β±2, Β±4 m x1 2p Β± 1 2, Β±1, Β±2, Β±4, Β±8, Β±16 f x x3 7x2 17x 15 81. B |
23.67, C 121.33, c 14.89 C 89, a 1.93, b 2.33 A 16.39, B 23.77, C 139.84 B 24.62, C 90.38, a 10.88 73. 77. 79. 83. 85. 87. 89. 1 2, 5 3, Β±2 β10 2 Section 10.3 (page 750) 29. 35. 41. x2 3 2 y x2 4y x 32 y 1 31. 37. β4 x2 6y y2 8x 43. y2 8x 33. y2 9x 39. y2 4x 4 Vocabulary Check (page 750) 1. ellipse; foci 3. minor axis 2. major axis; center 4. eccentricity 333200_10a_AN.qxd 12/9/05 2:42 PM Page A189 Answers to Odd-Numbered Exercises and Tests A189 2. c 3. d 4. f 1. b 7. Ellipse Center: Vertices: Foci: 0, 0 Β±5, 0 Β±3, 0 Eccentricity 11. Ellipse Center: Vertices: 0, 0 0, Β±3 0, Β±2 Foci: Eccentricity: y 2 3 4 2 1 β 15. Circle Center: Radius: 0, 1 2 3 5. a 9. Circle 6. e 0, 0 Center: Radius: 5 y 6 4 2 β6 β2 2 4 6 x β2 β4 β6 3, 5 13. Ellipse Center: Vertices: 3, 10, 3, 0 Foci: Eccentricity: 3, 8, 3, 2 3 5 y 12 8 6 4 2 β2 β1 1 β1 β2 β3 2, 4 17. Ellipse Center: Vertices: 3, 4, 1, 4 4 Β± 3 Foci:, 4 2 Eccentricity: 3 2 y β3 β2 β1 x 1 β1 β2 β3 β4 β5 19. Ellipse Center: Vertices: Foci: 2, 3 2, 6, 2, 0 2, 3 Β± 5 21. Circle 1, 2 Center: Radius: 6 y Eccentricity: β6 β4 β2 5 3 y 6 4 2 β2 x 2 β8 β6 6 2 β2 β2 β4 β6 β10 2 4 6 8 x |
23. Ellipse Center: Vertices: Foci: 3, 1 3, 7, 3, 5 3, 1 Β± 26 6 3 Eccentricity: y 8 4 2 β10 β8 β4 β2 2 4 x 25. Ellipse β6 y Center: Vertices: Foci: 3, 5 2, 3, 5 9, 5 2 2 3 Β± 33, 5 2 3 2 Eccentricity: y 3 2 β3 β2 β1 x 1 β1 β4 6 4 2 β2 β6 β8 2 4 6 10 x 29. Ellipse 2, 1 Center: 7 3, 1 3, 1, 5 Vertices: 15, 1 15, 1, 26 34 Foci: Eccentricity 27. Circle Center: Radius: 1, 1 2 3 333200_10a_AN.qxd 12/9/05 2:42 PM Page A190 A190 31. β6 Answers to Odd-Numbered Exercises and Tests 4 β4 33. β4 6 2 β4 Β± 5, 1 Center: Vertices: 1 Foci: 2 1 2, 1 1 2 Β± 2, 1 x2 y 2 36 11 y 32 9 1 39. 1 1 x2 36 y 2 32 x 22 1 0, 0 0, Β± 5 0, Β± 2 1 37. y 2 16 y 2 25 Center: Vertices: Foci: x2 4 21x2 400 x 22 16 x 22 4 1 43. y 32 9 y 42 1 1 1 1 53. x2 16 x2 25 y 42 12 y2 16 1 35. 41. 45. 47. 51. 55. 57. (a) y 59. (a) x2 321.84 y2 20.89 1 (b) 14 (0, 10) β21 21 (β25, 0) x (25, 0) β14 (c) Aphelion: (b) x2 625 (c) Yes x2 0.04 61. (a) (b) y2 100 1 y2 2.56 1 y 2 35.29 astronomical units Perihelion: 0.59 astronomical unit (c) The bottom half β 0.8 β 0.4 0.4 0.8 63. β 2 y x 654 ( β 3 5 5, β2 ) 2β y 4 ββ, ) 67. False. The graph of x24 y4 1 is not an ellip |
se. The 5 degree of y is 4, not 2. 69. (a) A a20 a (b) x2 196 y 2 36 1 (c) a 8 9 10 11 12 13 A 301.6 311.0 314.2 311.0 301.6 285.9 a 10, circle (d) 350 0 0 20 49. x2 48 x 22 4 y 42 64 y 22 1 1 1 The shape of an ellipse with a maximum area is a a 10 circle. The maximum area is found when so the equa(verified in part c) and therefore tion produces a circle. b 10, 71. Geometric 73. Arithmetic 75. 547 77. 340.15 Section 10.4 (page 760) Vocabulary Check (page 760) 1. hyperbola; foci 3. transverse axis; center 5. Ax2 Cy 2 Dx Ey F 0 2. branches 4. asymptotes 3. a 4. d 2. c 1. b 5. Center: 0, 0 Β±1, 0 Β± 2, 0 Vertices: Foci: Asymptotes: y y Β±x 7. Center: 0, 0 0, Β±5 0, Β± 106 Vertices: Foci: Asymptotes: 2 1 β1 β2 x 2 β8 β6 β2 9. Center: 1, 2 3, 2, 1, 2 Vertices: Foci: Asymptotes: 1 Β± 5 86 10 1 2 3 x y 10 8 6 4 2 β2 β4 β6 β10 y 3 2 1 β4 β5 333200_10a_AN.qxd 12/12/05 11:33 AM Page A191 Answers to Odd-Numbered Exercises and Tests A191 11. Center: 2, 6 Vertices: 2, 17 3 Foci:, 2, 19 3 2, 6 Β± 6 13 13. Center: Asymptotes: y 6 Β± 2 3 2, 3 3, 3, 1, 3 x 2 2 Β± 10, 3 Vertices: Foci: Asymptotes: y 3 Β± 3x 2 y 2 β2 2 4 6 β6 β10 β12 β14 y 2 β6 β4 β2 2 4 6 8 β4 β6 β8 x x 55. Parabola b to a, the larger the eccentricity of the hyperbola, c2 a2 b2. The larger |
3 15. y2 x 17. x 12 6 y 1 6 yβ² y 2 xβ² β 6 β 4 x 2 yβ² y 6 4 2 β2 β 4 β4 2 4 x 41. xβ² β2 7 37. (a) Hyperbola (b) (c) y 6x Β± 36x2 20x2 4x 22 10 β9 6 β6 9 39. (a) Parabola (b) y 4x 1 Β± 4x 12 16x2 5x 3 8 (c) 2 β4 y 6 43. y 4 3 1 β6 β4 β2 2 4 6 x β 4 β3 β2 β1 1 3 4 x β6 β2 β3 β 4 2 47. 49. 0, 8, 12, 8 55. No solution 8, 12 1, 3, 1, 3 2, 2, 2, 4 0, 4 53. 0, 3, 3, 0 45. 51. 57. 59. True. The graph of the equation can be classified by finding the discriminant. For a graph to be a hyperbola, the k β₯ 1 discriminant must be greater than zero. If then the 4, discriminant would be less than or equal to zero. 61. Answers will vary. 63. y 651 β1 β2 β6 β5 β4 β3 β2 x t β4 β3 β2 β1 1 2 3 4 x 1 2 β2 β3 β4 67. y 691 β1 β2 β3 β4 β3 β2 β1 β1 1 2 3 4 5 t Area 45.11 square units 71. 73. Area 48.60 square units 19. β15 45 23. 10 β10 4 β2 21. 15 β9 6 β6 9 26.57 25. 18 β6 6 β9 27 β6 33.69 31. d 32. c 29. b 30. a β4 31.72 28. f 27. e 33. (a) Parabola (b) y 8x 5 Β± 8x 52 416x2 10x 2 (c) β4 35. (a) Ellipse 2 1 β3 (b) (c) y 6x Β± 36x2 2812x2 45 14 β4 3 β3 5 333200_10a_AN.qxd 12/9/05 2:42 PM Page A193 Answers to Odd-Numbered Exercises and |
Tests A193 Section 10.6 (page 776) 7. (a) Vocabulary Check (page 776) 1. plane curve; parametric; parameter 2. orientation 3. eliminating the parameter y 4 3 2 1 1. (a) (bc) y 3 x2 The graph of the rectangular equation shows the entire parabola rather than just the right half. The graph of the rectangular equation continues the graph into the second and third quadrants. 5. (a) y 3. (a) y 6 5 4 2 1 βb) y 2 3 x 3 β2 β1 1 2 x β1 (b) y 16x2 9. (a) y 2 1 β1 β3 β2 β1 β2 (b) y x2 4x 4 11. (a) y 14 12 10 8 6 2 β2 2 4 6 8 10 12 14 x y x 2 3 (b) 15. (a) y 4 3 1 β3 β2 β1 1 2 3 β3 β4 (b) y x 1 x 13. (a) y 4 2 1 β4 β2 β1 1 2 4 β2 β4 (b) y2 9 x2 9 1 17. (a) y 3 2 1 β1 x 1 3 4 5 7 β2 β3 β4 β5 (b) x2 16 y2 4 1 19. (a) (b) x 42 4 21. (a) y 12 1 β1 y 4 3 2 1 β2 β1 β1 β2 β3 β4 1 2 3 4 5 6 x (b) y 1 x3 (b) y ln x 333200_10a_AN.qxd 12/9/05 2:42 PM Page A194 A194 Answers to Odd-Numbered Exercises and Tests 23. Each curve represents a portion of the line y 2x 1. (d) 200 Domain, 1, 1 0, 0, (a) (b) (c) (d) Orientation Left to right Depends on Right to left Left to right 27. 25. 29. 33. mx x1 y y1 x 6t y 3t x 4 cos y 7 sin 31. y k2 b 2 x h2 a2 x 3 4 cos y 2 4 sin x 4 sec y 3 tan 35. x t, y 3t 2 x t, y t 2 (b) x t, y |
