Consider a sphere of fixed radius . Find the right circular cone of maximum volume that can be inscribed in the sphere in terms of , and the radius and altitude of the cone, and , respectively.
We want to maximize the volume of the cone,
From the diagram we find the following expression for in terms of and ,
Thus, our expression for in terms of is
Taking the derivative of this we have
Setting this equal to 0 we have
(We used that a couple of times in the computation, which is fine since the cone does not have radius 0.) This critical point is a maximum since
Then, plugging this value of back into our expression for we have