Answers to All Exercises - CengageAnswers to All Exercises Ans3 AEE Appendix C.2 16.(page C16) 1. solution point 2. graph 3. Algebraic, graphical, numerical 4. 1.If possible, rewrite

Answers to All Exercises Ans1

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Appendix CSection C.1 (page C7) 1. Cartesian 2. Distance Formula 3. Midpoint Formula 4. (x − h)2 + (y − k)2 = r2, center, radius 5. c 6. f 7. a 8. d 9. e 10. b11. A: (2, 6); B: (−6, −2); C: (4, −4); D: (−3, 2)12. A: (1.5, −4); B: (0, −2); C: (−3, 2.5); D: (−6, 0)13.

−2−4 2 4 6

−2

−4

−6

2

4

6

x

y

(−4, 2)

(−3, −6)

(1, −4)

(0, 5)

14.

−1−2−3−4−5 1 2 3 4 5

−2

−3

−4

−5

1

2

3

4

5

x

y

(−5, −5)

(4, −2)

(−4, 0) (0, 0)

15.

x

y

(−2, −2.5) (0.5, −1)

(5, −6)

(3, 8)

−2−4−6−8 2 4 6 8

−4

−6

−8

2

4

6

8

16.

x

y

−1−2−3 2 3−1

−2

−3

1

2

352( ), 2−

32( ), 1

12( )1, −

(3, −3)

17. (−5, 4) 18. (2, −3) 19. (0, −6) 20. (−11, 0)21. Quadrant IV 22. Quadrant III23. Quadrant II 24. Quadrant I25. Quadrant III or IV 26. Quadrant I or IV27. Quadrant III 28. Quadrant III29. Quadrant I or III 30. Quadrant II or IV31. 8 32. 7 33. 5 34. 10 35. 13 36. 17

37. √277

6 38.

√45712

39. √71.78 40. √296.2

41. (a) 3, 4, 5 (b) 32 + 42 = 52

42. (a) 5, 12, 13 (b) 52 + 122 = 132

43. (a) 10, 3, √109 (b) 102 + 32 = (√109)244. (a) 4, 7, √65 (b) 42 + 72 = (√65)245. (√5)2 + (√45)2 = (√50)246. (√20)2 + (√20)2 = (√40)247. Two equal sides of length √2948. Two equal sides of length 2√1049. Opposite sides have equal lengths of 2√5 and √85.50. Opposite sides have equal lengths of 3√5 and √10.

51. The diagonals are of equal length (√58). The slope of the

line between (−5, 6) and (0, 8) is 25. The slope of the line

between (−5, 6) and (−3, 1) is −52. The slopes are negative

reciprocals, indicating perpendicular lines, which form a right angle.

52. The diagonals are of equal length (√10). The slope of the line between (2, 4) and (4, 3) is −1

2. The slope of the line between (2, 4) and (1, 2) is 2. The slopes are negative reciprocals, indicating perpendicular lines, which form a right angle.

53. (a)

x

y

−2 2 4 6 8

−2

2

4

6

8

(0, 0)

(8, 6)

(b) 10 (c) (4, 3)

54. (a)

(9, 0)

(1, 12)12

10

8

6

4

2

2−2 64 8 10x

y (b) 13 (c) (7

2, 6)

55. (a)

2

4

6

8

10

2 4 6 8 10

(2, 10)

(10, 2)

y

x

(b) 17 (c) (0, 52)

56. (a)

(−7, −4)

(2, 8)8

6

2

−10 −8 −6−2

−4

2−2x

y (b) 15 (c) (−5

2, 2)

57. (a)

(5, 4)

(−1, 2)

−1−1

2 3 4 51

3

4

5

x

y (b) 2√10 (c) (2, 3)

Answers to All Exercises

Ans2 Answers to All Exercises

58. (a)

2

4

6

8

10

2 4 6 8 10

(2, 10)

(10, 2)

y

x

(b) 8√2 (c) (6, 6)

59. (a)

( )( ), 1

−1 −−− −2

252 1

2

343

2

12

52

12

12

32

52

,−

x

y (b) √82

3

(c) (−1, 76)

60. (a)

1

1

3

236

6

6

66−

−

26

−

−

−−x

y

( )13

13

, −−

( )16

12

, −−

(b) √26

(c) (−14

, −512)

61. (a)

2

4

6

8

2 4−2

−2

−4 6

(6.2, 5.4)

(−3.7, 1.8)

y

x

(b) √110.97 (c) (1.25, 3.6)

62. (a)

5

−5

10

15

20

−5−10−15−20 5

(−16.8, 12.3)

(5.6, 4.9)

y

x

(b) √556.52 (c) (−5.6, 8.6)

63. $1182.5 million 64. $1272.5 million65. x2 + y2 = 25 66. x2 + y2 = 3667. (x − 2)2 + (y + 1)2 = 16 68. (x + 5)2 + ( y − 3)2 = 469. (x + 1)2 + (y − 2)2 = 5 70. (x − 3)2 + ( y + 2)2 = 2571. (x − 3)2 + ( y − 4)2 = 25 72. x2 + y2 = 1773. (x + 2)2 + ( y − 1)2 = 1 74. (x − 3)2 + (y + 2)2 = 975. (x − 3)2 + ( y + 6)2 = 16 76. (x + 2)2 + y2 = 10077. (x − 2)2 + (y + 1)2 = 1678. (x + 3)2 + (y − 1)2 = 25

79. Center: (0, 0) 80. Center: (0, 0) Radius: 3 Radius: 4

−1 1−2−4 42−1

−2

−4

1

2

4

x

y

−1 1−2−3 32−1

−2

−3

1

2

3

x

y

81. Center: (1, −3) 82. Center: (0, 1) Radius: 2 Radius: 7

−2 −1 1 2 3 4

−5

−3

−1

1

x

y

−2−4−8 42 8 10

−4

−8−10

2

4

6

10

x

y

83. Center: (12, 12) 84. Center: (2

3, −14)

