2003 AMC 10B Problem 17

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

\frac{\pi r^2 h}{3},

and a sphere with radius

r

has volume

\frac{4\pi r^3}{3}.

)

2 : 1

3 : 1

4 : 1

16 : 3

6 : 1

Answer: B

Concepts:volume sphere cone

Difficulty rating: 1370

Small hint:

The cone and sphere share the same radius

r

Big hint:

Set

\tfrac34 \cdot \tfrac43 \pi r^3

equal to the cone volume

\tfrac13 \pi r^2 h

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.