# 12-7 - z = 2 45 Distributing ρ 2 gives x 2 z 2 = 4 which...

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Solutions to Homework Assignment 6 MATH 249-01 and -02 Section 12.7, Page 842 3, 7, 9, 10, 15, 20, 31-36, 37-47 odd, 48, 49, 53, 56 3. Find the graph in the back of the book. I will express the point in rectangular coordinates. x = 2cos( π/ 4) = 2 ,y = 2sin( π/ 4) = 2 ,z = 2 . 9. We have r = p 1 2 + ( - 1) 2 = 2 . tan θ = y x = - 1 . This means that θ = 3 π 4 or θ = - π 4 . Since the xy -coordinates are (1 , - 1) , the projection into the xy -plane is in the fourth quadrant, so our angle must be - π/ 4 . The cylindrical coordinates are thus ( 2 , - π/ 4 , 4) . 15. Rectangular coordinates: x = 1sin( π/ 6)cos( π/ 6) = 3 4 ,y = 1sin( π/ 6)sin( π/ 6) = 1 4 ,z = 1cos( π/ 6) = 3 2 . 31. This is a cylinder of radius 3 whose axis is the z -axis. 32. This is a sphere of radius 3 whose center is the origin. Note the ρ versus the r in 31. 33. This is just the positive z -axis. 39. This is the horizontal plane
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Unformatted text preview: z = 2 . 45. Distributing ρ 2 gives x 2 + z 2 = 4 , which is a cylinder with axis on the z-axis. 48. Factoring gives ( ρ-4)( ρ-2) = 0 , so either ρ = 4 or ρ = 2 . This is the union of two concentric spheres, one of radius 2 and one of radius 4, both centered at the origin. 49. (a) In cylindrical coordinates, this is z = r 2 . (b) In spherical coordinates, this is ρ cos φ = ρ 2 sin 2 φ (cos 2 θ + sin 2 θ ) = ρ 2 sin 2 φ. This can also be rewritten as ρ sin 2 φ = cos φ. 53. (a) In cylindrical coordinates, this hyperboloid of two sheets has equation r 2 (cos 2 θ-sin 2 θ )-2 z 2 = 4 . (b) In spherical coordinates, this is ρ 2 (sin 2 φ cos 2 θ-sin 2 φ sin 2 θ-2cos 2 φ ) = 4 ....
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