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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.
Solution
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`.
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