Answer
Area of window = $\frac{\pi x^2}{8} + x(\frac{30-x-\pi x/2}{2})$ = $f(x)$
Work Step by Step
As is clear from the diagram, width of the window = diameter of the semicircular part.
Let the length of the rectangular portion of the window be $l$.
Total perimeter of the window = length of the semicircular arc + length of the rectangle *2 + $x$ =
$\pi(x/2) + 2l + x = 30$
So, $l = \frac{30-x-\pi x/2}{2}$
Area of window = area of semicircle + area of rectangle =
$\frac{\pi (x/2)^2}{2} + xl$
Area of window = $\frac{\pi x^2}{8} + x(\frac{30-x-\pi x/2}{2})$ = $f(x)$
