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A cone is sliced by a plane parallel to its base at one-fourth of its height measured from the apex. What is the ratio of the volume of the small top cone to the volume of the remaining bottom frustum?

1. 1 : 15
2. 1 : 63
3. 1 : 64
4. 1 : 27

Correct answer: 1 : 63

Solution

Cutting at one-fourth of the height from the top makes the small top cone similar to the full cone with height ratio 1/4, so its volume is (1/4)^3 = 1/64 of the whole. The frustum is the remaining 64 - 1 = 63 parts, giving top : frustum = 1 : 63.

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