EVNREV10 - R eview Exercises for Chapter 10 102. 0 ≤ z2...

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Unformatted text preview: R eview Exercises for Chapter 10 102. 0 ≤ z2 ≤ ≤2 r2 z 4 3 −5 5 4 4 y x 5 y 2 503 100. 2 ≤ ≤ 2 104. 0 ≤ 6r 8 4 ≤ ≤2 ≤ 2 0≤r≤3 0 ≤ z ≤ r cos z 4 3 2≤r≤4 0≤ ≤1 z −4 −2 −2 x 2 x 2 y 106. Cylindrical: 0.75 ≤ r ≤ 1.25, z 108. Cylindrical 1 2 8 z 110. 2 sec 4 sphere ⇒ cos 2⇒z 2 plane ≤r≤3 ≤2 9 r2 −4 4 0≤ 9 The intersection of the plane and the sphere is a circle. 4 x −4 y r2 ≤ z ≤ Review Exercises for Chapter 10 2. P (a) u (b) v (c) 2u v 2, \ 1, Q 7, 0 42 14i 52 5, 1R 7i, v 41 PR 2, 4 \ 4. v 4, 5 4i 5j v cos i v sin j 1 cos 225 i 2 2 i 4 2 j 4 1 sin 225 j 2 PQ 4i 5j 18i 5j y 6. (a) The length of cable POQ is L. \ OQ L 9i yj y2 ⇒ \ O 2 92 L2 4 y2 −9 x 9 81 y θ P 500 lb Q Tension: T Also, cy c OQ c 81 250 y 250 ⇒ T 81 y2 ⇒ T 250 L2 4 81 L 2 250L L2 324 18 in. Domain: L > 18 inches (b) L T (c) 1000 19 780.9 20 573.54 21 485.36 22 434.81 23 401.60 24 377.96 25 360.24 (e) lim T L→ (d) The line T curve at L 400 intersects the 250 23.06 inches. The maximum tension is 250 pounds in each side of the cable since the total weight is 500 pounds. 18 0 25 504 8. x Chapter 10 z 0, y Vectors and the Geometry of Space 7: 0, 7, 0 10. Looking towards the xy-plane from the positive z-axis. The point is either in the second quadrant x < 0, y > 0 or in the fourth quadrant x > 0, y < 0 . The z-coordinate can be any number. 12. Center: Radius: x 14. x2 x 2 2 0 2 2 y 40 , 0 3 2 2 2 64 , 3 z y2 0 2 0 2 6y 2 2 2, 3, 2 2 17 9 z2 4 4z 4 34 25 9 4 6 4 2 z 4 2 4 9 4 17 10x 5 2 25 y 3 2 z 2 2 Center: 5, Radius: 2 3, 2 2 4 6 8 x y 16. v 3 6, z 8 7 6 5 4 3 2 1 1 2 1 3 2, 8 0 3, 5, 8 18. v w 8 11 5, 5 4, 5 4, 3 7 7 3, 1, 4 2 (3, −3, 8) 5, 6 6, 10, Since v and w are not parallel, the points do not lie in a straight line. v 3 y 6 x 5 4 (6, 2, 0) 20. 8 6, 3, 2 49 8 6, 7 3, 2 48 , 7 24 16 , 77 22. P (a) u v (b) u (c) v 2, 1, 3 , Q \ 0, 5, 1 , R 2, 6, 2 3 66 9 3, 2, 54 2 3i 2i 6j 2 5, 5, 0 6j 3k 3 36 2k, PQ \ PR v v 4, 9 3, 6, 23 36 24. u Since v 4, 3, 6,v 16, 12, 24 26. u u v 1, 5 , v 4u, the vectors are parallel. 