Question

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?

$2 : 1$

$3 : 1$

$4 : 1$

$16 : 3$

$6 : 1$

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 $\pi r^3 = \dfrac{1}{3}\pi r^2 h,$ so $h = 3r,$ a ratio of $3 : 1.$

Thus, the correct answer is B.

2003 AMC 12B Problem 13

Solution:

Problem 13 in Other Years