t 2 1 x t, y 1 t (b) 37. (a) 39. (a) 41. (a) 43. (a) 45. 34 x t 2, y 3t 4 (b) x t 2, y t 2 4t 4 (b) x t 2, y 1 x t 2, y t 2 4t 5 t 2 47. 6 0 β6 51. 51 6 β6 55. d 18 6 4 β4, 49. 0 0 β6 53. b 4 β4 2, 2 Domain: Range: 1, 1 Domain: Range:, 57. (a) 100 Maximum height: 90.7 feet Range: 209.6 feet Maximum height: 136.1 feet Range: 544.5 feet 1 0 0 600 59. (a) x 146.67 cos t y 3 146.67 sin t 16t 2 (b) 50 No 0 0 60 (c) 450 Yes 0 0 500 (d) 19.3 61. Answers will vary. x a b sin 63. y a b cos 65. True x t y t 2 1 β y x2 1 x 3t y 9t 2 1 β y x2 1 67. Parametric equations are useful when graphing two functions simultaneously on the same coordinate system. For example, they are useful when tracking the path of an object so that the position and the time associated with that position can be determined. 5, 2 1, 2, 1 71. 69. 73. 75 y 75. 3 y Maximum height: 204.2 feet Range: 471.6 feet 105Β° ΞΈβ² x x ΞΈβ² β 2Ο 3 Maximum height: 60.5 feet Range: 242.0 feet 0 0 (b) 220 0 0 (c) 100 250 500 0 0 300 333200_10b_AN.qxd 12/9/05 2:43 PM Page A195 Section 10.7 (page 783) Vocabulary Check (page 783) 1. pole 3. polar 2. directed distance; directed angle x r cos tan y x y r sin r2 x2 y2 4. 1. Ο 4, 5. Ο 3. 1 2 3 4 0 Ο Ο 2 3Ο 2 Ο 2 3Ο, 4 3 5 6, 0, 13 6 0, 7. Ο 2 3Ο 2 1 2 3 4 0 Ο Ο 2 3Ο 2 1 2 3 4 |
0 2, 8.64, 2, 0.78 2 2 2 0, 3 11. 2, 9. 15. 1.1340, 2.2280 22, 10.99, 22, 7.85 2, 2 6, 19. 17. 13. 2, 5 25. 4 7, 0.8571 4 6, 29. r 4 csc 33. r 3 39. r 35. 2 3 cos sin 313, 0.9828 23. 21. 27. 31. 37. 5, 2.2143 13, 5.6952 17 6, 0.4900 r 10 sec 41. r2 16 sec csc 32 csc 2 4 1 cos or r 4 1 cos r a 47. 3x y 0 y 4 57. x2 y 22 6x2y 2y 3 4x2 5y 2 36y 36 0 r 2a cos 53. x2 y2 x23 0 49. x2 y2 16 43. 45. 51. 55. 59. 63. x2 y 2 4y 0 61. x2 4y 4 0 Answers to Odd-Numbered Exercises and Tests A195 65. The graph of the polar equation consists of all points that are six units from the pole. x2 y2 36 y 8 4 2 β8 β4 β2 2 4 8 67. The graph of the polar equation consists of all points on the line that make 6 an angle of positive polar axis. 3 x 3y 0 with the β4 β3 β2 69. The graph of the polar equation is not evident by simple inspection, so convert to rectangular form. x 3 0 β4 β8 y 4 3 2 1 β1 β2 β3 β4 β3 β2 β1 1 2 4 β2 β3 β4 x x x r 73. 71. True. Because is a directed distance, the point r, Β± 2n. be represented as x h2 y k2 h2 k2 Radius: Center: h2 k2 h, k 75. (a) Answers will vary. r, can b) (c) 2 1, r2, r1, and the pole are collinear. r1 d r1 2 r2 2 2r1r2 This represents the distance between two points on the line d r1 This is the result of the Pythagorean Theorem. 2. 2 1 r2 2 r2 (d) Answers will vary |
. For example: 3, 6, 4, 3 Points: Distance: 2.053 Points: Distance: 2.053 3, 76, 4, 43 77. 79. 2 log6 x log6 z log6 3 log6 y x ln x 2 lnx 4 3y 8 2, 3, 3 7, 88 2, 3 log7 81. 83. 35, 8 89. 93. Collinear 5 87. 85. 91. Not collinear ln xx 2 333200_10b_AN.qxd 12/9/05 2:43 PM Page A196 A196 Answers to Odd-Numbered Exercises and Tests Section 10.8 (page 791) Vocabulary Check (page 791) 2. polar axis 3. convex limaΓ§on 5. lemniscate 6. cardioid 1. 2 4. circle 1. Rose curve with 4 petals 5. Rose curve with 4 petals 3. LimaΓ§on with inner loop 7. Polar axis 9. 2 11., 2 13. Maximum: r 20 when polar axis, pole 3 2 0 1 2 3 31. Ο 35. Ο 2 Ο 2 3Ο 2 0 2 4 29. Ο 33. 37. Ο Ο Ο 2 3Ο 2 Ο 2 3Ο 2 Ο 2 3Ο 2 0 4 Ο Ο 39. 1 2 3 0 1 3 0 3Ο 2 Ο 2 0 4 10 5 3Ο 2 4 β10 5 β5 41. 6 0 2 45. β4 β4 β6 3 43. β11 14 47. 5 β10 3Ο 2 Ο 2 3Ο 2 Ο 2 β3 2 49. 0 β€ < 2 51. 3 β3 3 β4 5 2 4 6 8 0 3Ο 2 β2 0 β€ < 4 β3 0 β€ < 0,,, 6 2 3 5 6 19., 2 3 Ο 2 0 1 2 Zero: r 0 when 2 15. Maximum: r 4 when Zero: r 0 when 17. 21. 25. Ο Ο Ο Ο 2 3Ο 2 Ο 2 3Ο 2 Ο 2 3Ο 23. 27. Ο Ο Ο 333200_10b_AN.qxd 12/9/05 2:43 PM Page A197 53. β6 4 β4 55. 6 β3 4 β2 5 57. True. |
For a graph to have polar axis symmetry, replace r,. (b) r, 59. (a) r, or by 3Ο 2 3Ο 2 Upper half of circle Lower half of circle (c) Ο 2 (d 3Ο 2 3Ο 2 Full circle Left half of circle 61. Answers will vary. 63. (a) r 2 2 2 sin cos (b) r 2 cos (c) 65. (a) r 2 sin (d) r 2 cos Ο 2 (b 3Ο 2 k 0, k 1, k 2, k 3, circle convex limaΓ§on cardioid limaΓ§on with inner loop 3Ο 2 7 677 8 k = 0 β3 71. 13 5 69. Β±3 Answers to Odd-Numbered Exercises and Tests A197 73. x 12 9 y 22 4 1 y 5 3 2 1 β1 β1 β2 β3 β5 β4 β3 β2 x 1 2 3 Section 10.9 (page 797) Vocabulary Check (page 797) 1. conic 4. (a) iii 2. eccentricity; (b) i (c) ii e 3. vertical; right 1. e 1: r parabola 4 1 cos, 2 1 0.5 cos, 6 1 1.5 cos, e 0.5: r e 1.5: r e = 1 7 e = 0.5 β6 e = 1.5 15 ellipse hyperbola 7 3. e 1: r e 0.5: r e 1.5: r parabola 4 1 sin, 2 1 0.5 sin, 6 1 1.5 sin, ellipse hyperbola β16 e = 1 6 e = 0.5 17 e = 1.5 β16 5. f 6. c 7. d 8. e 9. a 10. b 333200_10b_AN.qxd 12/9/05 2:44 PM Page A198 A198 Answers to Odd-Numbered Exercises and Tests 11. Parabola Ο 2 13. Parabola Ο 2 31. β 15. Ellipse 3Ο 2 Ο 2 3Ο 2 17. Ellipse 3Ο 2 3Ο 2 19. Hyperbola 21. Hyperbola Ο 2 3Ο 2 Ο 0 1 25. β3 1 2 3 5 0 2 β2 2 Ο 2 3 |
Ο 2 Ο 23. Ellipse Ο β4 27. Ellipse 6 3 β7 33. r 37. r 41. r 45. r 1 1 cos 2 1 2 cos 10 1 cos 20 3 2 cos 35. r 1 2 sin 39. r 43. r 47. r 2 1 sin 10 3 2 cos 9 4 5 sin 49. Answers will vary. r 9.5929 107 51. 1 0.0167 cos 9.4354 107 9.7558 107 Perihelion: Aphelion: r 1.0820 108 1 0.0068 cos miles miles 1.0747 108 1.0894 108 Perihelion: Aphelion: r 1.4039 108 1 0.0934 cos kilometers kilometers 53. 55. Perihelion: Aphelion: 1.2840 108 1.5486 108 0.624 1 0.847 sin 2 ; 57. r miles miles r 0.338 astronomical unit 59. True. The graphs represent the same hyperbola. 61. True. The conic is an ellipse because the eccentricity is less 3 than 1. 63. Answers will vary. 65. r2 24,336 169 25 cos2 67. r2 144 25 cos2 9 69. r2 144 25 sin2 16 0 1 Ο 2 3Ο 2 1 β3 Parabola 71. (a) Ellipse (b) The given polar equation, r, the left of the pole. The equation, directrix to the right of the pole, and the equation, has a horizontal directrix below the pole. has a vertical directrix to has a vertical r2, r1, 29. 9 β3 β3 15 (c) r 2 = 4 1 β 0.4 sin ΞΈ 10 β12 12 β6 r 1 = 4 1 + 0.4 cos ΞΈ r = 4 1 β 0.4 cos ΞΈ 333200_10b_AN.qxd 12/12/05 11:35 AM Page A199 Answers to Odd-Numbered Exercises and Tests A199 n 75. n, 3 2 10 2 3 n 81. 72 10 79. n 2 sin 2u 24 25 cos 2u 7 25 6 73. 77. 83. 85. tan 2u 24 7 4 n 1 1 an 4 87. an 9n 89. 220 91. 720 Review Exercises (page 801) 1 |
