Problem:
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 of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius?
Answer Choices:
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Solution:
Let be the radius of the sphere and cone, and let be the height of the cone. Then the conditions of the problem imply that
Therefore, the ratio of to is .
The problems on this page are the property of the MAA's American Mathematics Competitions