Embed Size (px)
Citation preview
Precalculus with Limits, Answers to Section 11.1 1
Chapter 11Section 11.1 (page 817)
Vocabulary Check (page 817)1. three-dimensional2. -plane, -plane, -plane 3. octants
4. Distance formula 5.
6. sphere 7. surface, space 8. trace
1.2.3. 4.
5. 6.
7. 8. 9.
10. 11. Octant IV 12. Octant VI
13. Octants I, II, III, and IV
14. Octants III, IV, VII, and VIII
15. Octants II, IV, VI, and VIII
16. Octants I, II, VII, and VIII
17. units 18. units 19. units
20. units 21. units 22. 5 units
23. units 24. units
25.
26. 27.
28. 29. isosceles triangle
30. isosceles triangle 31.
32. 33. 34.
35. 36. 37.
38. 39.
40.
41.
42.
43.
44.
45.
46.
47. Center: radius:
48. Center: radius: 4
49. Center: radius: 2
50. Center: radius: 2
51. Center: radius: 1
52. Center: radius:
53. Center: radius: 3
54. Center: radius: 1
55. Center: radius: 1
56. Center: radius: 3
57. 58.
59. 60.
2
2
4
46x
y
z
−4 −4−2 (0, 1, −1)
−4
−6x2 + (y − 1)2 = 3y
x
2
2 2
z
(−2, 3, 0)
(y − 3)2 + z2 = 5
2
4
6
86
2
x
y
z
(0, −3, 0)
−10
−4
−6
−6−8
−2
(y + 3)2 + z2 = 25x
y(1, 0, 0)
z
(x − 1)2 + z2 = 36
4
2
6
2
8
−2
−6−8
−2
4
10
�12, 4, �1�;
�13, �1, 0�;
�12, 32, 1�;
�1, 13, 4�;2�3�0, 4, 3�;
��2, 0, 4�;
�3, �2, 0�;
�2, �1, 3�;
�0, 4, 0�;
52�5
2, 0, 0�;�x �
12�2
� �y � 1�2 � �z � 4�2 �614
�x �32�2
� y2 � �z � 3�2 �454
x2 � �y � 5�2 � �z � 9�2 � 16
�x � 3�2 � �y � 7�2 � �z � 5�2 � 25
�x � 2�2 � �y � 1�2 � �z � 8�2 � 36
x2 � �y � 4�2 � �z � 3�2 � 9
�x � 3�2 � �y � 4�2 � �z � 3�2 � 4
�x � 3�2 � �y � 2�2 � �z � 4�2 � 16�9, �72, �3
2��2.5, 2, 6���9
2, 92, 132 ��1, 0, 5.5�
�1, 6, �2��0, �1, 7��32, 72, 12�
�32, �1, 2�3, 3, 4�2;
6, 6, 2�10;32 � 22 � ��13 �2
32 � 62 � �3�5 �2�2�14�2� ��6�2
� ��62�2
�2�5�2 � 32 � ��29�2
�113�110
�114�13
�292�13�65
�0, 2, 8�
�10, 0, 0��6, �1, �1���3, 3, 4�
y
x
32
−4
65
1
1
2345
(4, 0, 4)
z
(0, 4, −3)
−4 −3 5 6
−3−4−5
y
x
12
4
1
2
3
−2
−3
−4
−5
1 2 3−2
−3−4
−3−4−5
(3, −1, 0)
(−4, 2, 2)z
y
x
12
4
3
321−2
−4
−3−4−5
(3, 0, 0)
(−3, −2, −1)
z
−2
−3
−4
−5
y
x
5432−2
−2
−3
1
3
5
2
4
1
2
3
4
5
z
(−1, 2, 1)(2, 1, 3)
��2, 3, 0�C:�2, �1, 2�;B:�6, 2, �3�;A:��3, 0, �2�C:�1, 3, �2�,B:��1, 4, 4�,A:
�x1 � x2
2,
y1 � y2
2,
z1 � z2
2 �yzxzxy
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 1
(Continued)
61. 62.
63. 64. 65.
66. (a)(b) trace:
trace:These traces would form circles.
(c) trace:This trace would form a circle.
(d) trace (e) trace67. False. is the directed distance from the plane to 68. False. The trace could also be a point or may not exist.69. 0; 0; 070. No, the graph of the equation is a
plane.71. A point or a circle (where the sphere and the plane meet)72. A straight line in the -plane73.