. 4 radian, 45 3. 1.1071 radians, 63.43 5. 0.4424 radian, 25.35 7. 0.6588 radian, 37.75 9. 22 11. Hyperbola 13. y 2 16x 15. y 22 12 43 5 x β4 β3 β2 β1 β2 β 3 17. 21. y 2x 2; 1, 0 x 22 y2 21 25 y 1 19. 86 meters x 22 4 y 23. y 12 1 35. Center: 3, 5 7, 5, 1, 5 3 Β± 25, 5 Vertices: Foci: Asymptotes: y 5 Β± 1 2 x 3 37. Center: 1, 1 5, 1, 3, 1 6, 1, 4, 1 Vertices: Foci: Asymptotes: y 1 Β± 3 4 y 2 β2 2 4 6 8 β8 β10 y 6 4 2 x 1 β6 β4 4 6 8 β4 β6 β8 39. 72 miles x2 8 45. y2 8 yβ² y 4 3 2 41. Hyperbola 1 47. xβ² yβ² 43. Ellipse x2 3 y2 2 y 2 1 1 xβ² β4 β3 β2 x 2 3 4 β2 β1 1 2 x β2 β3 49. (a) Parabola β1 β2 y 24x 40 Β± 24x 402 3616x2 30x 18 (b) (c 10 10 x β 6 β 8 β 10 4 3 2 1 β2 β1 β1 β2 β3 1 2 3 4 5 x β3 7 β1 9 25. The foci occur 3 feet from the center of the arch on a line connecting the tops of the pillars. 27. Center: 2, 1 29. Center: 1, 4 1, 0, 1, 8 1, 4 Β± 7 7 4 Vertices: Foci: Eccentricity: Vertices: 2, 11, 2, 9 Foci: 2, 1 Β± 19 Eccentricity: 19 10 31. y2 x2 8 1 33. 5x 42 16 5y2 64 1 51. (a) Parabola (b) y (c) 2x 22 Β± 2x 222 4x2 22x 2 2 7 1 β1 β11 333200_10b_AN.qxd_pg A200 1/9/06 9:01 |
AM Page A200 A200 Answers to Odd-Numbered Exercises and Tests 53. t x y 3 11 2 8 1 5 0 2 19 15 11 69. 75. 81. 85. 3 2 1, 2 213, 0.9828 r2 10 csc 2 x2 y2 3x 71., 32 2 32 2 r 7 79. 77. x2 y 2 25 83. x2 y2 y23 87. 73. 2, 2 r 6 sin y 20 16 12 4 x 8 12 β 12 β 8 β 4 β 4 β 8 55. (a 57. (a 2x (b) 59. (ab) (b) y 4x x2 y 2 36 61. 65 cos y 4 6 sin 63. x 3 tan y 4 sec Ο 67 3Ο 2 2,, 2, 9 4 5 4 7, 1.05, 7, 10.47 89. Symmetry:, 2 Maximum value of r No zeros of Ο 2 polar axis, pole r: r 4 for all values of Ο 0 2 3Ο 2 91. Symmetry:, 2 polar axis, pole Maximum value of r: r 4 when, Zeros of r: r 0 when 0,, 2, 3 2 Ο Ο 2 3Ο 2 0 4 93. Symmetry: polar axis r: Maximum value of r: when Zeros of r 4 r 0 Ο 2 when 0 2 4 6 8 0 Ο 0 2 3Ο 2 333200_10b_AN.qxd 12/9/05 2:44 PM Page A201 Answers to Odd-Numbered Exercises and Tests A201 95. Symmetry: 2 Maximum value of r: Zeros of r: r 0 Ο 2 when when r 8 2 3.4814, 5.9433 Ο 2 4 6 0 3Ο 2 97. Symmetry:, 2 polar axis, pole Maximum value of r: r 3, 4 2,, when 0, 3, 4 5, 4 7 4 when Zeros of r: r 0 Ο Ο 2 3Ο 2 0 4 99. LimaΓ§on 101. Rose curve β16 8 β8 8 β6 4 β4 103. Hyperbola 105. Ellipse Ο 2 1 3Ο 2 Ο Ο 0 Ο 2 3Ο 2 1 3 4 0 107. r 4 1 cos 109. r 5 3 2 cos |
111. r 7978.81 1 0.937 cos ; 11,011.87 miles 113. False. When classifying an equation of the form Ax2 Bxy Cy2 Dx Ey F 0, its graph can be determined by its discriminant. For a graph to be a parabola, its discriminant, must equal zero. So, if C 115. False. The following are two sets of parametric equations B2 4AC, A then or equals 0. B 0, for the line. x t, y 3 2t x 3t, y 3 6t 117. 5. The ellipse becomes more circular and approaches a circle of radius 5. 119. (a) The speed would double. (b) The elliptical orbit would be flatter; the length of the major axis would be greater. 121. (a) The graphs are the same. (b) The graphs are the same. 3 2 Chapter Test 1. 0.2783 radian, (page 805) 15.9 2. 0.8330 radian, 47.7 3. 72 2 4. Parabola: Vertex: Focus: y 1, 0 2, 0 y2 4x 2 β1 2 3 4 5 6 x β2 β3 β4 6 5. Hyperbola: x 22 4 y2 1 2, 0 0, 0, 4, 0 2 Β± 5, 0 Center: Vertices: Foci: Asymptotes2, 0) β4 2 6 8 β4 β6 x 1 16. Ο 2 17. Ο 2 333200_10b_AN.qxd 12/9/05 2:44 PM Page A202 A202 Answers to Odd-Numbered Exercises and Tests 6. Ellipse: y 12 9 x 32 16 3, 1 1, 1, 7, 1 3 Β± 7, 1 Center: Vertices: Foci: y 6 4 2 β8 β4 β2 2 x β2 β4 7. Circle: Center: y x 22 y 12 1 2 2. 5 y 22 4 5x2 16 1 8. x 32 3 2 45 10. (a) (b) yβ² y 6 4 x 12. x 6 4t y 4 7t 11 3Ο 2 Parabola 18. Ο Ο 2 3Ο 2 3Ο 2 Ο 2 Ellipse 19. Ο 0 3 4 0 2 4 3Ο |
is failure to meet the βbudget variance test.β $37,335 $37,640 $305 < $500 0.05$37,640 $1882 Because the difference between the actual expenses and the budget is less than of the budgeted amount, there is compliance with the βbudget variance test.β and less than $500 5% 65. (a) (e) 2 1. (a) 5, 1, 2 2, 2 (b) 0, 5, 1, 2 9, 5, 0, 1, 4, 2, 11 7 3, 9, 5, 0, 1, 4, 2, 11 13, 1, 6 (b) 1 (c) 2.01, 13, 1, 6, 0.666... 0.010110111... 6 (b) 3, 8 1 6 3, 8 3, 7.5, 1, 8, 22 11. 0.123 3, 6 (c) 9. 6 (c) (d) 3. (a) 1 (d) (e) 5. (a) (d) 7. 0.625 13. β8 β 7 β 6 β5 β4 15. 3 2 3, 1, 8, 22 (e) 1 < 2.b 17. 19. (a) (b) 21. (a) (b) 23. (a) (b) 25. (a) (b) 27. (a) (b) 1 0 x β€ 5 equal to 5. denotes the set of all real numbers less than or 0 1 2 3 4 5 6 x (c) Unbounded x < 0 denotes the set of all real numbers less than 0. x (c) Unbounded 1 2 denotes the set of all real numbers greater than 0 β 1 β 2 4, or equal to 4c) Unbounded 2 < x < 2 2 than and less than 2. denotes the set of all real numbers greater x 2 (c) Bounded 1 0 β 1 β2 1 β€ x < 0 than or equal to denotes the set of all real numbers greater 1 and less than 0. β 1 0 x (c) Bounded Year 1960 1970 1980 1990 2000 240 192 144 96 48 ( Expenditures (in billions) Surplus or deficit (in billions) $0.3 (s) $2.8 (d) $73.8 (d) $221. |
