Hello, Evan.Kimia!

Your answer is "close", so you're doing most of it Right.

You probably made a very simpler error.

A Norman window has the shape of a rectangle surmounted by a semicircle.

If the perimeter of the window is 20ft, find the value of $\displaystyle x$

so that the greatest possible amount of light is admitted.

For my critical number, i get 80/(pi + 4) . It shoud be: .$\displaystyle {\color{blue}\frac{20}{\pi + 4}}$ Code:

* * *
* *
* *
* *
* - - - - * - - - - *
| r r |
| |
y| |y
| |
| |
* - - - - - - - - - *
2r

The radius of the semicircle is $\displaystyle r.$

Hence, the width of the rectangle is $\displaystyle 2r.$

The height of the rectangle is $\displaystyle y.$

The perimeter of the semicircle is: .$\displaystyle \pi r$

The perimeter of the rectangle is: .$\displaystyle 2r + 2y$

The total perimeter is 20:

. . $\displaystyle \pi r + 2r + 2y \:=\:20 \quad\Rightarrow\quad y \:=\:10 - \left(\frac{\pi+2}{2}\right)r $ .[1]

The area of the semicircle is: .$\displaystyle \tfrac{1}{2}\pi r^2$

The area of the rectangle is: .$\displaystyle (2r)(y) $

The total area is: .$\displaystyle A \;=\;\tfrac{1}{2}\pi r^2 + 2ry $ .[2]

Substitute [1] into [2]: .$\displaystyle A \;=\;\tfrac{1}{2}\pi r^2 + 2r\bigg[10 - \frac{\pi+2}{2}\,r\bigg]$

. . And we have: .$\displaystyle A \;=\;\tfrac{1}{2}\pi r^2 + 20r - (\pi+2)r^2$

Differentiate and equate to zero: .$\displaystyle A' \;=\;\pi r + 20 - 2(\pi+2)r \;=\;0$

. . . $\displaystyle (\pi+4)r \:=\:20 \quad\Rightarrow\quad\boxed{ r \:=\:\frac{20}{\pi+4}}$

Substitute into [1]: .$\displaystyle y \:=\:10 - \left(\frac{\pi+2}{2}\right)\left(\frac{20}{\pi+4} \right) \quad\Rightarrow\quad\boxed{y \;=\;\frac{20}{\pi+4}} $