Radius: 32 Radius: 56

−1 1 2 3

1

3

y

x

x

y

−1 1 2

−1

−2

1

85. (0, 1), (4, 2), (1, 4) 86. (3, 3), (1, 0), (3, −3), (5, 0)87. (−1, 5), (−4, 8), (−6, 5), (−3, 2)88. (−2, −3), (0, 0), (−2, −4) 89. 5√74 ≈ 43 yd90. 301.0 ft; 307.0 ft 91. 192.1 km92. True. The lengths of the sides from (−8, 4) to (2, 11) and from

(2, 11) to (−5, 1) are both √149.93. False. It could be a rhombus.94. False. You would have to use the Midpoint Formula 15 times.95. 0; 096. No. The scales depend on the magnitudes of the quantities

measured.97. (2xm − x1, 2ym − y1); (a) (7, 0) (b) (9, −3)

98. (3x1 + x2

4,

3y1 + y2

4 ), (x1 + x2

2,

y1 + y2

2 ), (x1 + 3x2

4,

y1 + 3y2

4 ) (a) (74, −

74), (

52

, −32), (

134

, −54)

(b) (−32

, −94), (−1, −

32), (−

12

, −34)

99. Proof 100. (a) ii (b) iii (c) iv (d) i


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Appendix C.2 (page C16) 1. solution point 2. graph 3. Algebraic, graphical, numerical 4. 1. If possible, rewrite the equation so that one of the variables

is isolated on one side of the equation. 2. Make a table of values showing several solution points. 3. Plot these points on a rectangular coordinate system. 4. Connect the points with a smooth curve or line. 5. (a) Yes (b) Yes 6. (a) Yes (b) Yes 7. (a) No (b) Yes 8. (a) No (b) Yes 9. (a) No (b) Yes 10. (a) No (b) Yes11. (a) Yes (b) No 12. (a) Yes (b) No13.

x −2 0 23

1 2

y −4 −1 0 12

2

Solution point (−2, −4) (0, −1) (2

3, 0) (1, 12) (2, 2)

−1−2−3−4−5 1 2 3 4 5

−4

−5

1

2

3

4

5

x

y

14. x −3 −5

20 1 2

y −1 0 5 7 9

Solution point (−3, −1) (−5

2, 0) (0, 5) (1, 7) (2, 9)

x

y

−4−6 2 4 6−2

2

6

8

10

15. x −1 0 1 2 3

y 3 0 −1 0 3

Solution point (−1, 3) (0, 0) (1, −1) (2, 0) (3, 3)

−1−2−3 2 3 4 5 6 7

−2

−3

4

5

6

7

x

y

16. x −4 −3 −2 0 1

y 4 0 −2 0 4

Solution point (−4, 4) (−3, 0) (−2, −2) (0, 0) (1, 4)

x

y

−1−2−4−5 1 2 3

−3

1

2

3

4

5

17. b 18. d 19. c 20. a 21. e 22. f23.

x

y

−1−2−3 1 2 3−1

−2

1

3

4

24.

x

y

−1−2−3−4−5 2 3 4 5

−2

−6−7

1

32

25.

1 2 3 4 5 6−1

1

2

3

4

5

x

y 26.

−4 −3 −2 −1 1 2−1

2

3

4

5

x

y

27.

−1 1 2 3 4 5−1

1

2

3

4

5

x

y 28.

x

y

−1−2−4 −3 1 42 3

−3

−2

−1

1

2

3

4

5

29.

−2 1 2 3 4

−3

−2

2

3

x

y 30.

−1 1 2 3 5 6 7x

−4

−3

−2

−1

1

2

3

4

y


31.

−10

−10 10

10 32.

−10

−10 10

10

Intercepts: (103 , 0), (0, 5) Intercepts: (3

2, 0), (0, −1)33.

−10

−10 10

10

Intercepts: (−5, 0), (1, 0), (0, −1)34.

−10

−10 10

10

Intercepts: (2, 0), (4, 0), (0, −2)35.

−10

−10 10

10 36.

−10

−10 10

10

Intercept: (0, 0) Intercept: (0, 5)37.

−10

−10 10

10 38.

−10

−10 10

10

Intercepts: (−3, 0), (0, 0) Intercepts: (6, 0), (0, 0)39.

−10

−10 10

10 40.

−10

−10 10

10

Intercepts: (0, −2), (8, 0) Intercepts: (−1, 0), (0, 1)41.

−10

−10 10

10 42.

−10

−10 10

10

Intercepts: (0, 3), (1, 0), Intercepts: (−4, 0), (0, −4), (3, 0) (2, 0)

43.

−10

−10 10

10 44.

−10

−10 10

10

Intercepts: (0, 0), (2, 0) Intercepts: (0, 1), (1, 0)

45. Xmin = -10Xmax = 10Xscl = 2Ymin = -50Ymax = 100Yscl = 25

46. Xmin = -5Xmax = 1Xscl = 1Ymin = -3Ymax = 1Yscl = 1

47. The graphs are identical. Distributive Property48. The graphs are identical. Associative Property of Addition49. The graphs are identical. Associative Property of Multiplication50. The graphs are identical. Multiplicative Inverse Property51.

−2

−3 6

4 (a) (3, 1.41) (b) (−4, 3)

52.

−4

−6 6

4 (a) (−1, −4) (b) (3.49, 6)

53.

−6

−9 9

6 (a) (−0.5, 2.47) (b) (−1.58, −2), (0.40, −2),

(1.37, −2)

54.

