Problem:
A container in the shape of a right circular cone is inches tall and its base has a -inch radius. The liquid that is sealed inside is inches deep when the cone is held with its point down and its base horizontal. When the cone is held with its point up and its base horizontal, the liquid is inches deep, where , and are positive integers and is not divisible by the cube of any prime number. Find .
Solution:
When the cone is held point down, the liquid in the container forms a cone that is similar to the container, the ratio of similarity being . Thus the volume of the liquid is times the volume of the container. When the cone is held point up, the air in the container forms a cone whose height is and whose volume is times the volume of the container. Because the cone of air is similar to the container, , so . It follows that the depth of the liquid is . Thus .
The problems on this page are the property of the MAA's American Mathematics Competitions