74. 75.
76. 77.
78. 79.
80. 81.
82. 83.
84. 85. 86. 2
87. 1, 2, 6, 15, 31First differences: 1, 4, 9, 16Second differences: 3, 5, 7Neither
88.
First differences:
Second differences: 0, 0, 0
Linear
89. 2, 5, 8, 11
First differences: 3, 3, 3, 3
Second differences: 0, 0, 0
Linear
90.
First differences:
Second differences:
Quadratic
91.
92.
93.
94.
95.
96.
97.
98.�y � 5�2
16�
�x � 3�2
9� 1
�x � 6�2
4�
y2
32� 1
x2
45�4�
�y � 3�2
81�4� 1
�x � 3�2
9�
�y � 3�2
4� 1
�x � 2�2 � �20� y � 5�
�y � 1�2 � �12�x � 4�
�x � 3�2 � � y � 6�2 � 81
�x � 5�2 � �y � 1�2 � 49
�2, �2, �2
�4, �6, �8, �10
4, 0, �6, �14, �24
�1,
�1, �1, �1, �1
0, �1, �2, �3, �4
�7�149, 325.01�
�41, 51.34��5, 116.57�
3�2, 7�
4x � �
5 ± �894
y � �1 ± �10
2x � �
3 ± �132
x �5 ± �5
2z �
7 ± 5�52
v � �3 ± �17
2�7, 16, 12�
�x2, y2, z2� � �2xm � x1, 2ym � y1, 2zm � z1�xy
yz-
ax � by � cz � 0
xy-P.xy-z
xy-yz-
x2 � y2 � 39632xy-
y2 � z2 � 39632yz-x2 � z2 � 39632;xz-
x2 � y2 � z2 � 39632
x2 � y2 � z2 �1652
4�4, 4, 8��3, 3, 3�
x
y2
4
−6
−4
−2 −2
−42
z
yx
2 233
4455
5
6
z
Precalculus with Limits, Answers to Section 11.1 2
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 2
Precalculus with Limits, Answers to Section 11.2 3C
opyr
ight
©H
ough
ton
Mif
flin
Com
pany
. All
righ
ts r
eser
ved.
Section 11.2 (page 825)
Vocabulary Check (page 825)1. zero 2.3. component form 4. orthogonal 5. parallel
1. (a) 2. (a)(b) (b)
3. (a) (b) (c)
4. (a) (b) (c)
5. (a) (b)
(c) (d)
6. (a) (b)
(c) (d)
7. 8.
9. 10. 11.
12. 13. 14. 15. 16.
17. (a) (b)
18. (a)
(b)
19. 20. 28 21. 0 22. 0 23.
24. 25. 26.
27. Parallel 28. Neither 29. Orthogonal
30. Parallel 31. Not collinear 32. Collinear
33. Collinear 34. Not collinear 35.
36. 37. 38.
39. 40.
41. or
42. or
43. 226.52 newtons, 202.92 newtons, 157.91 newtons
44. (a)
(b)
(c)
Horizontal asymptote:Vertical asymptote:The minimum tension in each cable is 8; the minimumcable length is 18.
(d) 30 inches
L � 18T � 8
0 1000
30
T = 8
L = 18
T �8L
�L2 � 182, L > 18
��5�3, 0, 5�5�3, 0, 5�0, 2�2, �2�2�0, 2�2, 2�2
±�6±3�14
14
�112 , 32, 72��6, 52, �7
4��10, �5, 2��3, 1, 7�
65.47�109.92�49.80�
124.45��4
��134134
��3i � 5j � 10k�
�134134
��3i � 5j � 10k�
��7474
�8i � 3j � k��7474
�8i � 3j � k�
�14�34�41�74�29
9�2z � �0, �1, 0z � � 12, 6, 32
z � ��7, 19, 13z � ��3, 7, 6
y
x
12
34
2
1
3
4
5
6
321 4 5−2
−3−4
−2
−3
z52
− , 5, 5
y
x
1
2
2
1
21−1
−2
−1
−2
−2
z
12
− , 1, 1
y
x
12
34
2
1
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z ⟨−2, 4, 4⟩
y
x
2
1
3
4
21
−3−4
−3
−2
−4
−3−4−5−6
z
⟨1, −2, −2⟩
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−2
−4
−3−4
z
⟨0, 0, 0⟩
y
x
12
34
2
1
3
4
5
32 4−2
−3
−2
−3−4
z
32
, 32
, 92
y
x
12
34
2
3
4
321 4−2
−3−4
−3
−4
−3−4
z
⟨−1, −1, −3⟩y
x
12
34
1
2
3
4
5
6
31 4−2
−2
−3−4
z
⟨2, 2, 6⟩
�6767
�7, �3, �3�67�7, �3, �3
�1133
�7, �5, 53�11�7, �5, 5
y
x
3
12
34
−2
−3−2
−1
−4
−3−4
23
4
2
−4
1
−2
−3
z
(0, 0, −4)y
x
3
2
1
1
−1
−2−3
−2−3
12
3
23
z
(−2, 3, 1)
�0, 0, �4��2, 3, 1
v � v1i � v2 j � v3k
20 25 30 35 40 45 50
18.4 11.5 10 9.3 9.0 8.7 8.6T
L
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 3
(Continued)