0 ⇒ is orthogonal to v. 2 \ \ 28. u v u u v cos v 1, 0, 2, 3 2, 1 1 30. W F PQ F PQ cos 75 8 cos 30 300 3 ft lb 10 3 uv uv 83.9 1 3 10 R eview Exercises for Chapter 10 505 In Exercises 32–40, u < 3, 2, 1 , v > < 2, 4, 3,w > < 1, 2, 2 . > 32. cos uv uv arccos 11 14 29 11 14 29 56.9 34. Work u w 3 4 2 5 36. u v i 3 2 i 2 3 v j 2 4 j 4 2 v k 1 3 k 3 1 u. 10i 11j 8k v u 10i 11j 8k Thus, u 38. u v w i 3 2 i 3 1 u j 2 4 3, 2, 1 k 1 3 1, 2, 1 i 3 1 j 2 2 k 1 1 4i 4j 4k u v 10i k 1 2 11j 8k u u w v j 2 2 w 1 v 2 4i 6i 4j 1 2 4k 7j u 4k v w 5 (See Exercise 35) 2 40. Area triangle w 2 2 1 2 42. V u v w 2 0 0 1 2 1 0 1 2 25 10 44. Direction numbers: 1, 1, 1 (a) x (b) x 1 1 t, y y 2 2 z t, z 3 3 t 46. u v i 2 3 j 5 1 k 1 4 21 i 11j 13k 48. P \ 3, 0, 8, \ 4, 2 , Q \ 3, 4, 1 , R 4, 5, j 8 5 4 27x k 1 4 32 z 4y 2 32z 4 27i 1, 1, 2 PQ 4 13t n 1 , PR \ Direction numbers: 21, 11, 13 (a) x (b) x 21 21t, y y 11 1 1 z 13 11t, z 4 27 x 3 PQ PR i 0 4 4y 4j 32k 0 33 506 Chapter 10 Vectors and the Geometry of Space 50. The normal vectors to the planes are the same, n 5, 3, 1 . 52. Q u P \ 5, 1, 3 point 1, 2, 1 direction vector Choose a point in the first plane, P 0, 0, 2 . Choose a point in the second plane, Q 0, 0, 3 . \ 1, 3, 5 point on line 6, u \ PQ \ 2, i 6 1 2 j 2 2 k 2 1 264 6 2, 8, 14 PQ D 0, 0, \ 5 5 35 5 35 35 7 PQ PQ n n D PQ u u 2 11 54. y z2 56. y cos z 58. 16x 2 Cone 16y 2 9z 2 0 Since the x-coordinate is missing, we have a cylindrical surface with rulings parallel to the x-axis. The generating curve is a parabola in the yz-coordinate plane. z 2 1 2 x Since the x-coordinate is missing, we have a cylindrical surface with rulings parallel to the x-axis. The generating curve is y cos z. z 4 xy-trace: point 0,0, 0 xz-trace: z yz-trace: z z 4, x 2 z 4 ± 4x 3 4y 3 9 ± y2 3 4 y −2 x 2 2 y −3 3 23 y −3 x 60. x2 25 y2 4 z2 100 1 12 z 62. Let y r x the x-axis. −5 2 x and revolve the curve about Hyperboloid of one sheet xy-trace: xz-trace: yz-trace: x2 25 x2 25 y2 4 y2 4 z2 100 z2 100 1 x 5 y 1 1 64. 333 3 ,, , rectangular 442 (a) r 3 4 3 4 2 3 4 3 4 2 3 , 2 33 2 2 arctan 3 30 , 2 3 ,z 33 , 2 arccos 3 33 ,, , cylindrical 222 3 , 10 30 , , arccos 23 3 , spherical 10 2 2 (b) 3 , P roblem Solving for Chapter 10 507 66. 81, 5 , 27 3 , cylindrical 6 6561 5 6 arccos 27 3 54 3 arccos 1 2 2187 54 3 68. 12, r2 2 , , spherical 23 12 sin 2 3 2 ⇒r 63 2 3 z cos 6 3, , 12 cos 2 3 6 54 3, 5 , , spherical 63 2 6 , cylindrical 70. x 2 y2 z2 16 z2 4 16 (a) Cylindrical: r 2 (b) Spherical: Problem Solving for Chapter 10 x 2. f x 0 t4 y 1 dt (b) f x f0 x 2 4 (a) 4 2 −4 −2 −2 −4 x4 1 4 tan 1 u 2 2 1 i 2 j 2 2 , 2 2 t, y t. (c) ± 2 , 2 (d) The line is y x: x 4. Label the figure as indicated. \ S R PR \ a b b b a b a b 2 SQ a a a a 2 0, because P b Q b in a rhombus. \ \ 6. n PP0 n PP0 n + PP0 n n − PP0 n Figure is a square. → n and the points P form a circle of radius Thus, PP0 n