Problem:
An ice-cream novelty item consists of a cup in the shape of a 4-inchtall frustum of a right circular cone, with a 2-inch-diameter base at the bottom and a 4-inch-diameter base at the top, packed solid with ice cream, together with a solid cone of ice cream of height 4 inches, whose base, at the bottom, is the top base of the frustum. What is the total volume of the ice cream, in cubic inches?
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Solution:
A frustum is constructed by removing a right circular cone from a larger right circular cone. The volume of the given frustum is the volume of a right circular cone with a 4-inch-diameter base and a height of 8 inches, minus the volume of a right circular cone with a 2-inch-diameter base and a height of 4 inches. (The stated heights come from considering similar right triangles.) Because the volume of a right circular cone is , the volume of the frustum is
The volume of the top cone of the novelty is . The requested volume of ice cream is the sum of the volume of each part of the novelty, namely .
Note: In general, the volume of a frustum with height and base radii and is .
The problems on this page are the property of the MAA's American Mathematics Competitions