2003 AMC 12B Problem 13

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

Answer: B

Concepts:cone sphere volume

Difficulty rating: 1490

Small hint:

The sphere and cone share the same radius

r;

sphere volume is

\dfrac{4}{3}\pi r^3

Big hint:

Set

75\%

of the sphere’s volume equal to the cone’s volume

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

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.