−30

9

8 (a) (2, 3) (b) (0.65, 1.5), (1.42, 1.5),

(4.58, 1.5), (5.35, 1.5)

55. y1 = √16 − x2 56. y1 = √36 − x2

y2 = −√16 − x2 y2 = −√36 − x2

−6

−9 9

6

−8

−12 12

8

57. y1 = 2 + √49 − (x − 1)2 58. y1 = 1 + √25 − (x − 3)2 y2 = 2 − √49 − (x − 1)2 y2 = 1 − √25 − (x − 3)2

−10

−15 15

10

−6

−5

12

7

59. b, c, d 60. c, d


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61. (a)

00 9

500,000 62. (a)

00 10

9000

(b) $227,400 (b) 3.93 yr (c) 7.3 yr (c) $4460.5063. (a)

Year 2006 2007 2008 2009

New houses (in thousands) 410.5 290.9 198.1 132.1

Year 2010 2011 2012 2013

New houses (in thousands) 93.0 80.7 95.3 136.7

The model fits the data well. (b)

06 13

420

The model fits the data well. (c) 2015: 300,000; 2017: 570,680; Yes. Answers will vary. (d) 2009, 201264. (a)

00 70

100

The model fits the data well. (b) 63.6; The life expectancy in 1940 (c) t = 24 or 1964 (d) About 72.8 yr65. False. y = x2 − 1 has two x-intercepts.66. False. y = 0 has an infinite number of x-intercepts.67. Use the equations y = 3400 + 0.05x and y = 3000 + 0.07x

to model the situation. When the sales level x equals $20,000, both equations yield $4400.

2000

4000

6000

8000

40,00020,000

y

x

y = 3400 + 0.05x

y = 3000 + 0.07x

68. (a) Xmin = -9Xmax = 9Xscl = 1Ymin = -6Ymax = 6Yscl = 1

(b) (−1, 0), (3, 0), (0, −3) (c) (−4, 1): No

(2, −3): Yes Answers will vary.

Appendix C.3 (page C31) 1. equation 2. solve 3. extraneous 4. point of intersection 5. (a) Yes (b) No (c) No (d) No 6. (a) No (b) Yes (c) No (d) No 7. (a) Yes (b) No (c) No (d) No 8. (a) No (b) No (c) Yes (d) No 9. (a) No (b) No (c) No (d) Yes10. (a) No (b) Yes (c) No (d) No11. Identity 12. Identity 13. Identity 14. Identity15. Conditional equation 16. Conditional equation17. x = 12 18. x = 3

7 19. y = 1 20. y = 23

21. y = −9 22. z = 53 23. x = −10 24. x = −4

25. z = −65 26. x = 6 27. z = 17

48 28. y = −16

29. u = 10 30. y = 12 31. x = 4 32. x = 0

33. x = 5 34. x = 74 35. x = 11

6 36. No solution37. x = 5

3 38. z = 0 39. No solution40. No solution 41. (5, 0), (0, −5)42. (−4, 0), (0, −3) 43. (−2, 0), (1, 0), (0, −2)44. (−2, 0), (2, 0), (0, 4) 45. (−2, 0), (0, 0)46. (−3, 0), (0, 0) 47. No intercepts 48. (1

3, 0)49. (−2, 0), (6, 0), (0, −2) 50. (5, 0), (−7, 0), (0, 2.5)51. (1, 0), (0, 12) 52. (0, 0)53.

8−4

−4

4 54.

4−8

−4

4

(3, 0) (−52, 0)

55.

16−2

−6

6 56.

4−8

−4

4

(10, 0) (−3, 0)57. 5(4 − 4) = 0 58. 3(2 − 5) + 9 = 0

−6

−6

12

6

−4

−4

8

4

59. (0)3 − 6(0)2 + 5(0) = 0 60. (0)3 − 9(0)2 + 18(0) = 0 (5)3 − 6(5)2 + 5(5) = 0 (3)3 − 9(3)2 + 18(3) = 0 (1)3 − 6(1)2 + 5(1) = 0 (6)3 − 9(6)2 + 18(6) = 0

−2

−15

6

2

−4

−12

10

12

61. 1 + 2

3−

1 − 15

− 1 = 0

−3

−2

3

2


62. (−2) − 3 −10−2

= 0

−6

−4

6

4

5 − 3 −105

= 0

63. 1223 64. 8

3 65. 8913 66. 950 67. 6

68. 517 69. −1.796 70. 4.672 71. 3, 12 72. 89

13

73. 1, −1.2 74. −103 , 3 75. − 3

10 76. 15

77. 0.5, −3, 3 78. ±√2, −3 79. ±1 80. −15.625, 1 81. −1.333 82. ±√10 83. −1, 7 84. −7, 5 85. 11 86. 68 87. (1, 1) 88. (−1, 3) 89. (−1, 3), (2, 6) 90. (1, −1), (−2, 8) 91. (−1, 3) 92. (−1, −3) 93. (4, 1) 94. (6, 4) 95. (1.449, 1.899), (−3.449, −7.899) 96. (1.670, 1.660) 97. (0, 0), (−2, 8), (2, 8) 98. (0, 0), (3, −3) 99. 0, −1

2 100. ±13 101. 4, −2 102. 9, 1

103. 3, −12 104. −3

2, 11 105. 2, −6 106. 2, 6107. −a ± b 108. −a 109. ±7 110. ±12111. 16, 8 112. 0, 10 113. No solution114. No solution 115. 1

2 ± √3; 2.23, −1.23

116. −7 ± 2√11

4; −3.41, −0.09 117. 2

118. −92; −4.50 119. −8, 4 120. −1, 3

121. −3 ± √7 122. −4 ±√2 123. 1 ±√63

124. −92

, 112

125. No solution 126. 12±√52

127. −5 ± √89

4 128.

2 ± 3√23

129. 1 ± √3

130. 5 ± √3 131. −4 ± 2√5 132. 12±√52

133. No solution 134. No solution 135. 27

136. −43

137. −83±√13

3 138.

