## anonymous 5 years ago A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 16 ft, express the area A of the window as a function of the width x of the window.

1. watchmath

Let $$h$$ be the height of the rectangle. Here $$x/2$$ is the radius of the semicirlce. So perimeter= $$2h+x+\pi\cdot (x/2)=2h+(2\pi+1)x=160$$ Solving for $$h$$ we have $$h=80-(2\pi+1)(x/2)$$. Now the area of the window is $$A=hx+\frac{1}{2}\pi(x/2)^2$$ $$=80x-(2\pi+1)(x^2/2)-\pi(x^2/4)$$ You can simplify more if you want :)

2. anonymous

is the 160 suppose to be 16 ?

3. watchmath

yes sorry, that was a typo then $$h=16-(2\pi+1)(x/2)$$ and you continue from there :).

4. anonymous

okay thanks

5. watchmath

I mean $$h=8-(2\pi+1)(x/2)$$