45. True 46. True
47. The angle between u and v is an obtuse angle.
48. A line
49. (a) (b)
50. (a) (b)
51. (a) (b)
52. (a) (b)y � 4�t � 1�3y � 4t 3
x � t � 1x � ty � t 2 � 2t � 7y � t 2 � 8x � t � 1x � t
y �2
t � 1y �
2t
x � t � 1x � t
y � 3t � 1y � 3t � 2
x � t � 1x � t
Precalculus with Limits, Answers to Section 11.2 4
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 4
Precalculus with Limits, Answers to Section 11.3 5
Section 11.3 (page 832)
Vocabulary Check (page 832)1. cross product 2. 0 3.4. triple scalar product
1. 2.
3. 4.
5. 6. 7.
8. 9.
10. 11. 12.
13. 14.
15. 16.
17.
18. 19.
20. 21. 1 22. 3 23.24. 25. 14 26.27. (a) and is parallel to
and is parallel to (b) Area is (c) The dot product is not 0 and therefore the parallelo-
gram is not a rectangle.28. (a) and is parallel to
and is parallel to (b) Area is
(c) The dot product is not 0 and therefore the parallelo-gram is not a rectangle.
29. 30. 31. 32.
33. 34. 6 35. 2 36. 6 37. 2
38. 9 39. 12 40. 16 41. 84 42. 3
43. (a)
(b)
44. foot-pounds
45. True. The cross product is not defined for two-dimensionalvectors.
46. False. The cross product is not commutative.
47. The magnitude of the cross product will be four times aslarge.
48. Answers will vary. 49. 50. 51.
52. 53. 54. 55. 56.
57.
The minimum value of is 0 and found at point The maximum value of is 52 and found at point
58.
The minimum value of is 46 and found at the point There is no maximum value of on this region.z
�3, 4�.z
x
y
−2 2 4 6 8
−2
2
4
6
8
�6, 4�.z�0, 0�.z
x
y
−2 2 4 6 8
−2
2
4
6
8
�3�1��32
�12
��32
�12��3�
12
160�3
T �p2
cos 40�
�16
8�512�42903�11
3�132
�AB\
� AC\
� � 2�83.AC
\
� �5, 4, 1.BD\
� �5, 4, 1CD
\
� �1, 2, 3.AB\
� �1, 2, 3
�AB\
� AD\
� � 6�10.BC
\
� ��3, 4, 4.AD\
� ��3, 4, 4DC
\
� �1, 2, �2.AB\
� �1, 2, �23�30�213
�80623i �
23 j �
13k
�22
�i � j��4291287
�10i � 25j � 56k�
�76027602
��71i � 44j � 25k�
17
��6i � 3j � 2k��1919
�i � 3j � 3k�
�2i � j � k�i � 2j � k
2j �29k�18i � 6j�
32i �
32 j �
32k
�7i � 13j � 16k��7, 37, �20�0, 42, 0��29, 36, �38�3, �3, �3
y
x
1
22
1
2
(0, 1, 0)
z
−2
−1
−2
−2
−1
−1
y
x
1
22
1
2
z
−2−2
−2
−1
−1
(0, −1, 0)
j�j
y
x
1
22
1
2
z
−2
−2
−2
−1
−1
(−1, 0, 0)
y
x
1
2
1
−1
2
z
−2
−2
−2
−1
−1
−1
(0, 0, −1)
�i�k
�u � �v � sin �
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
15 20 25 30 35 40 45
5.75 7.66 9.58 11.49 13.41 15.32 17.24T
p
332522CB_1100_AN.qxd 4/4/06 5:30 PM Page 5
Section 11.4 (page 841)
Vocabulary Check (page 841)
1. direction, 2. parametric equations
3. symmetric equations 4. normal5.