2 (d) $236.4 (s92.2 $195.6 $590.9 $1253.2 $1788. Year and 4 are the terms; 7 is the coefficient. 3 are the terms; and 8 are the 1 are the terms; 4 and are the coefficients. 2 6 0 (b) 0 (b) 14 2 y β₯ 6 67. 71. 73. 75. 69. x 5 β€ 3 326 351 25 miles 7x 3x2, 8x, and 11 coefficients. 4x3, x2, and 5 81. (a) is undefined. 77. 10 79. (a) (b) 83. (a) Division by 85. Commutative Property of Addition 87. Multiplicative Inverse Property 89. Distributive Property 91. Multiplicative Identity Property 333200_App_AN.qxd 12/12/05 11:41 AM Page A204 A204 Answers to Odd-Numbered Exercises and Tests 93. Associative Property of Addition 95. Distributive Property 97. 1 2 105. (a) 99. 3 8 n 5n 1 5 101. 48 103. 5x 12 0.5 0.01 0.0001 0.000001 10 500 50,000 5,000,000 (b) The value of 5n 1 b 109. (a) No. If one variable is negative and the other is approaches infinity as n approaches 0. 1 a 107. False. If a b 0. a < b, where then >, (b) positive, the expressions are unequal. u v β€ u v The expressions are equal when u and v have the same sign. If u and v differ in sign, is less than u v. u v 111. The only even prime number is 2, because its only factors are itself and 1. 113. (a) Negative 115. Yes. a a if a < 0. (b) Negative Appendix A.2 (page A20) Vocabulary Check (page A20) 1. exponent; base 3. square root 5. index; radicand 7. conjugates 9. power; index 2. scientific notation 4. principle nth root 6. simplest form 8. rationalizing 5x3 y 2 222 (b) 59. (a) 0.011 (b) 0.005 61. (a) 4 (b) 2 53 x 63. (a) 22 (b) 3 32 |
65. (a) 6x2x (b) 18z z2 67. (a) 2x 32x 2 (b) 71. (a) 2x (b) 4y 342 13x 1 (b) 69. (a) 73. (a) 75 11 81. 185x 77. 83. 5 > 32 22 2 2 87. 912 89. 532 79. 85. 91. 2 35 3 21613 93. 8134 99. (a) 3 (b) 3x 12 101. (a) 2 42 (b) 82x 103. 95. 2 x 97.57 seconds 105. (a.93 5.48 7.67 9.53 11.08 12.32 8 9 10 11 12 13.29 14.00 14.50 14.80 14.93 14.96 (b) t β 8.643 14.96 107. True. When dividing variables, you subtract exponents. 109. using the property am an amn: a0 1, a 0, am am amm a0 1. (b) 81 9. (a) 243 64 1600 54 27. 13. 21. 5x6 1 (b) 15. 2.125 23. 1 24y2 (b) 3x2 111. When any positive integer is squared, the units digit is 0, 1, 4, 5, 6, or 9. Therefore, 5233 is not an integer. Appendix A.3 (page A31) Vocabulary Check (page A31) 8 8 8 8 8 1. 4.96 3. 5. (a) 27 9 7. (a) 1 (b) 5 11. (a) (b) 4 6 24 25. (a) 19. 6 (b) 17. 125z3 7 x (b) 29. (a) 33. (a) 2x3 x y2 4 3 31. (a) 1 (b) 1 4x 4 (b) 10 x square miles 35. (a) 33n (b) b5 a 5 gram per cubic centimeter 5.73 107 8.99 105 37. 39. 41. 4,568,000,000 ounces 43. 0.00000000000000000016022 coulomb 45. (a) 50,000 47. (a) 954.448 49. (a) 67,082.039 3 51. (a) 3 2 55. (a |
) (b) 2 3.077 1010 (b) 39.791 53. (a) (b) 57. (a) 7.550 (b) 200,000 4 (b) (b) 27 8 1 8 n; an; a0 2. descending 1. 3. monomial; binomial; trinomial 4. like terms 5. First terms; Outer terms; Inner terms; Last terms 6. factoring 7. completely factored 2. e 3. b 4. a 1. d 7. 11. (a) 2x3 4x2 3x 20 2x5 14x 1 5. f 15x 4 1 6. c 9. (b) Degree: 5; Leading coefficient: (c) Binomial 7.225 (b) 1 2 333200_App_AN.qxd 12/9/05 2:45 PM Page A205 13. (a) 3x4 2x2 5 (b) Degree: 4; Leading coefficient: (c) Trinomial x5 1 15. (a) 3 (b) Degree: 5; Leading coefficient: 1 (c) Binomial 17. (a) 3 (b) Degree: 0; Leading coefficient: 3 (c) Monomial 19. (a) 4x5 6x4 1 (b) Degree: 5; Leading coefficient: (c) Trinomial 4x3y 21. (a) 4 (b) Degree: 3; Leading coefficient: 4 (c) Monomial 23. Polynomial: 25. Not a polynomial because it includes a term with a nega- 3x3 2x 8 29. 2x 10 8.3x3 29.7x2 11 39. 1 45. 15z 2 5z 2x2 12x 6x 2 7x 5 x2 100 55. x2 4y 2 4x 2 20xy 25y 2 63. 8x3 12x 2y 6xy 2 y 3 m 2 n 2 6m 9 71. 4r 4 25 2.25x2 16 x y 1 53. 43. 49. 59. 67. 75. 37. y 4 y 3 y 2 33. 3x3 6x 2 3x 7.5x3 9x tive exponent 27. Polynomial: 31. 35. 41. 47. 51. 57. 61. 65. 69. 73. 77. 81. 87. 91. 95 |
. 99. 103. 107. 111. 115. 121. 125. 127. 129. 133. 137. 141. 145. 149. 153. 157. 3x3 2x 2 12z 8 4x 4 4x x 2 7x 12 x 4 x 2 1 4x 2 12x 9 x3 3x 2 3x 1 16x6 24x3 9 x2 2xy y2 6x 6y 9 1 4x2 3x 9 9x2 4 1.44x2 7.2x 9 79. 2x2 2x u4 16 83. 85. 3x 2 x2 25 x 5 89. 2xx 2 3 x 1x 6 x 3x 1 x 8 97. 2xx2 4x 10 1 2 3 x 9x 9 4x 1 4x 1 3u 2v3u 2v 2t 12 117. x 2 2 123. y 4 y 2 4y 16 2t 14t 2 2t 1 u 3vu2 3uv 9v2 s 3s 2 x 20x 10 5x 1x 5 x 1x2 2 3 x2 x3 x 23x 4 3x 15x 2 139. 143. 147. 151. 155. 159. 1 2 101. 109. 135. 105. 93. 3 3 3 x 6x 3 24y 34y 3 x 1x 3 x 22 119. 113. 5y 12 x 2x2 2x 4 3u 4v2 x 2x 1 131. y 5 y 4 3x 2x 1 3z 23z 1 2x 1x2 3 3x2 12x 1 2x 13x 2 6x 3x 3 Answers to Odd-Numbered Exercises and Tests A205 x 2x 4x 2x 4 3 4x23 60x 1 2x2 9x 1x 1 3x 1x2 5 x2 3x 12 185. 165. 177. x 12 169. 173. 1 4 x2x 4 163. 2xx 1x 2 x 36x 18 1 81 xx 4x2 1 t 6t 8 181. 5x 2x2 2x 4 51 x23x 24x 3 x 22x 137x 5 3x6 143x 2233x6 20x5 3 14, 14, 2, 2 161. 167. 171. 175. 179. 183. 187. 189. 191. 193. 195. 197. Two possible answers: 2, 199. Two possible answers |
: P 22x 25,000 201. (a) 500r 2 1000r 500 203. (a) (b) r % 21 2 11, 11, 4, 4, 1, 1 12 2, 4 (b) $85,000 3% 4% 5001 r 2 $525.31 $530.45 $540.80 r 41 2 % 5% 5001 r 2 $546.01 $551.25 (c) The amount increases with increasing r. 205. (a) (b) V 4x3 88x2 468x x (cm) 1 2 3 cm3 V 384 616 720 207. 211. 44x 308 x 209. (a) 3x 2 8x (b) 30x 213 215. 4r 1 217. 46 x6 x 219. (a) hR rR r (b) V 2R r 2 R rh 221. False. 4x2 13x 1 12x3 4x2 3x 1 333200_App_AN.qxd 12/12/05 11:42 AM Page A206 A206 Answers to Odd-Numbered Exercises and Tests a2 b2 a ba b 229. 223. True. 227. 231. 233. Answers will vary. Sample answer: x 3 8x 2 2x 7 x3n y2n is completely factored. x2 3 225. m n xn ynxn yn Appendix A.4 (page A42) Vocabulary Check (page A42) 1. domain 4. smaller 2. rational expression 5. equivalent 6. difference quotient 3. complex 81. (a) 9.09% (b) 1 x 4x h 4 73. 1 x 2 x 71. 75. 77. x 0 x 22x 1, 1 16 79. (a) minute (b) x 16 minute(s) (c) 60 16 15 4 minutes 288MN P NMN 12P; 9.09% 83. (a 10 75 55.9 48.3 45 43.3 42.3 12 14 16 18 20 22 41.7 41.3 41.1 40.9 40.7 40.6 (b) The model is approaching a T-value of 40. 85. False. In order for the simplified expression to be equivalent to the original expression, the domain of the simplified expression needs to be restricted. If n is even, x 1, 1. x 1. |
