20CFinalDec2008

20CFinalDec2008 - Math 20C Final Exam 150 points December 9...

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Math 20C Final Exam 150 points December 9, 2008 -------------------------------------------------------------------------------------------------------- Directions: Show all work and justify all answers. -------------------------------------------------------------------------------------------------------- (1) A solid cone of height 5 is cut into two pieces A and B, via a slice parallel to the base of the cone. Piece A is a cone of height 4. Which piece has the greater volume, piece A or piece B? Justify. (15 pts) ------------------------------------------------------------------------------------------------------------ (2) Use a triple integral to show that the volume of a sphere of radius 1 equals /3. (20 pts) ------------------------------------------------------------------------------------------------------------ (3) A mountainous surface z = f(x,y) is defined (implicitly) in the neighborhood of the point P = (1,4,1) by the equation 2x 2 + y 2 + 6z 2 = 6xyz . An aunt standing at
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