13±√38

3139. 1 ± √2 140. −3, 0 141. 6, −12142. No solution 143. 7 144. No solution145. 1

2 ± √3 146. −32 ± √3 147. −1

2

148. ±√ba

149. ±2, 0 150. ±32

, 0 151. −3, 0

152. 0, 43

153. 0, ±3√2

2 154. 0, ±

52

155. 3, 1, −1 156. ±2, −1 ± √3i 157. ±1, ±√3

158. ±2 159. ±12

, ±4 160. ±√76

161. −15, −1

3 162. 16 163. 2, −3

5 164. −3, 13165. 1

4 166. 94 167. 26 168. No solution

169. 0 170. −11 + √57

8 171. −256.5

172. −54 173. 9 174. 36 175. −59, 69

176. −5, 6, 1 ± √57

2 177. 1 178. 0, 1,

35

179. −3 ± √21

6 180.

1 ± √313

181. 2, −32

182. 34, −1 183. 3, −2 184. 5

3, −3185. √3, −3 186. 10, −1

187. (a)

−9

−7

9

5 (b) and (c) x = 0, 3, −1

(d) They are the same.

188. (a)

−5

−20

5

20 (b) and (c) x = ±3, ±1


189. (a)

−1

−5

11

3 (b) and (c) x = 5, 6


190. (a)

−4

−5

8

3 (b) and (c) x = 3

2


191. (a)

−4

−24

4

24 (b) and (c) x = −1


192. (a)

−10

−18

10

8 (b) and (c) x = 2


193. (a)

−10

−4

8

8 (b) and (c) x = 1, −3


194. (a)

−7

−5

11

7 (b) and (c) x = −1, 5


195. (a)

50001 13

6000

(2.9, 5479.4); In 2002, both states had the same population.

(b) (2.9, 5479.4); In 2002, both states had the same population.

(c) Change in population per year; Maryland’s population is growing faster.

(d) Maryland: 6,049,400; Wisconsin: 5,852,400 Answers will vary.


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196. (a) 2006 (b) Answers will vary. (c)

00 15

60

(d) 2010 (e) Answers will vary.197. (a)

1050

25

300

(b) 16.8°C (c) 2.5198. (a)

5150

40

300

(b) 211.6°F (c) 24.725 lb�in.2

199. False. The lines could be identical.200. False. See Example 14. 201. c = 5

8 202. c = 52

203. (a) 0, −ba

(b) 0, 1

Appendix C.4 (page C44) 1. double 2. −a ≤ x ≤ a 3. x ≤ −a, x ≥ a 4. zeros, undefined values 5. No 6. Transitive Property 7. d 8. a 9. f 10. b 11. e 12. c 13. (a) Yes (b) No (c) Yes (d) No 14. (a) No (b) No (c) Yes (d) Yes 15. (a) No (b) Yes (c) Yes (d) No 16. (a) Yes (b) Yes (c) Yes (d) No 17. x > 7 18. x ≥ −4

0 1 2 3 4 5 6 7 8 9

x −4−5 −1−2−3 0 1

x

19. x < −2 20. x ≥ 12

−4 −1−2−3 0 1

x x

12

−1 0 1 2

21. x > −1 22. x > 19 x

−2 −1 0 1

18

x

19 20 21

23. x ≥ 4 24. x < 7

2 3 4 5

x 5 6 7 8

x

25. −1 ≤ x ≤ 3 26. −83 < x ≤ 13

3

−2 −1 0 1 32

x

−3 −2 −1 1 3 50 2 4

x

133

83

−

27. −2 < x ≤ 5 28. −7 < x ≤ −13

−2 −1 0 1 2 3 4 5

x

0

x

−1−2−3−4−5−6−7−8

13

−

29. −92 < x < 15

2 30. −3 ≤ x < 7

−2−4−6

−

0 2 4 6 8

92

152

x

76543210−1−2−3−4

x

31.

−9

−6

9

6 32.

−9

−6

9

6

x ≤ 2 x > 72

33.

−9

−6

9

6 34.

−6

−9 9

6

x < 2 x > 435.

−4

−5

8

3 36.

−5

−2

7

6

(a) x ≥ 2 (b) x ≤ 32 (a) x ≤ 6 (b) x ≥ −3

2

37.

−2

−2

7

4 38.

−6

−2

6

6

(a) 53 ≤ x ≤ 3 (a) −2 ≤ x ≤ 4

(b) x ≥ 83 (b) x ≤ 4

39. x < −2, x > 2 40. −2 ≤ x ≤ 2

−3 −2 −1 0 1 2 3

x

4

x

3210−1−2−3−4

41. 1 ≤ x ≤ 13 42. x < 16, x > 24

4 620 8 10 12 14

1 13x

242220181614 26

x

43. x ≤ −7, x ≥ 13 44. x ≤ −28, x ≥ 0

x

−9 −6 −3 15129630

−7 13 −35 −28 −21 −14 −7 70

x

45. 12 < x < 3

2 46. 15 < x < 7

5

−1 0 1 2

x

32

12

210

x

75

15

47.

−3

−1

9

7 48.

−14

−4

10

12

(a) 1 ≤ x ≤ 5 (a) −10 ≤ x ≤ 6 (b) x ≤ −1, x ≥ 7 (b) x ≤ −4, x ≥ 049. ∣x∣ ≤ 3 50. ∣x∣ > 1 51. ∣x + 1∣ ≥ 452. ∣x − 7∣ < 3 53. ∣x − 6∣ < 10 54. ∣x + 5∣ ≤ 855. ∣x + 1∣ > 3 56. ∣x − 3∣ ≥ 557. Positive on: (−∞, −1) ∪ (5, ∞) Negative on: (−1, 5)


58. Positive on: (−∞, −1) ∪ (4, ∞) Negative on: (−1, 4)

59. Positive on: (−∞, 2 − √10

2 ) ∪ (2 + √102

, ∞) Negative on: (2 − √10

2,

2 + √102 )

60. Positive on: (1 − √414

, 1 + √41

4 ) Negative on: (−∞,

1 − √414 ) ∪ (1 + √41

4, ∞)

61. Negative on: (−∞, ∞) 62. Positive on: (−∞, ∞)63. (−∞, −5], [1, ∞) 64. (−1, 7) x

−6 −5 −4 −3 −2 −1 0 1 2

−2 20 4 6 8

x−1 7

65. (−7, 3) 66. (−∞, 2], [4, ∞)

