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

Let the length of rectangle be a and the breadth of rectangle be b.
The perimeter of the window is 2b + a + πa/2 = 2b + a(1 + π/2) = 10
⇒ b = 5 - a(1 + π/2)/2
The area of the window is A = ab + πa²/8 = a(5 - a(1 + π/2)/2) + πa²/8 = 5a - a²(1 + π/2)/2 + πa²/8
= 5a - a²(1 + π/4)/2
Now differentiate A and equate it to zero , we get
dA/da = 5 - a(1 + π/4) = 0
a = 5/(1 + π/4) = 20/(4 + π)
For the window to admit maximum light through the opening, the dimensions should be a = 20/(4 + π) and b = 5 - a(1 + π/2)/2 = 5 - [20/(4 + π)](1 + π/2)/2