87. Completely factor each polynomial in the numerator and in the denominator. Then conclude that there are no common factors. If n is odd, Appendix A.5 (page A56) Vocabulary Check (page A56) 2. solve ax b 0 1. equation 4. 6. quadratic equation 7. factoring; extracting square roots; completing the 3. identities; conditional 5. extraneous square; Quadratic Formula 3. Conditional equation 9. Conditional equation 1. Identity 7. Identity 9 13. 6 4 21. 5 27. No solution. The x-terms sum to zero. 31. 4 15. 5 23. 25. 9 17. 9 33. 3 35. 0 19. No solution 5. Identity 11. 4 29. 10 37. No solution. The variable is divided out. 39. No solution. The solution is extraneous. 41. 2 45. 0 49. 53. 59. 67. 43. No solution. The solution is extraneous. 47. All real numbers x x2 6x 6 0 51. 55. 57. 20 2, 6 Β± 11 2x2 8x 3 0 3x2 90x 10 0 5 61. a 69. 65. 73. Β±33 3, 1 2 Β±7 0, 1 2 71. 63. 4, 2 3, 4 1. All real numbers 5. All real numbers x such that 7. All real numbers x such that x 2 x β₯ 1 9. 3. All nonnegative real numbers 3x, x 0 4y, y 1 2 5 19. 13., x 0, x 5 3x 2 1 2 xx 3 x 2 x2 1 x 2, x 2, x 2 3y y 1, x 0 15. y 4, y 4, y 3 23. y 4 y 6 27 2x 3 x 3 x 1 1 2 3 Undef The expressions are equivalent except at x 3. 31. The expression cannot be simplified. 5x 2, x 1, r 0 35. 33. 1 4 r 1 r, r 1 39. x 6x 1 x2, x 6 t 3 t 3t 2, t 2 x 5 x 1 43. 45. 6x 13 x 3 2 x 2 49. x2 3 x 1x 2x 3 2 x x2 1, x 0 53. The error was incorrect subtraction in the numerator. 57. xx 1, x 1, 0, x 2 |
1 2 2x 1 2x, x > 0 61. 2x3 2x2 5 x 112 1 xx h, h 0 67. 63. 1 x2 15 x7 2 x2 3x 1 3, x 0 11. 17. 21. 25. 29. 37. 41. 47. 51. 55. 59. 65. 69. 333200_App_AN.qxd 12/9/05 2:45 PM Page A207 75. 8, 16 77. 2 Β± 14 79. 1 Β± 32 2 81. 2 87. 1 Β± 83. 6 3 4, 8 85. 11 6, 11 6 89. 2 Β± 23 91. 5 Β± 89 4 93. 1 2, 1 95. 1 4, 3 4 97. 1 Β± 3 99. 7 Β± 5 101. 4 Β± 25 103. Β± 2 3 7 3 105. 4 3 107. 1 2 113. 6 Β± 11 115. 119. 123. 1.355, 14.071 0.290, 2.200 Β± 2 109. 265 8 Β± 3 8 121. 125. 2 7 111. 2 Β± 6 2 117. 0.976, 0.643 1.687, 0.488 1 Β± 2 127. 6, 12 129. 1 2 Β± 3 131. 1 2 133. Β± 3 4 97 4 Answers to Odd-Numbered Exercises and Tests A207 195. 500 units 197. False. x3 x 10 3x x2 10 The equation cannot be written in the form ax b 0. 199. False. See Example 14 on page A55. 201. Equivalent equations have the same solution set, and one is derived from the other by steps for generating equivalent equations. 2x 5, 2x 3 8 203. Yes. The student should have subtracted 15x from both sides to make the right side of the equation equal to zero. Factoring out an x shows that there are two solutions, x 0 x 6. and x2 3x 18 0 x2 2x 1 0 x 0, b a x2 22x 112 0 a 9, b 9 207. 211. x 0, 1 205. 209. 213. (a) (b) 137. Β±3 139. 6 141. 3, 0 Appendix A.6 (page A66) 135. 143. 149. 157. 165. 173. 0, Β± 32 2 3, 1, 1 Β± 1 2, Β±4 16 159. 3 Β± 162 3 |
Β± 21 6 145. 151. Β±1 1, 2 147. Β± 3, Β±1 153. 50 155. 26 2, 5 167. 161. 0 163. 9 Β± 14 169. 1 171. 2, 3 2 175. 4, 5 177. 1 Β± 31 3 1 17 2 179. 3, 2 181. 3, 3 183. 3, 185. (a) 61.2 inches (b) Yes. The estimated height of a male with a 19-inch femur is 69.4 inches. (c) Height, x Female femur length Male femur length 60 70 80 90 100 110 15.48 19.80 24.12 28.44 32.76 37.08 14.79 19.28 23.77 28.26 32.75 37.24 (d) 100 inches x 100.59; There would not be a problem because it is not likely for either a male or a female to be 100 inches tall (which is 8 feet 4 inches tall). 187. 189. 191. after about 28 hours y 0.25t 8; 6 inches 6 inches 2 inches 203 3 11.55 inches 193. (a) 1998 (b) During 2007 A P P E N D I X A Vocabulary Check (page A66) 1. solution set 4. solution set 2. graph 5. double 3. negative 6. union Bounded 3. x > 11. Unbounded Unbounded 1 β€ x β€ 5. x < 2. 8. f 1. 5. 7. b 13. (a) Yes 15. (a) Yes 17. (a) Yes 19. x < 3 9. d (b) No (b) No (b) Yes 1 2 3 4 5 12. a 10. c (c) Yes (c) No (c) Yes 21. x 11. e (d) No (d) Yes (d) No x < 3 2 3 2 β2 β1 0 1 2 3 x 23. x β₯ 12 25. x > 2 10 11 12 13 14 27 31. x β₯ 4 2 3 4 5 6 35. x β₯ 4 β 6 39 < 15 9 2 β 9 2 β 2 15 29. x < 5 3 4 5 6 7 33. x β₯ 2 0 1 2 3 4 37. 1 < x < 3 β1 416 β4 β 333200_App_AN.qxd 12/9/05 2:45 PM Page A208 |
A208 Answers to Odd-Numbered Exercises and Tests 45. 6 < x < 6 x β6 β4 β2 0 2 4 6 x 49. No solution 53. x β€ 3 2, x β₯ 3 β 3 2 β2 β1 1 0 4 < x < 5 57 43. 10.5 β€ x β€ 13.5 10.5 13.5 10 11 12 13 14 47. x < 2, x > 2 2 3 β3 51. 0 β1 β2 1 14 β€ x β€ 26 26 14 10 15 20 25 30 55. x β€ 5, x β₯ 11 11 5 10 15 β15 59. 0 β10 β5 x β€ 29 2, x β₯ 11 2 β 29 2 β 11 2 β16 β12 β8 β4 x x x x 61. 10 63. β10 10 β10 β10 x > 2 65. 10 x β€ 2 67. β10 24 β15 10 β 10 β10 6 β€ x β€ 22 x β€ 27 2, x β₯ 1 2 βa) (b) (a) (b) β5 x β₯ 2 x β€ 3 2 8 73. β5 10 β2 1 β€ x β€ 5 x β€ 1, x β₯ 7 (a) (b) 5, 77. 3,, 7 75. 2 81. All real numbers within eight units of 10 83. 85. x β€ 3 x 7 β₯ 3 x 12 < 10 79. 87. x 3 > 4 r > 3.125% 134 β€ x β€ 234 89. 93. 97. 99. (a) 91. x > 6 x β₯ 36 95. 5 75 0 150 x β₯ 129 1 β€ t β€ 10 t > 16 106.864 square inches β€ area β€ 109.464 square inches (b) 101. (a) 103. 105. Might be undercharged or overcharged by $0.19. 107. 13.7 < t < 17.5 (b) 13.7 17.5 t 12 13 14 15 16 17 18 19 20 β€ h β€ 80 109. 111. False. c has to be greater than zero. 113. b 10 Appendix A.7 (page A75) 10 1 β10 Vocabulary Check (page A75) 1. numerator 2. reciprocal 1. Change all signs when distributing the minus sign. 2x 3y 4 2x 3y 4 3. Change all signs when distributing the minus sign. 4 14x 1 4 16x 2x 1 numer |