0 2 4−2−4−6−8

x−7 3 x

5430 1 2

67. 0, [4, ∞) 68. (−∞, 3]

0 1 2 3 4 5 6 7

x

−1−2

2 3 4 51

x

69. (−3, −1), (32, ∞) 70. (−∞, −3), (−1

2, 2)

−1 0 1 2 3

x

4 5−2−3−4

32

−

−1 0 1 2 3

x

−2−3−4−5−6

12

71. (−1, 1), (3, ∞) 72. [−132 , −2], [2, ∞)

−1 0 1 3 42

x

0 2 4−2−4−6−8

x

132

−

73. No solution 74. −32

0

x

1

32

−1−2−3

−

75. (−∞, 12), (12, ∞) 76. All real numbers x

210−3 −2 −1 3

x

12

−3−4 −2 −1 0 1 2 3 4

x

77. (a) x = 1 (b) x ≥ 1 (c) x > 178. (a) x = −1, 3 (b) x ≤ −1, x ≥ 3 (c) x < −1, x > 379.

−5

−2

7

6 80.

−10

−24

10

48

(a) x ≤ −1, x ≥ 3 (a) −∞ < x ≤ −4, (b) 0 ≤ x ≤ 2 1 ≤ x ≤ 4 (b) x = −2, 5 ≤ x < ∞81. (−∞, −1), (0, 1) 82. (−∞, 0), (1

4, ∞)

−1−2 210

x

10−1

x

14

83. (−∞, −1), [4, ∞) 84. (−2, 3]

−2 −1 10 2 3 4 5

x

−1−2 0 1 2 3

x

85.

−6

−4

12

8 86.

−6

−4

6

4

(a) 0 ≤ x < 2 (a) 1 ≤ x ≤ 4 (b) 2 < x ≤ 4 (b) −∞ < x ≤ 0 87. [5, ∞) 88. [−5

2, ∞) 89. [−3, 4] 90. (−∞, −2], [2, ∞) 91. (−∞, −1

3], [7, ∞) 92. (−∞, ∞) 93. (a) 2005 (b) (2003, 2005); (2005, 2012) 94. (a) 2009 (b) (2009, 2012); (2003, 2009) 95. (a) 10 sec (b) (4, 6) 96. (a) 8 sec (b) (0, 1.17), (6.83, 8) 97. (a)

00 22

100 (b) (1996, 2001) (c) 6.605 < x < 11.646

98. (a) and (b)

1000

300

350

(c) 181.5 lb (d) Answers will vary. 99. t ≥ 6.45; Starting in 2007, there were at least 900 Bed Bath

& Beyond stores.100. t ≥ 10.97, t ≤ 6.83; From 2000 to 2006 and 2011 to 2013,

there were at most 600 Williams-Sonoma stores.101. t ≈ 1.04; In 2001, there were about the same number of Bed

Bath & Beyond stores as Williams-Sonoma stores.102. t ≥ 1.04; Starting in 2002, the number of Bed Bath &

Beyond stores exceeded the number of Williams-Sonoma stores.

103. 33313 vibrations�sec 104. 3.6 mm

105. 1.2 < t < 2.4 106. 0 < v < 500107. False. 10 ≥ −x108. True. 32x2 + 3x + 6 is always positive.109. False. Cube roots have no restrictions on the domain.110. iv, ii, iii, i 111. (a) iv (b) ii (c) iii (d) i112. (a) a, b (b) Positive on: (−∞, a) ∪ (b, ∞) Negative on: (a, b) When x < a or x > b, the factors have the same sign

so the product is positive. When a < x < b, the factors have opposite signs so the product is negative.

(c) x = a and x = b

Appendix D (page D5) 1. directly proportional 2. constant, variation 3. directly proportional 4. inverse 5. combined 6. jointly proportional 7. y = 12

5 x 8. y = 7x 9. y = 205x 10. y = 290

3 x11. Model: y = 33

13x; 25.38 cm, 50.77 cm12. Model: y = 53

14x; 18.93 L, 94.64 L


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13. y = 0.0368x; $7360 14. y = 0.07x; $37.8415. (a) 0.05 m (b) 1762

3 N 16. 29313 N

17. x 2 4 6 8 10

y = kx2 4 16 36 64 100

2 4 6 8 10

20

40

60

80

100

x

y

18. x 2 4 6 8 10

y = kx2 8 32 72 128 200

2 4 6 8 10

40

80

120

160

200

x

y

19. x 2 4 6 8 10

y = kx2 2 8 28 32 50

2 4 6 8 10

10

20

30

40

50

x

y

20. x 2 4 6 8 10

y = kx2 1 4 9 16 25

2 4 6 8 10

5

10

15

20

25

x

y

21. 0.61 mi/h 22. k(2v)2

kv2 = 4

23. x 2 4 6 8 10

y = k�x2 12

18

118

132

150

xx2 4 6 8 10

10

10

10

10

10

1

2

3

4

5

y

24. x 2 4 6 8 10

y = k�x2 54

516

536

564

120

xx2 4 6 8 10

1

2

3

5

4

4

4

4

1

y

25. x 2 4 6 8 10

y = k�x2 52

58

518

532

110

xx2 4 6 8 10

1

3

5

2

2

2

2

1

y

26. x 2 4 6 8 10

y = k�x2 5 54

59

516

15

2 4 6 8 10

1

2

3

4

5

x

y

27. y =5x 28. y =

25

x 29. y = −710

x 30. y =120

x

31. A = kr2 32. V = ke3 33. y =kx2 34. h =

k√ss

35. F =kgr2 36. z = kx2y3 37. P =

kV

38. R = kS(L − S) 39. R = k(Te − T) 40. F =km1m2

r2


41. The area of a triangle is jointly proportional to its base and height.

42. The surface area of a sphere varies directly as the square of its radius.

43. The volume of a sphere varies directly as the cube of its radius.

44. The volume of a right circular cylinder is jointly proportional to the product of its height and the square of its radius.

