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

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