Question

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

• A. length $=\frac{20}{\pi +2}$, breadth $=\frac{10}{\pi +2}$
• B. length $=\frac{20}{\pi +3}$, breadth $=\frac{10}{\pi +3}$
• C. length $=\frac{40}{\pi +2}$, breadth $=\frac{20}{\pi +2}$
• D. length $=\frac{20}{\pi +4}$, breadth $=\frac{10}{\pi +4}$

Answer 1

The correct option is D length $=\frac{20}{\pi +4}$, breadth $=\frac{10}{\pi +4}$

Let the length of the window be x.
Let its breadth be y.
Hence total perimeter will be
$2y+x+\frac{\pi \left(x\right)}{2}=10$
Or
$y=\frac{20-x(2+\pi )}{4}$ ...(i)
Hence Area of the window will be
=Area of rectangle + Area of semicircle.
$A=xy+\frac{1}{2}\left(\frac{\pi \left(x^2\right)}{4}\right)$
$=\frac{20x-x^2(2+\pi )}{4}+\frac{\pi \left(x^2\right)}{8}$
Or
$=\frac{1}{8}[40x-2x^2(\pi +2)+\pi .x^2]$
Or
$\frac{1}{8}[40x-x^2(\pi +4\left)\right]$
Now
$\frac{dA}{dx}=0$
Or
$40-2x(\pi +4)=0$
Or
$20-x(\pi +4)=0$
Or
$x=\frac{20}{\pi +4}$.
Substituting in the expression of y, we get
$5-\frac{(2+\pi )}{4}\left(\frac{20}{\pi +4}\right)$
$=5-\frac{(10+5\pi )}{\pi +4}$
Or
$\frac{5\pi +20-10-5\pi }{\pi +4}$
Or
$y=\frac{10}{\pi +4}$.
Hence
$Length=\frac{20}{\pi +4}$units
And $Breadth=\frac{10}{\pi +4}$units.