45. Average speed is directly proportional to the distance and inversely proportional to the time.

46. ω varies directly as the square root of g and inversely as the square root of W.

47. A = πr2 48. y =75x

49. y =28x

50. z = 2xy 51. F = 14rs3 52. P =18xy2

53. z =2x2

3y 54. v =

24pq287s2 55. 506.27 ft

56. 0.054 in. 57. 1470 J58. No. The 15-inch pizza is the best buy.59. The velocity is increased by 4.60. (a) The safe load is unchanged. (b) The safe load is eight times as great. (c) The safe load is four times as great. (d) The safe load is one-fourth as great.61. (a)

Tem

pera

ture

(in

°C)

d

C

2000 4000

1

2

3

4

5

Depth (in meters)

(b) Yes. k1 = 4200, k2 = 3800, k3 = 4200, k4 = 4800, k5 = 4500

(c) C =4300

d (d)

00

6000

6 (e) About 1433 m

62. (a)

F

1

2

3

4

5

7

6

Force (in pounds)

y

Len

gth

(in

cent

imet

ers)

2 4 6 8 10 12

(b) Yes. (c) 15.65 lb63. False. y will increase if k is positive and y will decrease if k is

negative.64. False. E is jointly proportional to the mass of an object and

the square of its velocity.

65. Inversely 66. Directly

Appendix E (page E3) 1. linear 2. equivalent inequalities 3. 4 4. 6 5. 7 6. 6 7. 4 8. 3 9. 20 10. 2011. 4 12. 4 13. 3 14. 8 15. 5 16. −517. −10 18. 3

4 19. No solution20. All real numbers 21. −6

5 22. 6 23. 924. 50 25. x < 2 26. x > −13 27. x < 928. x < 22 29. x ≤ −14 30. x ≥ 4 31. x > 1032. x < −6 33. x < 4 34. x > 12 35. x < 336. x > −4 37. x ≥ 2 38. x ≤ −3 39. x ≤ −540. x ≤ −1 41. x < 6 42. x ≥ 1

2 43. x ≥ 444. x < 7 45. x ≥ −4 46. x > 16

Appendix FAppendix F.1 (page F8) 1. solution 2. graph 3. linear 4. point, equilibrium 5. g 6. d 7. a 8. h 9. e 10. b11. f 12. c13.

−1−2−3−4 1 2 3 4−1

−2

−3

−4

−5

−6

1

x

y 14.

−2−4−6−8 2

2

8

10

12

14

4 6 8−2

x

y

15.

−1

2

1

3

4

2 3 4 5 6 7−1

−2

−3

−4

x

y 16.

-1 1 2 3 4 5

-3

-2

-1

1

2

3

x

y

17.

−2−4−6−8 2 6 8−2

−4

−6

−8

6

8

4

2

x

y 18.5

4

3

2

1

−3

−2

−4

−5

1−1−2−3−4−6−7−8x

y

19.

−3 −2 −1 1 2 3

−2

1

2

3

4

x

y 20.

−3 −2 −1 1 2 3

−2

−1

1

2

4

x

y


AP

PE

ND

ICE

S

21.

−4 −3 −2 −1 1

−2

−1

1

3

4

x

y 22.

−6 −4 2 4

−8

−6

2

x

y

23.

−1−2−3 1

1

3

4

2 3 5

−2

−3

−4

−1

x

y 24.

−1−2−3 1

1

3 4 5

−2

−3

−4

−5

−1

x

y

25.

−2−4−6 2

2

4 6 8 10

−4

−6

−8

−12

−14

−2

x

y 26.

−2 2

2

6

8

10

4 6 8 10 12 14

−4

−6

−2

x

y

27.

x

y

−1−2−3−4 1 2 3 4−1

1

3

4

5

6

7

28.

−6−9−12

3

6

9

6 9 12

−9

−12

−15

x

y

29.

−1 1

1

2

3

4

2 3 4 5 6 7

−2

−3

−4

−1

x

y 30.5

4

3

1

−3

−4

−5

54321−1−2−3−4−5x

y

31.

431−1−2−3−5−6

5

4

2

1

−2

−4

−5

x

y 32.

654321−1−2−3

9

8

7

6

5

4

3

1x

y

33.

−4

−6 6

4 34.

−2

−6 9

10

35.

−4

−6 6

4 36.

−25

−30 30

15

37.

−9

−9 9

3 38.

−4

−6 6

4

39.

−2

−3 3

2 40.

−4

−6 6

4

41.

−1

0 6

3 42.

−1

−6 6

7

43.

−1

−6 6

7 44.

−4

−6 6

4

45. x3+

y2> 1 46. y ≥ x2 − 4 47. x2 + y2 ≤ 9

48. x > 5 49. (a) Yes (b) No (c) No (d) No50. (a) No (b) No (c) No (d) Yes51.

−2 1 2

−1

2

3

x(−1, 0) (1, 0)

(0, 1)

y 51.

1 3

1

2

3

x(0, 0) (2, 0)

(0, 3)

y

53.

−3 −1 1 3 4

−3

−2

1

3

5

x(−2, 0)

109

79

,( )

y 54.

2 4 6

2

6

x

(1, 0)

(6, 6)

y


55.5

4

2

1

−4

−5

54321−1−4−5

(−3, 0)(0, −3)

x

y 56.5

4

2

3

1

−4

−5

98765432−1

(1, −1)

(4, 2)

x

y

57.

−1−2−3−4 1 2−1

−2

1

4

5

6

y

x

58.

x

y

−1−2−3 1 2 3−1

−2

−3

1

2

3

No solution No solution59.

x

y

−1−2−3−4 1 2−1

−2

−3

−4

4

225

45

,( )

60.

x

y

−1 1 2 3 4 5 6 7

−2

−3

1

2

3

4

5

(4, 4)

(0, 0)

61.

