Homework SourseA norman window has the shape of a rectangle surmounted by a
A norman window has the shape of a rectangle surmounted by a semicircle(that is, the diameter of the semicircle is equal to the width of the rectangle). if the perimeter of the window is 6 meteres, find the dimensions of the window so that it allows the greatest possible light into the house
Solution
let length of rectangle be l and breadth be b...so the radius of semicircle is l/2
perimeter, P = l + 2b + l/2 = 6
2b = 6 - l - l/2
b = 3 - l/2 - l/4
area, A = l^2/4 + lb
A = l^2/4 + l(3-l/2 - l/2)
A = l^2/4 + 3l - l^2/2 - l^2/2
A = 3l - l^2/2 - l^2/4
dA/dl = 3 - l - l/2 = 0
3 = l + l/2
6 = (2+)l
l = 6/(2+)....................(1)
b = 3 - l/2 - l/4
b = 3 - 3/(2+) - 3/2(2+)
b = [6+6 - 6 - 3]/2(2+)
b = 3/2(2+)...............(2)
(1) & (2) are the dimensions
