1985 AIME Problems/Problem 2
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 cm) of the hypotenuse of the triangle?
Let one leg of the triangle have length and let the other leg have length . When we rotate around the leg of length , the result is a cone of height and radius , and so of volume . Likewise, when we rotate around the leg of length we get a cone of height and radius and so of volume . If we divide this equation by the previous one, we get , so . Then so and so . Then by the Pythagorean Theorem, the hypotenuse has length .
Let , be the legs, we have the equations Thus . Multiplying gets Adding gets Let be the hypotenuse then
Let and be the two legs of the equation. We can find by doing . This simplified is . We can represent the two legs as and for and respectively.
Since the volume of the first cone is , we use the formula for the volume of a cone and get . Solving for , we get .
Plugging in the side lengths to the Pythagorean Theorem, we get an answer of .
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