A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 metres. Find the dimensions of the window to admit maximum light through the whole opening. How having large windows help us in saving

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 metres. Find the dimensions of the window to admit maximum light through the whole opening. How having large windows help us in saving electricity and conserving environment ?

Answers (1)

Let ABCD be a rectangle and let the semi circle is described on the side AB as its diameter
Let AB = 2x and AD = 2y
Let P = 10 m be the given perimeter of window
$\therefore 10= 2x+4y+\pi x$
$\Rightarrow 4y= 10-2x-\pi x---(i)$
Area of the window , $A= \left ( 2x \right )\left ( 2y \right )+\frac{1}{2}\pi x^{2}$
$\Rightarrow A= 4xy+\frac{1}{2}\pi x^{2}$
$\Rightarrow A= 10x-2x^{2}-\pi x^{2}+\frac{1}{2}\pi x^{2}\left [ By(i) \right ]$
$\Rightarrow A= 10x-2x^{2}-\frac{1}{2}\pi x^{2}$
On differentaiating w.r.t x both side
$\frac{dA}{dx}= 10-4x-\pi x$
Again differentaiating w.r.t x both side
$\frac{d^{2}A}{dx^{2}}= -\left ( 4+\pi \right )$

for local points of maxima & minima , $\frac{dA}{dx}= 0\Rightarrow 10-4x-\pi x$
$= 0\Rightarrow x= \frac{10}{4+\pi }$
$\therefore \frac{d^{2}A}{dx^{2}}_{dx= \frac{10}{4+\pi }}= -\left ( 4+\pi \right )< 0,$ so, A is maximum at
$x= \left ( \frac{10}{4+\pi } \right )M$
Now, length of the window is $2x= \left ( \frac{20}{4+\pi } \right )M$
and width is $2y= \left ( \frac{10}{4+\pi } \right )M$
Also the radius of the semi-circular opening
$x= \left ( \frac{10}{4+\pi } \right )M$
VBQ: If the window is large maximum light will bw powerd and we don,t require to one electric bulb, so we can save electricity and have sources the enviornment