Question

A window is in the form of a rectangle surmounted by a semi-circular opening. The total perimeter of the window is 10 meters. Find the dimensions of the rectangle so as to admit maximum light through the whole opening.

Answer 1

Correct Answer - breadth `= (20)/((pi + 4))m`, height `= (10)/((pi + 4))m`
Let the length and breadth of the rectangle be x and y metres respectively. Then, radius of the semicircle `= (x//2)` metres
So, the perimeter of the window is given by
`10 = x + 2y + (pix)/(2) rArr y = 5 - ((2 + pi)/(4))x`
`:. A = xy + (1)/(2) pi ((x)/(2))^(2) rArr A = x [5- ((2 + pi))/(4) .x] + (pi x^(2))/(8)`
`:. (dA)/(dx) = (5 - x - (pix)/(4)) and (d^(2)A)/(dx^(2)) = - (1 + (pi)/(4)) lt 0`
Now, `(dA)/(dx) = 0 rArr x = (20)/((pi + 4))`.
`:.` A is maximum when `x = (20)/((pi + 4)) and y = (10)/((pi + 4))`