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332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 1
Precalculus with Limits, Answers to Section 11.1 1
3 7 1
Chapter 11 32. 2, 2, 2 33. 0, 1, 7 34. 1, 6, 2
Section 11.1 (page 817) 9 9 13
35. 1, 0, 5.5 36. 2, 2, 2 37. 2.5, 2, 6
7 3 2 2 2
Vocabulary Check (page 817) 38. 9, 2, 2 39. x 3 y 2 z 4 16
2 2 2
1. three-dimensional 40. x 3 y 4 z 3 4
2. xy-plane, -plane, xz yz-plane 3. octants 2 2 2
41. x y 4 z 3 9
x x y y z z
4. Distance formula 5. 1 2, 1 2, 1 2 2 2 2
2 2 2 42. x 2 y 1 z 8 36
6. sphere 7. surface, space 8. trace 2 2 2
43. x 3 y 7 z 5 25
2 2 2
1. A: 1, 4, 4, B: 1, 3, 2, C: 3, 0, 2 44. x y 5 z 9 16
2. A: 6, 2, 3; B: 2, 1, 2; C: 2, 3, 0 3 2 2 2 45
z 45. x 2 y z 3 4
3. z 4.
1 2 2 2 61
5 3 4 46. x 2 y 1 z 4 4
4 (3, 2, 1) 47. Center: 5, 0, 0 ; radius: 5
3 (2, 1, 3) 2 2
2 (1, 2, 1) 5 4 3 2 1 2 3 y
21 48. Center: 0, 4, 0; radius: 4
1 y (3, 0, 0)
2 1 2 3 4 5 4 2 49. Center: 2, 1, 3; radius: 2
32 x 3
4 2 4
5 3 5 50. Center: 3, 2, 0; radius: 2
x
5. z (4, 2, 2) 6. z 51. Center: 2, 0, 4; radius: 1
3 4 5 52. Center: 0, 4, 3; radius: 2 3
2 3 4 1
(4, 0, 4) 3 4 53. Center: 1, , 4 ; radius: 3
1 2 3
y 1
5 4 3 2 21 123 y 1 3
54. Center: , , 1 ; radius: 1
(3, 1, 0) 43 1 2 3 56 2 2
4 2 1
x 3 55. Center: , 1, 0 ; radius: 1
65 3 3
4 4 (0, 4, 3)
5 x 56. Center: 1, 4, 1 ; radius: 3
5 2
7. 3, 3, 4 8. 6, 1, 1 9. 10, 0, 0 57. 58.
z z
10. 0, 2, 8 11. Octant IV 12. Octant VI (x 1)2 + z2 = 36 (0, 3, 0)
6
13. Octants I, II, III, and IV 4 10 4
86 6 8
2 4 2
14. Octants III, IV, VII, and VIII 2 2 2
4 2 2
15. Octants II, IV, VI, and VIII (1, 0, 0) 6 6 y
All rights reserved. 10 8 y x 8
. 16. Octants I, II, VII, and VIII x 6
2 2
(y + 3) + z = 25
17. 65 units 18. 2 13 units 19. 29 units 59. 60.
z
flin Company20. 13 units 21. 114 units 22. 5 units z
(y 3)2 + z2 = 5
23. 110 units 24. 113 units 4
2 2 2 2 4 2 4
25. 2 5 3 29
Houghton Mif (2, 3, 0) 2 (0, 1, 1)
2 2 2 2 2 2
26. 2 14 6 62 27. 3 6 3 5 2
x 2 2 4
2 2 2 x 6
28. 3 2 13 29. 6, 6, 2 10; isosceles triangle 4 y
3 y 2 2
x + (y 1) = 3
Copyright ©30. 3, 3, 4 2; isosceles triangle 31. 2, 1, 2 6
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 2
Precalculus with Limits, Answers to Section 11.1 2
(Continued) 82. 83.
5, 116.57 41, 51.34
84. 85. 86. 2
61. z 62. z 149, 325.01 7
6 87. 1, 2, 6, 15, 31
5 First differences: 1, 4, 9, 16
Second differences: 3, 5, 7
6 Neither
4 2 4 88. 0, 1, 2, 3, 4
2 2 First differences: 1, 1, 1, 1
3 2 2 3
5 4 4 2 y Second differences: 0, 0, 0
x 5 y 4
x Linear
2
63. 3, 3, 3 64. 4, 4, 8 65. x2 y2 z2 165 89. 2, 5, 8, 11
4 1,
66. (a) 2 2 2 2 First differences: 3, 3, 3, 3
x y z 3963
(b) trace: 2 2 2 Second differences: 0, 0, 0
xz- x z 3963 ;
trace: 2 2 2
yz- y z 3963 Linear
These traces would form circles. 90. 4, 0, 6, 14, 24
(c) trace: 2 2 2
xy- x y 3963 First differences: 4, 6, 8, 10
This trace would form a circle.
