# Thread: Norman window problem

1. ## Norman window problem

A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 38 ft, express the area A of the window as a function of the width x of the window.

What I have is A = pi(x)/2+xh

I know it gives you the 38 ft for a reason. And the fact its highlighted in red also hints that I should have used it somewhere.

2. Hello, leftyguitarjoe!

Your answer has the area in terms of both $x$ and $h.$
Your answer should have $x$ only.

A Norman window has the shape of a rectangle surmounted by a semicircle.
If the perimeter of the window is 38 ft, express the area $A$ of the window
as a function of the width $x$ of the window.
Code:
* * *
*           *
*               *
*                 *

*                   *
* - - - - * - - - - *
|                   |
|                   |
h |                   | h
|                   |
|                   |
* - - - - - - - - - *
x

The semicircle has diameter $x$ and radius $\frac{x}{2}$

The circumference of the semicircle is: . $\frac{1}{2} \times \pi x \:=\:\frac{\pi}{2}x$

The perimeter is: . $P \;=\;\frac{\pi}{2}x + x + 2h \:=\:38 \quad\Rightarrow\quad 2h + \left(\frac{\pi+2}{2}\right)x \:=\:38$

. . $2h \:=\:38 - \left(\frac{\pi+2}{2}\right)x \quad\Rightarrow\quad h \:=\:19 - \left(\frac{\pi + 2}{4}\right)x$ .[1]

The area of the semicircle is: . $\frac{1}{2}\pi r^2 \:=\: \frac{\pi}{2}\left(\frac{x}{2}\right)^2 \:=\: \frac{\pi}{8}x^2$

The area of the rectangle is: . $xh$

The total area is: . $A \;=\;\frac{\pi}{8}x^2 + xh$ .[2]

Substitute [1] into [2]: . $A \;=\;\frac{\pi}{8}x^2 + x\left[19 - \left(\frac{\pi+2}{4}\right)x\right]$

. . which simplifies to: . $A \;=\;19 x - \left(\frac{\pi +4}{8}\right)x^2$

But check my work . . . please!
.