Given a semicircle of radius , find the largest rectangle (in terms of volume) that can be inscribed in the semicircle, with base lying on the diameter.
Let be the radius of the semicircle, one half of the base of the rectangle, and the height of the rectangle. We want to maximize the area, . Referencing the diagram we have
Setting this derivative equal to 0 and solving for ,
This is a maximum of the area since
Since we then have
Thus, the base of the rectangle has length and its height has length .