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


Let the radius of semicircle, length and breadth of rectangle be r, x and y respectively


AE = r


AB = x = 2r (semicircle is mounted over rectangle) …1


AD = y


Given: Perimeter of window = 10 m


x + 2y + πr = 10


2r + 2y + πr = 10


2y = 10 – (π + 2).r


y = … 2


To admit maximum amount of light, area of window should be maximum


Assuming area of window as A


A = xy +


A = (2r) () +


A = 10r - πr2 – 2r2 +


A = 10r – 2r2 -


Condition for maxima and minima is



10 – 4r - πr = 0



= - 4 - π < 0


For r = A will be maximum.


Length of rectangular part = (from equation 1)


Breath of rectangular part = (from equation 2)




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