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
Using the AM-GM inequality
With equality if and only if
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
and this happens when