Question

Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.

Answer 1

Let x and y be the window dimensions and x be the side of the equilateral portion.

Let A be the complete area of the window (through which light enters):

A = `xy + sqrt(3)/4 x^2`

Also, x + 2y + 2x = 12   ...(Given)

`\implies` 3x + 2y = 12

`\implies y = (12 - 3x)/2`

Then, A = `x xx ((12 - 3x)/2) + sqrt(3)/4x^2`

= `6x - (3x^2)/2 + sqrt(3)/4x^2`

Then, `(dA)/dx = 6 - 3x + sqrt(3)/2x`

For maximum light to enter, the area of the window should be the maximum

Put `(dA)/dx = 0`

`6 - 3x + sqrt(3)/2x = 0`

`x = 12/(6 - sqrt(3))`

Again, `(d^2A)/(dx^2) = -3 + sqrt(3)/2 < 0`  ...(For any value of x)

i.e., A is maximum if `x = 12/(6 - sqrt(3))` and 

`y = (12 - 3(12/(6 - sqrt(3))))/2`

= `(18 - 6sqrt(3))/(6 - sqrt(3))`

Hence dimensions are `(12/(6 - sqrt(3)))m`.

and `((18 - 6sqrt(3))/(6 - sqrt(3)))m`.