Volume of a cube inscribed into a cone?

Question

A right circular cone has a cube inscribed in it. (This means that the four upper vertices of the cube touch the inside of the cone at an unknown point) If the radius of the base of the cone is 1 and its height is 3, what is the volume of the cube?

This problem is something that I've been trying to figure out for a while, and am very curious about. It was given to me just as a sort of brain jump start in my Calculus III class, and it's been driving me crazy ever since! Anyone have any ideas as to how to solve this one?

Answer 1

the volume is .61094. the length of one side of the cube is .8485

i did it using similar triangles. for the smaller triangle. the height is 3-x, and the width is x root 2. The bigger triangle is height 3 and width 2. since they are similar, you can use a ration to solve for z. just let the ratios equal each other.