# A Norman window is constructed by adjoining a semicircle to

Khaleesi Herbert 2021-08-06 Answered
A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a Norman window of maximum area if the total perimeter is 22 feet. Radius is x/2.
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Let represent the measure of the vertical dimension of the rectangular portion of the window. Let represent the horizontal dimension of the window. is also the diameter of the semi-circular part of the window. The perimeter of the window is then: $22=x+2y+\frac{1}{2}\pi x$
${P}_{Total}=\left({P}_{Rectangle}-x\right)+\frac{{P}_{Circle}}{2}$
$10-\frac{1}{2}x-\frac{1}{5}\pi x=y$
$A=xy+\frac{\pi {\left(\frac{1}{2}x\right)}^{2}}{2}$
${A}_{Total}={A}_{Retangle}+\frac{{A}_{Circle}}{2}$
$A=x\left(10-\frac{1}{2}x-\frac{1}{5}\pi x\right)+\frac{\pi {\left(\frac{1}{2}x\right)}^{2}}{2}$
Plug in $10-\frac{1}{2}x-\frac{1}{5}\pi x$ for $y$
$A=10x-\frac{1}{2}{x}^{2}-\frac{\pi {x}^{2}}{10}$
$\frac{dA}{dX}=10-x-\frac{\pi x}{5}$
Power rule $\frac{d}{dx}{x}^{n}=n{x}^{n-1}$
$0=10-x-\frac{\pi x}{5}$
Set $\frac{dA}{dx}$ equal to 0
$x+\frac{\pi x}{5}=10$
Add $x+\frac{\pi x}{5}$ to get x alone
$x\left(1+\frac{\pi }{5}\right)=10$
$x=\frac{10}{1+\frac{\pi }{5}}$
$x=\frac{40}{4+\pi }$
$\frac{{d}^{2}A}{{dx}^{2}}=-\frac{\left(4+\pi \right)}{4}<0$ Solve for y:
$y=10-\frac{\left(2+\pi \right)x}{4}$
Hence $x=\frac{40}{4+\pi }$ is the x-coordinate of the maximum point. The y dimensoin and the semi-circle redius are calculated as above.