−4 −2 2 4

−4

−2

2

4

x

y 62. y

x−2−6−8 2 4 6 8

−4

−6

−8

2

4

6

8

(−3, −4)

(3, 4)

63.

x

y

−1−2−3−4 1 2 3 4−1

−2

2

3

4

5

6

(0, 1) ( 3 , ( 3) + 1)3 23

64. y

x−2−3 1 2 4 5

−1

−2

−3

4

(0, 3)

(3, 0)

65.

x

y

−1−2−3−4 3 42−1

−2

−3

−4

4

3

(1, −2)

(−1, 2)

(1, 0)

66. y

x2

−1

−2

2

(−1, 0)

(0, 1)

(1, 0)

67.

x

y

−1 1 2 3 5 6 7−1

1

2

3

5

6

7

(4, 4)

116

4,( )

12

, 4( )

68. y

x−1 1

−1

−2

2

(−2, 0) (2, 0)

(−2, e−2) (2, e−2)

69. {14x + 1

4yxy

<≥≥

100

70. {yyx

≤≥≥

6x1

−−

2x3 71.

{yyxy

≤≤≥≥

4 − x2 − 1

4x00

72.

{xxxy

++≥≥

3y < 6y < 400

73. x2 + (y − 2)2 ≤ 4

74. x2 + y2 > 4 75. {21 ≤≤

xy

≤≤

57 76. {4x

4x0

−−≤

yyy

≥≤≤

0164

77. {yyy

≤ 32x

≤ −x + 5≥ 0

78. {yyy

≤≤≥

x−x

0

++

11

79.

x

y

10 20 30 40 50 60 70 80 90100−10

10

20

30

40

50

60Consumersurplus

Producersurplus

(80, 10)

Consumer surplus: 1600 Producer surplus: 40080.

x

y

Consumersurplus

Producersurplus

(500, 75)

100 200 300 400 500 600−50

50

150

200

250

Consumer surplus: 6250 Producer surplus: 12,50081.

x

y

Consumersurplus

Producersurplus

50,000 100,000−50

50

100

150

200

250

350

( (750,0007

19507

,

Consumer surplus: ≈ 1,147,959.18 Producer surplus: ≈ 2,869,897.96


AP

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82.

x

y

Consumersurplus

Producersurplus

(2,000,000, 100)

1,000,000 2,000,000−50

50

100

200

250

Consumer surplus: 40,000,000 Producer surplus: 20,000,00083. (a)

{x + y

xyx

≤≥≥≥

30,00075007500

2y

(b)

x

y

20,000

10,000

20,000

30,000

84. (a)

{x

20x

+

+

yyx

35y

≥≥≥≥

20,0005000

10,000300,000

(b)

x

y

5,000−5,000

15,000

20,000

85. (a)

{20x15x10x

+++

10y10y20y

xy

≥≥≥≥≥

280160180

00

(b)

x

y

−5 5 10 20 30 35−5

5

15

25

30

86. (a)

{800x +x

1200yxy

≥≤≥≥

2y20,000

42

(b)

x

y

8 12 16 20 24−4

4

8

12

16

20

24

87. (a)

{2x +xyπy

xy

≥≥≥≥

500125

00

(b)

20

20

60

80

40 60 80x

y

88. (a) {πy2 − πx2

xy

≥ >>

100x

(b) y

x−1−2−3−4 1 2 3 4

−3

−4

1

3

4

(c) The line is an asymptote to the boundary. The larger the circles, the closer the radii can be and still satisfy the constraint.

89. True 90. False. 3x + y2 ≥ 2 is outside the parabola.91. Test a point on either side. 92. Answers will vary.

Appendix F.2 (page F17) 1. optimization 2. objective function 3. constraints, feasible solutions 4. The vertices 5. Minimum at (0, 0): 0 6. Minimum at (0, 0): 0 Maximum at (0, 6): 30 Maximum at (0, 4): 32 7. Minimum at (0, 0): 0 8. Minimum at (0, 0): 0 Maximum at (6, 0): 60 Maximum at (2, 0): 14 9. Minimum at (0, 0): 0 10. Minimum at (0, 2): 6 Maximum at (3, 4): 17 Maximum at (5, 3): 2911. Minimum at (0, 0): 0 12. Minimum at (3, 0): 3 Maximum at (4, 0): 20 Maximum at (0, 4): 2413. Minimum at (0, 0): 0 Maximum at (60, 20): 74014. Minimum at (0, 600): 21,000 Maximum at (900, 0): 45,00015. Minimum at (0, 0): 0 Maximum at any point on the line segment connecting

(60, 20) and (30, 45): 210016. Minimum at (675, 0): 10,125 Maximum at (0, 800): 16,000


17.

2 3 4 5

−1

1

3

4

x

(0, 2)

(5, 0)

(0, 0)

y 18.

x

y

2 4 6 8 10−2

2

4

6

8

(0, 0)

(0, 8)

(4, 0)

Minimum at (0, 0): 0 Minimum at (0, 0): 0 Maximum at (5, 0): 30 Maximum at (0, 8): 6419.

x

y

5 10 2015

5

15

20

25

30

(0, 10)

(0, 0)

(5, 8)

(7, 0)

20.

2 3

1

2

4

x

(0, 3)

(0, 0)

(4, 1)

(5, 0)

y

Minimum at (0, 0): 0 Minimum at (0, 0): 0 Maximum at (5, 8): 47 Maximum at (4, 1): 2121.

x

y

5 10 2015

5

15

20

25

30

(0, 10)

(0, 0)

(5, 8)

(7, 0)

Minimum at (0, 0): 0 Maximum at (5, 8): 2122.

2 3

1

2

4

x

(0, 3)

(0, 0)

(4, 1)

(5, 0)

y Minimum at (0, 0): 0 Maximum at any point on

the line segment connecting (0, 3) and (4, 1): 12

23.

x

y

5 10 2015

5

15

20

25

30

(0, 10)

(0, 0)

(5, 8)

(7, 0)

Minimum at any point on the line segment connecting (0, 0) and (0, 10): 0

Maximum at (7, 0): 14

24.

