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Unformatted text preview: Answer to 2 : 4 (3) Which of the following vector fields F : R 2 → R 2 in the plane are conservative, in the sense that the value of a path integral of the vector field always just depends on the endpoints of the path? The formulas below express the value of F at a given point ( x,y ) as a linear combination of i = (1 , 0) and j = (0 , 1) (I) F = x i + y j . (II) F = y i + x j . (III) F = y 2 i + x 2 j . (A) I only (B) II only (C) III only (D) I and III (E) I and II (F) none of the above Answer to 3: 5 (4) Find the area of the parallelogram in three space whose vertices are at (0 , , 0), (1 , 1 , 1), (1 , 1 , 0), (2 , , 1). (A) √ 6 (B) √ 3 (C) 1 (D) 2 (E) 4 (F) none of the above Answer to 4: 6 (5) A silo is to be built having a flat circular bottom, cylindrical side, and a hemispher ical top. The silo is to have total volume (including the top cap) V = 900 π cubic feet. If the cost of the metal is 5 dollars/sq.ft. for the roof, 2 dollars/sq.ft. for the sides, and 1 dollar/sq.ft. for the floor, what should the ratio of height ( h ) to radius ( r ) of the cylindrical portion of the silo be in order to minimize the total cost of the materials needed? You can use without proof the usual formulas for the volume and surface areas of spheres and cylinders. For example, a sphere of radius r has volume 4 πr 3 / 3 and surface area 4 πr 2 . The volume of a solid cylinder of radius r and height h is πr 2 h . 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: 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 ,...
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 Fall '07
 Temkin
 Math, Calculus, Critical Point

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