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 trough the whole opening.

Answer 1

Let the length of rectangle be $a$ and the breadth of rectangle be $b$

The perimeter of the window is $2b+a+\frac{\pi a}{2}=2b+a(1+\frac{\pi }{2})=10$

$\Rightarrow b=5-\frac{a}{2}(1+\frac{\pi }{2})$

The area of the window is $A=ab+\frac{\pi a^2}{8}=5a-\frac{a^2}{2}(1+\frac{3\pi }{8})$

Now differentiate $A$ and equate it to zero , we get $a=\frac{5}{1+\frac{3\pi }{8}}$

For the window to admit maximum light through the opening , the dimensions should be $a=\frac{5}{1+\frac{3\pi }{8}}$