1. (a)
(b)
2. (a)
(b)
3. (a)
(b)
4. (a)
(b)
5. (a)
(b)
6. (a)
(b)
7. (a)
(b)
8. (a)
(b)
9. (a)
(b)
10. (a)
(b)
11. (a)(b) No symmetric equations
12. (a)(b) No symmetric equations
13. (a)
(b)
14. (a)
(b)
15. 16.
17. 18.19.20.21.22.23.24.25.26.27. 28.29. Orthogonal 30. Parallel31. Orthogonal 32. Parallel33. (a) (b)34. (a) (b)35. (a) (b)
36. (a) and (b). The two planes are parallel because the nor-mal vector is a scalar multiple of the normal vector and the planes do not intersect.
37. 38.
39. 40.
y
x
32
6
32
−1 −1−2
−2
6
23
45
6
(0, 5, 0)
(0, 0, 5)
z
y
x
32
4
4
6
3
−1−2
564
32
56
(0, 2, 0)(4, 0, 0)
z
−1−2
y
x
3456
232
34
56
(0, 0, 1)
(2, 0, 0)
z
−2−4
−5
(0, −4, 0)
y
x
34
4
56
2
−2
56
23
(0, 0, 2)
(0, 3, 0)
(6, 0, 0)
z
n2��3, �6, 3,n1�2, 4, �2
x � 6t � 1, y � t, z � 7t � 177.8�
x � �5t �32, y � �t �
16, z � 2t66.9�
x � �t � 2, y � 8t, z � 7t60.7�
7x � y � 11z � 5 � 0y � 5 � 02x � 11y � 4z � 5 � 06x � 2y � z � 8 � 0�x � y � 4z � 7 � 0�3x � 9y � 7z � 0�x � y � 2z � 12 � 0�x � 2y � z � 2 � 0�3y � 5z � 0�2x � y � 2z � 10 � 0
z � 3 � 0x � 2 � 0
y
(5, 1, 5)
z
x
6
4
2
2
−6
−4−6
24
64
6y
(0, 2, 1)
z
x
3
2
1
1
−1
−2−3
−2−3
12
32
3
x � 39
�y � 3�13
�z � 4�12
x � �3 � 9t, y � 3 � 13t, z � 4 � 12t
2x � 13
� �2y � 4
5�
2z � 1�1
x � �12
� 3t, y � 2 � 5t, z �12
� t
x � 2, y � �1 � 2t, z � 5 � 8t
x � 3 � 4t, y � 1, z � 2 � 3t
x � 2�1
�y � 3�8
�z � 1
4
x � 2 � t, y � 3 � 8t, z � �1 � 4t
x � 34
�y � 8�10
� z � 15
x � �3 � 4t, y � 8 � 10t, z � 15 � t
x � 28
�y � 3
5�
z12
x � 2 � 8t, y � 3 � 5t, z � 12t
x � 2�1
�y4
�z � 2�5
x � 2 � t, y � 4t, z � 2 � 5t
x � 13
�y
�2� z � 1
x � 1 � 3t, y � �2t, z � 1 � t
x � 22
�y � 3�3
� z � 5
x � 2 � 2t, y � �3 � 3t, z � 5 � t
x � 54
�z � 10
3, y � 0
x � 5 � 4t, y � 0, z � 10 � 3t
x � 43
�y � 1
8�
z�6
x � �4 � 3t, y � 1 � 8t, z � �6t
x � 33
�y � 5�7
�z � 1�10
x � 3 � 3t, y � �5 � 7t, z � 1 � 10t
x �y2
�z3
x � t, y � 2t, z � 3t
a�x � x1� � b�y � y1� � c�z � z1� � 0
PQ\
t
Precalculus with Limits, Answers to Section 11.4 6
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 6
Precalculus with Limits, Answers to Section 11.4 7
(Continued)
41. 42.
43. 44. 45. 46.
47. (a) (b) The approximations are verysimilar to the actual values of
(c) Answers will vary.
48.
