A window is the in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is of constant length k then its maximum area is

Let r be the radius of the circle and dimensions of rectangle are 2x,y. $\Rightarrow 2r+2y+\pi r=k$

 $$\begin{gathered} Area=2r\cdot y+\frac{1}{2}\pi r^2. \\ f\left(r\right)=2r\left(\frac{{\displaystyle k-\pi r-2r}}{{\displaystyle 2}}\right)+\frac{1}{2}\pi r^2 \end{gathered}$$

$f\text{'}\left(x\right)=k-2\pi r-4r+\pi r=0\Rightarrow k=(\pi +4)r\Rightarrow r=\frac{k}{\pi +4}$

$2y=k-(\pi +2)\frac{k}{\pi +4}=\frac{2k}{\pi +4}\Rightarrow y=\frac{k}{\pi +4}$

maximum area $=2\cdot \frac{k}{\pi +4}\cdot \frac{k}{\pi +4}+\frac{1}{2}\pi \frac{k^2}{{(\pi +4)}^2}=\frac{k^2(4+\pi )}{2{(\pi +4)}^2}=\frac{k^2}{2(\pi +4)}$