Given a right circular cone with radius and height , find the right circular cylinder of maximum volume that can be inscribed in the cone.

The setup is the same as in the previous exercise, except this time we want to maximize the volume instead of the lateral surface area. Again, we have the following expression for ,

Then,

Taking the derivative of this with respect to ,

Setting this equal to 0 we have,

Plugging this into our expression for we then have