49. False. Lines that do not intersect and are not in the sameplane may not be parallel.
50. True
51. Parallel. is a scalar multiple of
52. (a) Sphere:
(b) Two planes:
53. 54.
55. 56.
57. 58.
59.
60. r �1
sin � � 2 cos �
r � 5 csc �
r � 4 cos �r � 7
3x2 � 4y2 � 2x � 1 � 0x2 � y2 � 3x � 0
y � �xx2 � y2 � 100
4x � 3y � z � 10 ± 2�26
�x � 4�2 � �y � 1�2 � �z � 1�2 � 4
��15, 27, �30.�10, �18, 20
89.1�
z.
3�14
�3�14
144�6
�2�6
3
�66
89
xy
2
2
−2
−2
−4
−6
6
z
(6, 0, 0)(0, 0, −2)
x y6
−1−2 −2
−1
−7−6
54
3
65
4
z
(0, 0, −6)
(0, 3, 0)(2, 0, 0)
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
Year
1999 7.85
2000 7.70
2001 7.50
2002 7.34
2003 7.22
z
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 7
Review Exercises (page 844)1. 2.
3. 4.
5. 6.
7.
8.
9.
10. 11. 12.
13.
14.
15.
16.
17. Center: radius: 3
18. Center: radius: 2
19. (a) (b)
20. (a) (b)
21. (a)
(b)
(c)
22. (a) (b)
(c)
23. (a) (b)
(c)
24. (a) (b)
(c)
25. 26. 0 27. 1 28. 29.
30. 31. Parallel 32. Orthogonal
33. Not collinear 34. Collinear
35. 159.1 pounds of tension115.6 pounds of tension115.6 pounds of tension
36. 106.1 pounds of tension77.1 pounds of tension77.1 pounds of tension
37. 38.
39. 40.
41.
42. 43. 75 44. 48
45. (a)
(b)
46. (a)
(b)x5
�y � 10
20�
z � 3�3
x � 5t, y � �10 � 20t, z � 3 � 3t
x � 14
�y � 3
3�
z � 5�6
x � �1 � 4t, y � 3 � 3t, z � 5 � 6t
Area � �8 � 2�2 2.83
Area � �172 � 2�43 13.11
j�71�7602
7602i �
22�76023801
j �25�7602
7602k
�15, 25, �105��10, 0, �10C:B:A:C:B:A:
47.61�
90��5�9
��19539
, �11�195
195,
7�195195
�195�5, �11, 7
��2�18537
, 6�185
185,
7�185185
�185��10, 6, 7
���35
7,
3�3535
, �3535 �35��5, 3, 1
��3333
, 4�33
33,
�4�3333
�33
�1, 4, �4
2
4
6
x
y
z(y − 1)2 + z2 = 5
(−2, 1, 0)
2
4
64x
y
z
(−2, 1, 0)
(x + 2)2 + (y − 1)2 = 9
2
2
4
4 42
6x
y(0, 3, 0)
z
(y − 3)2 + z2 = 16
−2 −2
2
2
4
4 46x
y
(0, 3, 0)
z
x2 + z2 = 7
−4−2
�5, �3, 2�;
�2, 3, 0�;
x2 � �y � 4�2 � �z � 1�2 �2254
�x � 1�2 � �y � 5�2 � �z � 2�2 � 36
�x � 3�2 � �y � 2�2 � �z � 4�2 � 16
�x � 2�2 � �y � 3�2 � �z � 5�2 � 1
��6, �6, �2��1, 2, �9��4, 0, �1�
�132 , 2, 5�
��29�2� ��13�2
� ��42�2
�29, �13, �42
��29�2� ��38�2
� ��67�2
�29, �38, �67
�61�41
�0, �7, 0���5, 4, 0�
y
x
12
34
1
2
−2
−3
3
4
5
31 2−2
−3−4
−3−4
z
(0, 0, 5)
(2, 4, −3)
y
x
12
34
1
2
3
−2
−3
−4−5
(5, −1, 2)
(−3, 3, 0)
z
1 2 3
−2
−3
−4
−5
Precalculus with Limits, Answers to Review Exercises 8
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 8
Precalculus with Limits, Answers to Review Exercises 9
(Continued)
47. (a) (b)
48. (a)
(b)
49. 50.
51. 52.
53.
54.
55.
56.
57. 58. 59. 60.
