## One Solution:

Using the AM-GM inequality, what is the maximum volume of a right circular cylinder that can be inscribed in a sphere of radius R.

We can argue easily that such a cylinder exists.

Take a cross-section of the sphere and the inscribed cylinder through the center of the sphere. Let r be the radius of the cylinder and h its height.

The volume is given by

and we have the Pythagorean relationship

We can solve for h to get

Now,

Using the AM-GM inequality

With equality if and only if

This gives

So the volume is always less than or equal to a constant and it can reach that constant, the maximum volume, if and only if

We have

and this happens when

Note that