imc

Question

An ice cream cone consists of a sphere of vanilla ice cream and a right circular cone that has the same diameter as the sphere. If the ice cream melts, it will exactly fill the cone. Assume that the melted ice cream occupies 75% of the volume of the frozen ice cream. What is the ratio of the cone's height to its radius? (Note: a cone with radius r and height h has volume r^2 h / 3 and a sphere with radius r has volume 4 r^3 / 3.)

Accepted Answer

Let r be the radius of both the cone and the sphere of ice cream, and let h be the height of the cone. The cone can hold a maximum of r^2h/3 melted ice cream. The volume of the frozen ice cream is 4 r^3/3. The volume of the melted ice cream is 75 % of that, which is r^3. Since the melted ice cream fills the cone exactly, align* r^3&= r^2h/3 r&=h/3 3&=h/ralign* Then the ratio of the cone's height to its radius is h:r=3:1