(d) yz-trace (e) xy-trace Second differences: 2, 2, 2
67. False. is the directed distance from the plane to Quadratic
z xy- P.
68. False. The xy-trace could also be a point or may not exist. 91. 2 2
x 5 y 1 49
69. 0; 0; 0 92. 2 2
70. No, the graph of the equation is a x 3 y 6 81
axbycz0
plane. 93. 2
y 1 12x 4
71. Apoint or a circle (where the sphere and the yz-plane meet) 94. 2
72. Astraight line in the xy-plane x 2 20y 5
73. x , y , z 2x x, 2y y , 2z z 2 2
2 2 2 m 1 m 1 m 1 95. x 3 y 3 1
74. 75. 3 ± 17 9 4
7, 16, 12 v 2
2 2
96. x y3 1
76. z 7 ± 5 5 77. x 5 ± 5 454 814
2 2 2 2
x 6 y
78. 3 ± 13 79. 1 ± 10 97. 1
x 2 y 2 4 32
2 2
80. 5 ± 89 81. 7 y 5 x 3
x 4 3 2, 4 98. 16 9 1
All rights reserved.
.
flin Company
Houghton Mif
Copyright ©
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 3
Precalculus with Limits, Answers to Section 11.2 3
Section 11.2 (page 825) (c) (d)
z z
5
, 5, 5
Vocabulary Check (page 825) 6 2
2 1
, 1, 1 5
2 2
1. zero 2. v v i v j v k 1 4
1 2 3 3
3. component form 4. orthogonal 5. parallel y 2 4
2 1 1 2 3
1 1
1. (a) 2, 3, 1 2. (a) 0, 0, 4 2 1 3 2 y
21 142 3 5
(b) (b) x 3
2 4 2
z z x
3 3 7. z 3, 7, 6 8. z 7, 19, 13
2 2 4
4 1 3 1 3
2
3(2, 3, 1) 3 9. z 2, 6, 2 10. z 0, 1, 0 11. 9 2
1 2 1
32 2 1
1 2 2 3 12. 29 13. 74 14. 41 15. 34 16. 14
1 1 4 3 2 4 y
2 2 3 x 17. (a) 748i 3j k (b) 748i 3j k
3 y 3 74 74
x 4 (0, 0, 4)
18. (a) 1343i 5j 10k
3. (a) (b) (c) 11 134
7, 5, 5 3 11 33 7, 5, 5
(b) 1343i 5j 10k
134
4. (a) (b) (c) 67
7, 3, 3 67 67 7, 3, 3 19. 4 20. 28 21. 0 22. 0 23.
124.45
5. (a) (b) 24.
49.80 25.
109.92 26.
65.47
z z 27. Parallel 28. Neither 29. Orthogonal
6 〈2, 2, 6〉 4 30. Parallel 31. Not collinear 32. Collinear
5 3 4
4 2 3 33. Collinear 34. Not collinear 35. 3, 1, 7
3 5 7 11 3 7
2 4 3 2 y 36. 10, 5, 2 37. 6, 2, 4 38. 2, 2, 2
21 142 3
1 3 〈1, 1, 3〉
y
4 3 2 1 143 4 39. 3 14 40.
2 x 3 ± 14 ± 6
4 3 2
4 41. or
x 0, 2 2, 2 2 0, 2 2, 2 2
(c) (d) 42. or
5 3, 0, 5 5 3, 0, 5
z z 43. 226.52 newtons, 202.92 newtons, 157.91 newtons
5 3, 3, 9 4
4 2 2 2 3 44. (a) T 8L , L > 18
3 2 4 2 2
3
L 18
2 〈0, 0, 0〉 (b)
1 4 3 2 y
y 21 142 3 L 20 25 30 35 40 45 50
4 3 2 21 243 3
All rights reserved.3 4 2 T 18.4 11.5 10 9.3 9.0 8.7 8.6
. x
4 2 3
x 3 4 (c)
30 L = 18
6. (a) (b)
flin Company z z 〈2, 4, 4〉
4 4 T = 8
3 4 3 4 0100
2 3 2 3 0
Houghton Mif 1 1 Horizontal asymptote: T 8
6 5 4 3 1 2 y 4 3 2 y Vertical asymptote: L 18
21 142 3
2 4 3 The minimum tension in each cable is 8; the minimum
〈1, 2, 2〉 2 cable length is 18.
x 3 x 3
Copyright © 4 4 (d) 30 inches
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 4
Precalculus with Limits, Answers to Section 11.2 4
(Continued) 50. (a) x t (b) x t 1
45. True 46. True y 2 y 2
47. The angle between u and v is an obtuse angle. t t 1
48. Aline 51. (a) x t (b) x t 1
y t2 8 y t2 2t 7
49. (a) x t (b) x t 1 52. (a) x t (b) x t 1
y 3t 2 y 3t 1 3 3
y 4t y 4t 1
All rights reserved.
.
flin Company
Houghton Mif
Copyright ©
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