Find the rectangle with the maximum area which can be inscribed in a semicircle.

You are given a semicircle of radius 1 ( see the picture on the left ).
It is possible to inscribe a rectangle by placing its two vertices on
the semicircle and two vertices on the *x*-axis. Start moving the mouse
pointer over the left figure and watch the rectangle being resized.

Let's compute the area of our rectangle. If (*x*,*y*) are the coordinates of
*P*, then we can express the area as

The picture on the right presents a graph of

We can express *A* as a function of *x* by eliminating *y*. Since
*P* lies on a semicircle of radius 1, *x*^{2}+*y*^{2}=1. Solving for *y* and substituting for *y* in *A*, we have

Now it is straighforward to find the exact value of

The Java applet which shows the graphs above was written by Marek Szapiel.

He also wrote most of the rest of the page.

The Javascript was written by Larry Husch.

He also wrote most of the rest of the page.

The Javascript was written by Larry Husch.