ator and denominator separately. ax y x 9 ax y cannot be simplified. 9. 11. Divide out common factors, not common terms. 2x2 1 5x cannot be simplified. 13. To get rid of negative exponents: 1 a1 b1 1 a1 b1 ab ab ab b a. 15. Factor within grouping symbols before applying exponent to each factor. x2 5x12 xx 512 x12x 512 69. 3 71. 6 β4 8 5. z occurs twice as a factor. 5z6z 30z2 7. The fraction as a whole is multiplied by a, not the 333200_App_AN.qxd 12/9/05 2:45 PM Page A209 17. To add fractions, first find a common denominator. Answers to Odd-Numbered Exercises and Tests A209, 29. x1 4x4 7x2x13 2x2 x 15 49 16 3x 1 3y 4x xy 21. 25 9 37. 4 3 y x 3x 2 1 2x 2 1 7x 4 3 4x83 7x53 1 x13 7x2 4x 9 x2 33x 14 1 x 323x 274 3x 21215x2 4x 45 2x2 512 55. 47. 51. 19. 27. 35. 41. 45. 49. 53. 59. 23. 1 3 25. 2 33. 1 5x 31. 1, 2 39. 3x22x 13 43. 16 x 5 x 3 x12 5x32 x72 27x2 24x 2 6x 14 4x 3 3x 143 61. (a) (b) (c) 63. 65. x t x 0.5 1.0 1.5 2.0 1.70 1.72 1.78 1.89 2.5 3.0 3.5 4.0 t 2.18 2.57 2.36 2.02 x 0.5 mile 3xx2 8x 20 x 4x2 4 6x2 4x2 8x 20 y2 x 1 xy2 y2 True. x1 y2 1 x True. x 4 x 16 57. x x2 67. Add exponents when multiplying powers with like bases. xn x3n x4n 69. When a binomial is squared, there is also a middle term. xn yn2 x2n 2xnyn y2n x |
2n y2n 71. The two answers are equivalent and can be obtained by factoring. 1 10 2x 132 2x 13262x 1 10 2x 13212x 4 2x 1323x 1 2x 1323x 1 2x 152 1 6 1 60 1 60 4 60 1 15 2x 332x 1 2 5 (a) (b) 8 15 4 x32x 1 333200_Index_SE.qxd 12/8/05 11:32 AM Page A211 Index A Absolute value of a complex number, 470 inequality, solution of, A63 properties of, A4 of a real number, A4 Acute angle, 283 Addition of a complex number, 163 of fractions with like denominators, A7 with unlike denominators, A7 of matrices, 588 vector, 449 properties of, 451 resultant of, 449 Additive identity for a complex number, 163 for a matrix, 591 for a real number, A6 Additive inverse, A5 for a complex number, 163 for a real number, A6 Adjacent side of a right triangle, 301 Adjoining matrices, 604 Algebraic expression, A5 domain of, A36 equivalent, A36 evaluate, A5 term of, A5 Algebraic function, 218 Algebraic tests for symmetry, 19 Alternative definition of conic, 793 Alternative form of Law of Cosines, 439, 490 Amplitude of sine and cosine curves, 323 Angle(s), 282 acute, 283 between two lines, 729 between two vectors, 461, 492 central, 283 complementary, 285 conversions between radians and degrees, 286 coterminal, 282 degree, 285 of depression, 306 of elevation, 306 initial side, 282 measure of, 283 negative, 282 obtuse, 283 positive, 282 radian, 283 reference, 314 of repose, 351 standard position, 282 supplementary, 285 terminal side, 282 vertex, 282 Angular speed, 287 Aphelion distance, 798 Arc length, 287 Arccosine function, 345 Arcsine function, 343, 345 Arctangent function, 345 Area common formulas for, 7 of an oblique triangle, 434 of a sector of a circle, 289 of a triangle, 622 Heronβs Area Formula, 442, 491 Argument of a complex number, 471 Arithmetic combination, 84 Arithmetic sequence, 653 common difference of, 653 |
nth partial sum, 657 nth term of, 654 recursion form, 654 sum of a finite, 656, 723 Associative Property of Addition for complex numbers, 164 for matrices, 590 for real numbers, A6 Associative Property of Multiplication for complex numbers, 164 for matrices, 590, 594 for real numbers, A6 Associative Property of scalar multiplication for matrices, 594 Astronomical unit, 796 Asymptote(s) horizontal, 185 of a hyperbola, 755 oblique, 190 of a rational function, 186 slant, 190 vertical, 185 Augmented matrix, 573 Average rate of change, 59 Average value of a population, 261 Index A211 Axis (axes) imaginary, 470 of a parabola, 129, 736 polar, 779 real, 470 rotation of, 763 of symmetry, 129 B Back-substitution, 497 Base, A11 natural, 222 Basic equation, 534 guidelines for solving, 538 Basic Rules of Algebra, A6 Bearings, 355 Bell-shaped curve, 261 Binomial, 683, A23 coefficient, 683 cube of, A25 expanding, 686 square of, A25 Binomial Theorem, 683, 724 Book value, 32 Bounded, A60 Bounded intervals, A2 Branches of a hyperbola, 753 Break-even point, 501 Butterfly curve, 810 C Cardioid, 789 Cartesian plane, 2 Center of a circle, 20 of an ellipse, 744 of a hyperbola, 753 Central angle of a circle, 283 Change-of-base formula, 239 Characteristics of a function from set A to set B, 40 Circle, 20, 789 arc length of, 287 center of, 20 central angle, 283 classifying by discriminant, 767 by general equation, 759 radius of, 20 sector of, 289 area of, 289 333200_Index_SE.qxd 12/8/05 11:32 AM Page A212 A212 Index standard form of the equation of, 20 unit, 294 Circumference, common formulas for, 7 Classification of conics by the discriminant, 767 by general equation, 759 Coded row matrices, 625 Coefficient binomial, 683 correlation, 104 equating, 536 leading, A23 of a polynomial, A23 of a |
variable term, A5 Coefficient matrix, 573 Cofactor(s) expanding by, 614 of a matrix, 613 Cofunction identities, 374 Collinear points, 13, 623 test for, 623 Column matrix, 572 Combination of n elements taken r at a time, 696 Combined variation, 107 Common difference, 653 Common formulas area, 7 circumference, 7 perimeter, 7 volume, 7 Common logarithmic function, 230 Common ratio, 663 Commutative Property of Addition for complex numbers, 164 for matrices, 590 for real numbers, A6 Commutative Property of Multiplication for complex numbers, 164 for real numbers, A6 Complement of an event, 708 probability of, 708 Complementary angles, 285 Completely factored, A26 Completing the square, A49 Complex conjugates, 165 Complex fraction, A40 Complex number(s), 162 absolute value of, 470 addition of, 163 additive identity, 163 additive inverse, 163 argument of, 471 Associative Property of Addition, 164 Associative Property of Multiplication, 164 Commutative Property of Addition, 164 Commutative Property of Multiplication, 164 Distributive Property, 164 equality of, 162 imaginary part of, 162 modulus of, 471 nth root of, 475, 476 nth roots of unity, 477 polar form, 471 product of two, 472 quotient of two, 472 real part of, 162 standard form of, 162 subtraction of, 163 trigonometric form of, 471 Complex plane, 470 imaginary axis, 470 real axis, 470 Complex zeros occur in conjugate pairs, 173 Component form of a vector v, 448 Components, vector, 463, 464 Composite number, A7 Composition, 86 Compound interest continuous compounding, 223 formulas for, 224 Conditional equation, A46 Conic(s) or conic section(s), 735 alternative definition, 793 classifying by the discriminant, 767 by general equation, 759 degenerate, 735 eccentricity of, 793 locus of, 735 polar equations of, 793, 808 rotation of axes, 763 Conjugate, 173, A17 of a complex number, 165 Conjugate axis of a hyperbola, 755 Consistent system of linear equations, 510 Constant, A5 function, 57, 67 of proportionality, 105 |
