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The surface area of the flat bottom of such a

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The surface area of the flat bottom of such a cylinder is πr 2 , while the surface area of the curved vertical part of the cylinder is equal to 2 πrh . (A) h : r = 1: 1 (B) h : r = 3: 2 (C) h : r = 4: 3 (D) h : r = 5: 3 (E) h : r = 5: 2 (F) h : r = 7: 2 (G) h : r = 9: 5 (H) None of the above Answer to 5:
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7 (6) Consider the spiraling curve in parametric cylindrical coordinates r ( t ) = t , θ ( t ) = πt , z ( t ) = t 2 . Which of the following computes the length of this curve from the origin to (2 , 0 , 4)? (A) Z 2 π t =0 p 1 + ( π 2 + 4) t 2 dt (B) Z 2 t =0 p 1 + ( π 2 + 4) t 2 dt (C) Z 2 π t =0 1 + π 2 + 4 t 2 dt (D) Z 2 t =0 1 + π 2 + 4 t 2 dt (E) Z 4 t =0 t 1 + π 2 + t 2 dt (F) Z 2 t =0 t 1 + π 2 + t 2 dt (G) Z 2 π t =0 t 1 + π 2 + t 2 dt (H) None of the above. Answer to 6:
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8 (7) A spaceship flies along the parameterized curve 0 t < x ( t ) = e t ; y ( t ) = ( t - 1) 2 ; z ( t ) = 1 + t The temperature of space is given by τ ( x, y, z ) = xyz . At the exact point at which the spaceship is coldest along its path, what is the dot product of the spaceship’s velocity vector ˆ v with the gradient vector τ of temperature? Answer to 7:
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9 (8) Which of the following is the equation of the tangent plane to the surface x 3 - xy 2 + 4 xz = 9 when x = 1 and y = 2? Answer to 8:
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