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? (Note: A cone with radius $r$ and height $h$ has volume $\pi r^2 h / 3,$ and a sphere with radius $r$ has volume $4\pi r^3 / 3.)$

$2 : 1$

$3 : 1$

$4 : 1$

$16 : 3$

$6 : 1$

Solution:

The melted volume equals the cone's volume, so $\dfrac34 \cdot \dfrac43 \pi r^3 = \dfrac13 \pi r^2 h.$

Simplifying gives $\pi r^3 = \dfrac13 \pi r^2 h,$ so $h=3r.$ The ratio of height to radius is $3:1.$

Thus, the correct answer is B.

2003 AMC 10B Problem 17

Solution:

Problem 17 in Other Years