Problem:
When a right triangle is rotated about one leg, the volume of the cone produced is . When the triangle is rotated about the other leg, the volume of the cone produced is . What is the length (in ) of the hypotenuse of the triangle?
Solution:
The volume of a cone with a circular base of radius and height is given by . Denoting the length of the legs of the right triangle by and , this implies that
Dividing the first of these equations by the second one yields . Hence, and . It follows that and that the triang1e's hypotenuse (by Pythagoras' Theorem) is .
The problems on this page are the property of the MAA's American Mathematics Competitions