Question

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is $10m$. Find the dimensions of the window to admit maximum light through the whole opening.

Answer 1

Let the width and height of window be $2x$ $m$ and $ym$ respectively. Then, perimeter $=10m$.

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

Let $A$ be the area of the window. then

$A=2xy+\frac{\pi }{2}{x}^2$

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


$\Rightarrow \frac{dA}{dx}=10-2x(\pi +2)+\pi x$ and,

$\frac{d^{2}A}{d{x}^2}=-2(\pi +2)+\pi =-\pi -4$

For maximum value of $A$, we must have

$\frac{dA}{dx}=\Rightarrow 10-2x(\pi +2)+\pi x=0\Rightarrow x=\frac{10}{\pi +4}$

Clearly, $\frac{d^{2}A}{d{x}^2}=-\pi -4<0$ for all $x$.

So, $A$ is maximum when $x=\frac{10}{\pi +4}$ and

$y=5-\frac{1}{2}\times \frac{10}{\pi +4}(\pi +2)$

$=\frac{5\pi +20-5\pi -10}{\pi +4}=\frac{10}{\pi +4}$

Hence, the dimensions of the window are

$2x=\frac{20}{\pi +4}$ m and $y=\frac{10}{\pi +4}$ m.