Hello, leftyguitarjoe!

Your answer has the area in terms of both $\displaystyle x$ and $\displaystyle h.$

Your answer should have $\displaystyle x$ 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 $\displaystyle A$ of the window

as a function of the width $\displaystyle x$ of the window. Code:

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

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

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

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

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

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

The area of the rectangle is: .$\displaystyle xh$

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

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

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

But check my work . . . please!

.