A window in the form of a rectangle is surmounted by a semi-circular opening.


A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.


Let the dimensions of the rectangular part be $x$ and $y$.

Radius of semi-circle $=\frac{x}{2}$

Total perimeter $=10$

$\Rightarrow(x+2 y)+\pi\left(\frac{x}{2}\right)=10$

$\Rightarrow 2 y=\left[10-x-\pi\left(\frac{x}{2}\right)\right]$

$\Rightarrow y=\frac{1}{2}\left[10-x\left(1+\frac{\pi}{2}\right)\right]$             ......(1)


Area, $A=\frac{\pi}{2}\left(\frac{x}{2}\right)^{2}+x y$

$\Rightarrow A=\frac{\pi x^{2}}{8}+\frac{x}{2}\left[10-x\left(1+\frac{\pi}{2}\right)\right]$        $[$ From eq. $(1)]$

$\Rightarrow A=\frac{\pi x^{2}}{8}+\frac{10 x}{2}-\frac{x^{2}}{2}\left(1+\frac{\pi}{2}\right)$

$\Rightarrow \frac{d A}{d x}=\frac{\pi x}{4}+\frac{10}{2}-\frac{2 x}{2}\left(1+\frac{\pi}{2}\right)$

For maximum or minimum values of $A$, we must have

$\frac{d A}{d x}=0$

$\Rightarrow \frac{\pi x}{4}+\frac{10}{2}-\frac{2 x}{2}\left(1+\frac{\pi}{2}\right)=0$

$\Rightarrow x\left[\frac{\pi}{4}-1-\frac{\pi}{2}\right]=-5$

$\Rightarrow x=\frac{-5}{\left(\frac{4-x}{4}\right)}$

$\Rightarrow x=\frac{20}{(\pi+4)}$

Substituting the value of $x$ in eq. $(1)$, we get


$\Rightarrow y=5-\frac{10(\pi+2)}{2(\pi+4)}$

$\Rightarrow y=\frac{5 \pi+20-5 \pi-10}{(\pi+4)}$

$\Rightarrow y=\frac{10}{(\pi+4)}$

$\frac{d^{2} A}{d x^{2}}=\frac{\pi}{4}-\frac{\pi}{2}-1$

$\Rightarrow \frac{d^{2} A}{d x^{2}}=\frac{\pi-2 \pi-4}{4}$

$\Rightarrow \frac{d^{2} A}{d x^{2}}=\frac{-\pi-4}{4}<0$

Thus, the area is maximum when $x=\frac{20}{\pi+4}$ and $y=\frac{10}{\pi+4}$.

So, the required dimensions are given below :

Length $=\frac{20}{\pi+4} \mathrm{~m}$

Breadth $=\frac{10}{\pi+4} \mathrm{~m}$

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