Question

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.

Answer 1

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 :)

Answer 2

is the 160 suppose to be 16 ?

Answer 3

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

Answer 4

okay thanks

Answer 5

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