Answer
$A=15x-\frac{x^2}{2}-\frac{x^2\pi}{8}$
Work Step by Step
The width of the rectangle is $x$. Let's define the height of the rectangle as $y$. Then, we know the area of the Norman window is the area of the rectangle plus the area of the semicircle. Knowing that the radius of the semicircle is $\frac{1}{2}x$ and that the area of a full circle is $\pi r^2$, the area for our Norman window is as follows.
$A=xy+\frac{1}{2}(\frac{1}{2}x)^2\pi$
The question asks us to express the area as a function of $x$, so we must replace $y$ by looking at the perimeter, $P$. Remember that the circumference of a whole circle is $2\pi r$. $P=2y+x+\frac{1}{2}x\pi=30 ft$
Now we solve for $y$.
$y=\frac{30-x-\frac{1}{2}x\pi}{2}=15-\frac{x}{2}-\frac{x\pi}{4}$
By plugging this into our area function above, we get:
$A=15x-\frac{x^2}{2}-\frac{x^2\pi}{4}+\frac{x^2\pi}{8}$.
$A=15x-\frac{x^2}{2}-\frac{x^2\pi}{8}$
