A Norman window has the shape of a rectangle surmounted by a semicircle. Suppose the outer perimeter of such a window must be 600 cm. In this problem you will find the base length [math] which will maximize the area of such a window. The applet above shows a plot of the area function. Use the slider to visualize how the area changes for different values for [math], and use the corresponding graph to estimate the optimal radius. Then use calculus to find an exact answer. (Correction: In the figure "r" should be "x").

When the base length is zero, the area of the window will be zero. There is also a limit on how large [math] can be: when [math] is large enough, the rectangular portion of the window shrinks down to zero height. What is the exact largest value of [math] when this occurs?
largest [math]: [math].

Determine a function [math] which gives the area of the window in terms of the parameter [math] (this is the function plotted above):
[math] [math].

Now find the exact base length [math] which maximizes this area:
[math] [math].