61. False. 62. True. See page 831.
63–66. Answers will vary.
u � v � ��v � u�
6�147
�11055
�66
�110110
x
y
(0, 0, −4)
(0, 3, 0)1 21
−1
−2
43
2
−2
−3
−1
1
z
x
y
(3, 0, 0)
1 21
−2 −1
3
3
4
1
−1
−2
2
z
(0, 0, −2)
y
x
24
2
4
2
4
(1, 0, 0)
z
−4
−4
−2
(0, 0, −1)
(0, −5, 0)
x
y
(0, 0, 2)
(2, 0, 0)1 1
2
3
1
−2
z
(0, −3, 0)
�x � y � 2z � 12 � 0z � 2 � 0
�2y � 5z � 14 � 0�2x � 12y � 5z � 0
x � 3 � y � 2 � z � 1
x � 3 � t, y � 2 � t, z � 1 � t
x�2
�y
5�2� zx � �2t, y �
52
t, z � t
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 9
Chapter Test (page 846)1.
2. No. 3.4.
5.6. (a) (b) 84 (c) 7.8. (a)
(b)
9. Neither 10. Orthogonal11. is parallel to
is parallel to
12. 13. 200
14.
15.
16. 17. 88.5�8
�14�
4�147
y
x
4
810
2
4
6
−6
−8
−10
2 4 6
−6
−10
(0, −10, 0)
(0, 0, −5)
(2, 0, 0)
z
y
x
1
65
1
3
4
21 4−2
−3
−3−4
z
(0, 0, 3)
(6, 0, 0)
(0, 4, 0)
27x � 4y � 32z � 33 � 0Area � 2�230
BD\
� �1, �3, 3.AC\
CD\
� �4, 8, �2.AB\
x � 8�2
�y � 2
6�
z � 5�6
x � 8 � 2t, y � �2 � 6t, z � 5 � 6t46.23��0, 62, 62�194
v � ��12, 5, �5u � ��2, 6, �6,
4
4
6
8
6
2
8
12
y
x sphere
z
−4
−8−10
−4−2
xz-trace
�x � 7�2 � �y � 1�2 � �z � 2�2 � 19�7, 1, 2���76�2
� ��102�2 ��194�2
y
x
24
2
−4
4
z
−2
−4
(−2, −2, 3)
(5, −2, 3)
Precalculus with Limits, Answers to Chapter Test 10
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 10
Precalculus with Limits, Answers to Problem Solving 11
Problem Solving (page 849)1. (a)
(b) Answers will vary.(c)(d) Answers will vary.
2. Sphere of radius 4 centered at 3. Answers will vary. See online website.4. Answers will vary.5. (a) Right triangle (b) Obtuse triangle
(c) Obtuse triangle (d) Acute triangle6.
7. 8–11. Proof
12. (a)
(b)(c) This is what should be expected. When
the pipe wrench is horizontal.
13. (a)(b)
(c) when
(d)
(e) The zero is the angle making parallelto
14. Proof
15. (a) when
(b)
(c) The distance between the two insects appears to lessenin the first 3 seconds but then begins to increase withtime.
(d) The insects get within 5 inches of each other.
16. (a) (b)
17. (a) (b)
18. (a)
(b)
The minimum is at
(c) Yes, there are slant asymptotes. Using we havethe slant asymptotes
y � ±�105
21�s � 1�.
s � x,
s � �1.D 2.2361
−11
−4
10
10
D ��PQ
\
� u ��u�
���7 � s�2 � ��6 � 2s�2 � 25
�21
D � �5D �3�2
2
D ��53
D ��149�17
��2533
17
−1 110
20
t � 0.d � �70
F.AB
\
� 141.34�;
� 51.34�
� � 30�.�AB\
� F � � 298.2
−300
0 180
400
�AB\
� F � � 25�10 sin � � 8 cos ��F � �200�cos �j � sin �k�AB
\
� �54 j � k,
� � 90�,� � 90�.540�2 763.68
00
180
1440
1080 sin �
�F � � 860.0 lbsF3 � ��5�3, 5, �40
F2 � �5�3, 5, �40,F1 � �0, �10, �40,
�x1, y1, z1�
a � b � 1
y
x
3
2
1
1
−1
−2−3
−2−3
23
23
z
uv
Cop
yrig
ht ©
Hou
ghto
n M
iffl
in C
ompa
ny. A
ll ri
ghts
res
erve
d.
332522CB_1100_AN.qxd 4/4/06 5:31 PM Page 11