2 3

1

2

4

x

(0, 3)

(0, 0)

(4, 1)

(5, 0)

y Minimum at any point on the line segment connecting (0, 0) and (5, 0): 0


25.

x

y

5 10 15 20 25−5

5

10

15

30

35

40

45

(36, 0)

(24, 8)(40, 0)

Minimum at (24, 8): 104 Maximum at (40, 0): 160

26.

x

y

−5 5 20 25−10

10

40

50

(0, 0)

(12, 0)

(0, 20) (10, 8)

(6, 16)



27.

x

y

5 10 15 20 25−5

5

10

15

30

35

40

45

(36, 0)

(24, 8)(40, 0)


28.

x

y

−5 5 20 25−10

10

40

50

(0, 0)

(12, 0)

(0, 20) (10, 8)

(6, 16)



29.

x

y

5 10 15 20 25−5

5

10

15

30

35

40

45

(36, 0)

(24, 8)(40, 0)




AP

PE

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S

30.

x

y

−5 5 20 25−10

10

40

50

(0, 0)

(12, 0)

(0, 20) (10, 8)

(6, 16)


31. (a) and (b) (c) (3, 6)

x

y

−3 3 12 15−3

3

6

9

(3, 6)

(5, 0)

(0, 10)

3x + y = 15

4x + 3y = 30

12 = 2x + y

32. (a) and (b) (c) (5, 0)

x

y

−2 2 4 6−2

2

4

6

8

14

(0, 10)

(3, 6)

(5, 0)

25 = 5x + y

4x + 3y = 30

3x + y = 15

33. (a) and (b) (c) (0, 10)

x

y

−3 3 15−3

3

6

9

15

(3, 6)

(5, 0)

(0, 10)

10 = x + y

3x + y = 15

4x + 3y = 30

34. (a) and (b)

x

y

−2 2 4 6−2

2

4

6

8

14

(0, 10)

(3, 6)

(5, 0)

3x + y = 15

4x + 3y = 30

(c) Maximum at any point on the line segment connecting (3, 6) and (5, 0).

35.

1 2 3 4

3

4

x

(0, 1)

(2, 3)

(0, 0)

y

The constraints do not form a closed set of points. Therefore, z = x + y is unbounded.

36.

(0, 0)x

1 3

1

2

( )2019

4519

(2, 0)

(0, 3) ,

y

z is maximum at any point on the line segment connecting (2, 0) and (20

19, 4519).

37.

−3 −2 1 2

−2

−1

3

x

y

The feasible set is empty.38.

x

(0, 7)

(0, 0)

(7, 0)

2 4 6

2

4

6

10

y

The constraint x ≤ 10 is extraneous. Maximum at (0, 7): 1439. (a) Four audits, 32 tax returns (b) Maximum revenue: $17,60040. 1000 units of model A; 500 units of model B Maximum profit: $76,00041. (a) Three bags of brand X, six bags of brand Y (b) Minimum cost: $19542. Three bags of brand X; two bags of brand Y Minimum cost: $10543. True 44. True 45. z = x + 5y46. z = x + y 47. z = 4x + y 48. z = −10x + y49. (a) t > 9 (b) 3

4 < t < 950. (a) −3 < t < 6 (b) t > 6

Appendix G (page G7) 1. mathematical induction 2. first 3. arithmetic

4. second 5. 5

(k + 1)(k + 2) 6. 4

(k + 3)(k + 4)


7. 2k+1

(k + 2)! 8. 2k

(k + 1)! 9. 1 + 6 + 11 + . . . + (5k − 4) + (5k + 1)10. 7 + 13 + 19 + . . . + (6k + 1) + (6k + 7)11–24. Answers will vary. 25. 1,625,62526. 25,333 27. 572 28. 671,58029 – 46. Answers will vary.47. 0, 3, 6, 9, 12 First differences: 3, 3, 3, 3 Second differences: 0, 0, 0 Linear48. 2, 0, 3, 1, 4 First differences: −2, 3, −2, 3 Second differences: 5, −5, 5 Neither49. 3, 1, −2, −6, −11 First differences: −2, −3, −4, −5 Second differences: −1, −1, −1 Quadratic50. −3, 6, −12, 24, −48 First differences: 9, −18, 36, −72 Second differences: −27, 54, −108 Neither51. 0, 1, 3, 6, 10 First differences: 1, 2, 3, 4 Second differences: 1, 1, 1 Quadratic52. 2, 4, 16, 256, 65,536 First differences: 2, 12, 240, 65,280 Second differences: 10, 228, 65,040 Neither53. 2, 4, 6, 8, 10 First differences: 2, 2, 2, 2 Second differences: 0, 0, 0 Linear54. 0, 4, 10, 18, 28 First differences: 4, 6, 8, 10 Second differences: 2, 2, 2 Quadratic55. an = n2 − 3n + 5, n ≥ 156. an = n2 − 4n + 10, n ≥ 1 57. an =

12n2 + n − 3

58. an =74n2 − 5n + 3

59. (a) an = 3(4)n−1

(b) an =√34 [1 +

13∑

n

k=2(49)

k−2

] (c) 3(43)n−1

60. (a) 7 (b) 15 (c) an = 2n − 1 (d) Answers will vary.61. False. Not necessarily62. False. The first differences are all the same.63. False. It has n − 2 second differences.64. (a) Pn is true for all integers n ≥ 3. (b) Pn is true for all integers 1 ≤ n ≤ 50. (c) P1, P2, and P3 are true. (d) P2n is true for any positive integer n.

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Answers to All Exercises - CengageAnswers to All Exercises Ans3 AEE Appendix C.2 16.(page C16) 1. solution point 2. graph 3. Algebraic, graphical, numerical 4. 1.If possible, rewrite