in the plane with center at P. P0 P 508 Chapter 10 Vectors and the Geometry of Space r 8. (a) V 2 0 r2 x2 dx d > 0, 2 r 2x x3 3 r 0 43 r 3 (b) At height z x a2 2 y2 b2 x2 a2 d2 c2 y2 b2 d2 1 1 1. d2 c2 c2 c2 d2 x2 a2 c2 c2 Area c y2 d2 b2 c2 c2 a2 c2 d 2 c2 b2 c2 d 2 c2 ab 2 c c2 d2 V 2 0 ab 2 c c2 d 2 dd d3 3 c 0 2 ab 2 cd c2 4 abc 3 10. (a) r 2 cos (b) z z2 r 2 cos 2 x2 y2 Cylinder Hyperbolic paraboloid 12. x (a) u P t 3, y 1 t 2 1, z 2t 1; Q 4, 3, s 2, 1, 4 direction vector for line 3, 1, \ 1 point on line 1 i 1 2 j 2s 1 7 s k 1 4 2 PQ \ 1, 2, s u \ PQ 7 si 6 2s j 5k D (b) PQ u u 10 6 21 2s 2 25 (c) Yes, there are slant asymptotes. Using s Ds 1 21 5 21 2.2361 at s 1. y ± x, we have x2 2x 1 22 5x2 x 1 10x 2 110 21 → ± −11 −4 5 21 10 5 x 21 The minimum is D 105 s 21 1 slant asymptotes. P roblem Solving for Chapter 10 14. (a) The tension T is the same in each tow line. 6000i T cos 20 2T cos 20 i ⇒T 6000 2 cos 20 3192.5 lbs 2T cos cos 20 i T sin 20 sin 20 j 509 (b) As in part (a), 6000i ⇒T 3000 cos < 90 20 Domain: 0 < (c) T (d) 10,000 10 3046.3 30 3464.1 40 3916.2 50 4667.2 60 6000.0 3192.5 0 0 90 (e) As increases, there is less force applied in the direction of motion. 16. (a) Los Angeles: 4000, Rio de Janeiro: 4000, (b) Los Angeles: x y z 118.24 , 55.95 43.22 , 112.90 118.24 118.24 4000 sin 55.95 cos 4000 sin 55.95 sin 4000 cos 55.95 1568.2, 2919.7, 2239.7 43.22 43.22 Rio de Janeiro: x y z 4000 sin 112.90 cos 4000 sin 112.90 sin 4000 cos 112.90 2523.3, 1556.5 2685.2, (c) cos uv uv 91.18 (d) s 4000 1.59 1568.2 2685.2 1.59 radians 6366 miles 2919.7 2523.3 4000 4000 2239.7 1556.5 —CONTINUED— 510 Chapter 10 Vectors and the Geometry of Space 16. —CONTINUED— (e) For Boston and Honolulu: a. Boston: 4000, Honolulu: 4000, b. Boston: x y z 71.06 , 47.64 157.86 , 68.69 71.06 71.06 4000 sin 47.64 cos 4000 sin 47.64 sin 4000 cos 47.64 2795.7, 2695.1 959.4, Honolulu: x y z 4000 sin 68.69 cos 4000 sin 68.69 sin 4000 cos 68.69 3451.7, 157.86 157.86 1404.4, 1453.7 3451.7 2795.7 1404.4 4000 4000 2695.1 1453.7 (f) cos uv uv 73.5 959.4 1.28 radians (g) s 4000 1.28 5120 miles 20. Essay. 18. Assume one of a, b, c, is not zero, say a. Choose a point in the first plane such as d1 a, 0, 0 . The distance between this point and the second plane is D a d1 a a2 d1 a2 b2 d2 c2 b0 b2 c0 c2 d1 a2 d2 b2 c2 . d2 ...
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This note was uploaded on 05/18/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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