term, A5, A23 of variation, 105 Constraints, 552 Consumer surplus, 546 Continuous compounding, 223 Continuous function, 139, 771 Conversions between degrees and radians, 286 Convex limaΓ§on, 789 Coordinate(s), 2 polar, 779 Coordinate axes, reflection in, 76 Coordinate conversion, 780 Coordinate system, polar, 779 Correlation coefficient, 104 Correspondence, one-to-one, A1 Cosecant function, 295, 301 of any angle, 312 graph of, 335, 338 Cosine curve, amplitude of, 323 Cosine function, 295, 301 of any angle, 312 common angles, 315 domain of, 297 graph of, 325, 338 inverse, 345 period of, 324 range of, 297 special angles, 303 Cotangent function, 295, 301 of any angle, 312 graph of, 334, 338 Coterminal angles, 282 Cramerβs Rule, 619, 620 Critical numbers, 197, 201 Cross multiplying, A48 Cryptogram, 625 Cube of a binomial, A25 Cube root, A14 Cubic function, 68 Curtate cycloid, 778 Curve butterfly, 810 plane, 771 rose, 788, 789 sine, 321 Cycloid, 775 curate, 778 D Damping factor, 337 Decreasing function, 57 Defined, 47 Definitions of trigonometric functions of any angle, 312 Degenerate conic, 735 Degree, 285 conversion to radians, 286 of a polynomial, A23 DeMoivreβs Theorem, 474 333200_Index_SE.qxd 12/8/05 11:32 AM Page A213 Denominator, A5 Domain rationalizing, 384, A16, A17 Dependent system of linear equations, 510 Dependent variable, 42, 47 Depreciated costs, 32 Descartesβs Rule of Signs, 176 Determinant of a matrix, 606, 611, 614 of a 2 2 matrix, 611 Diagonal matrix, 601, 618 Diagonal of a polygon, 700 Difference common, 653 of functions, 84 quotient, 46, A42 of two squares, A27 of vectors, 449 Differences first, 680 second, 680 Dimpled limaΓ§on, 789 Direct variation, 105 as an nth power, 106 Directed line segment, 447 initial point |
, 447 length of, 447 magnitude, 447 terminal point, 447 Direction angle of a vector, 453 Directly proportional, 105 to the nth power, 106 Directrix of a parabola, 736 Discrete mathematics, 41 Discriminant, 767 classification of conics by, 767 Distance between a point and a line, 730, 806 between two points in the plane, 4 on the real number line, A4 Distance Formula, 4 Distinguishable permutations, 695 Distributive Property for complex numbers, 164 for matrices, 590, 594 for real numbers, A6 Division of fractions, A7 long, 153 of real numbers, A5 synthetic, 156 Division Algorithm, 154 Divisors, A7 of an algebraic expression, A36 of cosine function, 297 of a function, 40, 47 implied, 44, 47 of a rational function, 184 of sine function, 297 Dot product, 460 properties of, 460, 492 Double-angle formulas, 407, 425 Double inequality, A63 Doyle Log Rule, 505 E Eccentricity of a conic, 793 of an ellipse, 748, 793 of a hyperbola, 793 of a parabola, 793 Effective yield, 251 Elementary row operations, 574 Eliminating the parameter, 773 Elimination Gaussian, 520 with back-substitution, 578 Gauss-Jordan, 579 method of, 507, 508 Ellipse, 744, 793 center of, 744 classifying by discriminant, 767 by general equation, 759 eccentricity of, 748, 793 foci of, 744 latus rectum of, 752 major axis of, 744 minor axis of, 744 standard form of the equation of, 745 vertices of, 744 Endpoints of an interval, A2 Entry of a matrix, 572 main diagonal, 572 Epicycloid, 778 Equal matrices, 587 Equality of complex numbers, 162 properties of, A6 of vectors, 448 Equating the coefficients, 536 Equation(s), 14, A46 basic, 534 conditional, A46 equivalent, A47 generating, A47 Index A213 graph of, 14 identity, A46 of a line, 25 general form, 33 intercept form, 36 point-slope form, 29, 33 slope-intercept form, 25, 33 summary |
of, 33 two-point form, 29, 33, 624 linear, 16 in one variable, A46 in two variables, 25 parametric, 771 position, 525 quadratic, 16, A49 second-degree polynomial, A49 solution of, 14, A46 solution point, 14 system of, 496 in two variables, 14 Equilibrium point, 514, 546 Equivalent equations, A47 generating, A47 expressions, A36 fractions, A7 generate, A7 inequalities, A61 systems, 509 operations that produce, 520 Evaluate an algebraic expression, A5 Evaluating trigonometric functions of any angle, 315 Even function, 60 trigonometric functions, 298 Even/odd identities, 374 Event(s), 701 complement of, 708 probability of, 708 independent, 707 probability of, 707 mutually exclusive, 705 probability of, 702 the union of two, 705 Existence theorems, 169 Expanding a binomial, 686 by cofactors, 614 Expected value, 726 Experiment, 701 outcome of, 701 sample space of, 701 Exponent(s), A11 properties of, A11 rational, A18 333200_Index_SE.qxd 12/8/05 11:32 AM Page A214 A214 Index Exponential decay model, 257 Exponential equation, solving, 246 Exponential form, A11 Exponential function, 218 f with base a, 218 natural, 222 one-to-one property, 220 Exponential growth model, 257 Exponential notation, A11 Exponentiating, 249 Expression algebraic, A5 fractional, A36 rational, A36 Extended principle of mathematical induction, 675 Extracting square roots, A49 Extraneous solution, A48, A54 F Factor Theorem, 157, 213 Factorial, 644 Factoring, A26 completely, A26 by grouping, A30 polynomials, guidelines for, A30 solving a quadratic equation by, A49 special polynomial forms, A27 Factors of an integer, A7 of a polynomial, 173, 214 Family of functions, 75 Far point, 216 Feasible solutions, 552 Finding a formula for the nth term of a sequence, 678 Finding intercepts of a graph, 17 Finding an inverse function, 97 Finding an inverse matrix, 604 Finding test intervals for a polynomial, 197 Finite sequence, 642 Fin |
