Consider a right circular cylinder with radius and altitude . Find the right circular cylinder of largest lateral surface area that can be inscribed in this cone.

The lateral surface area of the cylinder is given by , where is the radius of the cylinder and is the height of the cylinder. From the diagram we find a formula for in terms of the constants and , and the radius of the cylinder ,

Thus, letting denote the lateral surface area we have

Taking the derivative of this with respect to and setting it equal to zero,

Since

Therefore, this critical point is a maximum.

Then plugging this back into our expression for we have