Question 1198597: The lateral surface of a right circular cone is cut along an element and the surface is “spread out” on a plane. The resulting figure is a quadrant of a circle whose radius is 6 in. The volume of the cone may be expressed as V= (xπ√γ)/z 〖in〗^3 where, x, y, and z are positive integers. Find the smallest sum of x, y, and z.
Answer by greenestamps(13200)   (Show Source):
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The radius of the quarter circle is 6. The circumference of the whole circle would be 12pi, so the length of the curved part of the quarter circle is 3pi.
That 3pi is the circumference of the original cone; that means the radius of the base of the cone is (3pi)/(2pi) = 3/2.
The radius of the quarter circle is the slant height of the cone. Using the Pythagorean Theorem with the radius and slant height of the cone, the height of the cone is
The volume of the cone is then
So x=9, y=15, and z=8.
ANSWER: 9+15+8 = 32
NOTE: Expressing the volume of the cone with the radical NOT in simplest form would produce larger sums for x+y+z, so 32 is the smallest possible sum.
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