ite series, 647 First differences, 680 Fixed cost, 31 Fixed point, 397 Focal chord latus rectum, 738 of a parabola, 738 Focus (foci) of an ellipse, 744 of a hyperbola, 753 of a parabola, 736 FOIL Method, A24 Formula(s) change-of-base, 239 for compound interest, 224 double-angle, 407, 425 half-angle, 410 Heronβs Area, 442, 491 for the nth term of a sequence, 678 power-reducing, 409, 425 product-to-sum, 411 Quadratic, A49 reduction, 402 sum and difference, 400, 424 sum-to-product, 412, 426 Four ways to represent a function, 41 Fractal, 726 Fraction(s) addition of with like denominators, A7 with unlike denominators, A7 complex, A40 division of, A7 equivalent, A7 generate, A7 multiplication of, A7 operations of, A7 partial, 533 decomposition, 533 properties of, A7 rules of signs for, A7 subtraction of with like denominators, A7 with unlike denominators, A7 Fractional expression, A36 Frequency, 356 Function(s), 40, 47 algebraic, 218 arithmetic combination of, 84 characteristics of, 40 common logarithmic, 230 composition, 86 constant, 57, 67 continuous, 139, 771 cosecant, 295, 301 cosine, 295, 301 cotangent, 295, 301 cubic, 68 decreasing, 57 defined, 47 difference of, 84 domain of, 40, 47 even, 60 exponential, 218 family of, 75 four ways to represent, 41 graph of, 54 greatest integer, 69 of half-angles, 407 Heaviside, 126 identity, 67 implied domain of, 44, 47 increasing, 57 inverse, 93, 94 cosine, 345 sine, 343, 345 tangent, 345 trigonometric, 345 linear, 66 logarithmic, 229 of multiple angles, 407 name of, 42, 47 natural exponential, 222 natural logarithmic, 233 notation, 42, 47 objective, 552 odd, 60 one-to-one, 96 period of, 297 periodic, 297 piecewise-defined, 43 polynomial, 128 power, 140 product of, 84 quadratic, 128 quotient of, 84 range |
of, 40, 47 rational, 184 reciprocal, 68 secant, 295, 301 sine, 295, 301 square root, 68 squaring, 67 step, 69 sum of, 84 summary of terminology, 47 tangent, 295, 301 transcendental, 218 trigonometric, 295, 301, 312 undefined, 47 value of, 42, 47 Vertical Line Test, 55 zero of, 56 Fundamental Counting Principle, 692 Fundamental Theorem of Algebra, 169 of Arithmetic, A7 Fundamental trigonometric identities, 304, 374 G Gaussian elimination, 520 with back-substitution, 578 333200_Index_SE.qxd 12/8/05 11:32 AM Page A215 Gaussian model, 257 Gauss-Jordan elimination, 579 General form of the equation of a line, 33 Generalizations about nth roots of real numbers, A15 Generate equivalent fractions, A7 Generating equivalent equations, A47 Geometric sequence, 663 common ratio of, 663 nth term of, 664 sum of a finite, 666, 723 Geometric series, 667 sum of an infinite, 667 Graph, 14 of cosecant function, 335, 338 of cosine function, 325, 338 of cotangent function, 334, 338 of an equation, 14 of a function, 54 of an inequality, 541, A60 in two variables, 541 intercepts of, 17 of inverse cosine function, 345 of an inverse function, 95 of inverse sine function, 345 of inverse tangent function, 345 of a line, 25 point-plotting method, 15 of a rational function, guidelines for analyzing, 187 of secant function, 335, 338 of sine function, 325, 338 special polar, 789 symmetry, 18 of tangent function, 332, 338 Graphical interpretations of solutions, 510 Graphical method, 500 Graphical tests for symmetry, 18 Greatest integer function, 69 Guidelines for analyzing graphs of rational functions, 187 for factoring polynomials, A30 for solving the basic equation, 538 for verifying trigonometric identities, 382 H Half-angle formulas, 410 Half-life, 225 Harmonic motion, simple, 356, 357 Heaviside function, 126 Heronβs Area Formula, 442, 491 Horizontal asymptote, 185 Horizontal components of v, 452 Horizontal line, 33 Horizontal Line Test, 96 Horizontal shift, 74 Hor |
izontal shrink, 78 of a trigonometric function, 324 Horizontal stretch, 78 of a trigonometric function, 324 Horizontal translation of a trigonometric function, 325 Human memory model, 235 Hyperbola, 185, 753, 793 asymptotes of, 755 branches of, 753 center of, 753 classifying by discriminant, 767 by general equation, 759 conjugate axis of, 755 eccentricity of, 793 foci of, 753 standard form of the equation of, 753 transverse axis of, 753 vertices of, 753 Hypocycloid, 810 Hypotenuse of a right triangle, 301 I Idempotent square matrix, 639 Identity, A46 of the complex plane, 470 function, 67 matrix of order n, 594 Imaginary axis of the complex plane, 470 Imaginary number, 162 pure, 162 Imaginary part of a complex number, 162 Imaginary unit i, 162 Implied domain, 44, 47 Improper rational expression, 154 Inclination, 728 and slope, 728, 806 Inclusive or, A7 Inconsistent system of linear equations, 510 Increasing annuity, 668 Increasing function, 57 Independent events, 707 probability of, 707 Independent system of linear equations, 510 Independent variable, 42, 47 Index A215 Index of a radical, A14 of summation, 646 Indirect proof, 568 Inductive, 614 Inequality (inequalities), A2 absolute value, solution of, A63 double, A63 equivalent, A61 graph of, 541, A60 linear, 542, A62 properties of, A61 satisfy, A60 solution of, 541, A60 solution set of, A60 symbol, A2 Infinite geometric series, 667 sum of, 667 Infinite sequence, 642 Infinite series, 647 Infinite wedge, 545 Infinity negative, A3 positive, A3 Initial point, 447 Initial side of an angle, 282 Integer(s) divisors of, A7 factors of, A7 irreducible over, A26 Intercept form of the equation of a line, 36 Intercepts, 17 finding, 17 Intermediate Value Theorem, 146 Interval bounded, A2 on the real number line, A2 unbounded, A3 Invariant under rotation, 767 Inverse additive, A5 multiplicative, A5 Inverse function, |
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