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

Answer: 26

Concepts:cone volume Pythagorean Theorem

Difficulty rating: 1890

Small Hint:

Write the two cone volumes in terms of the triangle’s legs $a$ and $b$

Big Hint:

Dividing the volume equations gives the ratio of the two legs

Solution:

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

1985 AIME Problem 2

Answer: 26

Difficulty rating: 1890

Small Hint:

Big Hint:

Solution:

Problem 2 in Other Years