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 $x$ 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 $x$, 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 $x$ can be: when $x$ is large enough, the rectangular portion of the window shrinks down to zero height. What is the exact largest value of $x$ when this occurs?
largest $x$: $\mathrm{cm}$.

Determine a function $A(x)$ which gives the area of the window in terms of the parameter $x$ (this is the function plotted above):
$A(x) =$ $\mathrm{cm}^2$.

Now find the exact base length $x$ which maximizes this area:
$x =$ $\mathrm{cm}$.