2003 AMC 12B Problem 13

Problem 13 of 25IntermediateGeometry

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies

75\%

of the volume of the frozen ice cream. What is the ratio of the cone’s height to its radius?

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Solution

Let

r

be the common radius and

h

the cone’s height. The melted ice cream fills the cone, so

\frac{3}{4}\cdot\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2 h.

This simplifies to

\pi r^3 = \dfrac{1}{3}\pi r^2 h,

h = 3r,

a ratio of

3 : 1.

Thus, the correct answer is B.

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