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jmc

geometry intermediate

Problem

A right cone has a base with a circumference of inches and a height of 30 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fraction.
Solution
Let the cone have radius inches; we have , so . Let the new height of the cone be inches. We have ; solving yields . Thus the ratio of the new height to the original height is .
Final answer
\frac{3}{10}