1985 AIME Problem 2

Question

When a right triangle is rotated about one leg, the volume of the cone produced is

800\pi\text{ cm}^3.

When the triangle is rotated about the other leg, the volume of the cone produced is

1920\pi\text{ cm}^3.

What is the length (in cm) of the hypotenuse of the triangle?

Answer

Write the two cone volumes in terms of the triangle’s legs

a

and

b

Dividing the volume equations gives the ratio of the two legs

Let the legs be

a

and

b.

In a suitable order,

\begin{aligned} \frac13\pi b^2a&=800\pi,\\ \frac13\pi a^2b&=1920\pi. \end{aligned}

Their ratio gives

\frac{a}{b}=\frac{12}{5},

so write

a=12k

and

b=5k.

The first equation becomes

100k^3=800,

hence

k=2.

The legs are

24

and

10,

so the hypotenuse is

\sqrt{24^2+10^2}=26.

Problem 2 in other years