A log is the shape of the frustum of a right circular cone with height 12 feet, base with diameter feet, and top diameter of 4 feet. Find the volume of the largest right circular cylinder that can be cut from the log if the axis of the cylinder coincides with the axis of the log.

Let denote the radius of the right circular cylinder, and its height. Then, the volume is given by

First, we find an equation for in terms of . A graph depicting the edge of the frustum will help:

So, in the graph, lies on the line connecting the origin to the top edge of the frustum. The equation of this line is since it has slope (since the height of the frustum is 12 feet and the horizontal distance between the lower edge and upper edge is ). Therefore,

Hence, we have the volume of the cylinder we are cutting out given by

Since we want to maximize this volume, we take the derivative with respect to and look for critical points,

Setting this equal to 0 we have,

We also have an additional constraint from the problem which is that we should require since the maximum height of the cylinder we are cutting is 12 feet and

Since this implies that this value is valid for . So if we have

Plugging this into the formula for the volume of the cylinder we obtain a maximum for the volume of

If then we have and so . Therefore,

Putting this together into one equation we have