The dimension of rectangle window to admit the most light is x = 16 / ( 8 - √3) and y = 24 - 8√3 / 8 - √3
According to the question,
The perimeter of the window is 16 feet
Also, Window is surmounted by an equilateral triangle
Let x and y be the dimensions of rectangular part of window and x be side of equilateral part. If A be the total area of window, then
A = x.y + (√3/4)x² -------------(1)
Also, x + 2y + 2x = 16
=> 3x + 2y= 16
=> y = 16-3x / 2
Replacing the value of y in equation (1)
=> A = x.(16-3x / 2) + (√3/4)x²
=> A = 8x - 3x²/2 + (√3/4)x²
Differentiating A w.r.t x
=> A' = 8 - 3x + (√3/2)x
For critical value , we put A'=0
=> 8 - 3x + (√3/2)x = 0
=> x = 16 / ( 8 - √3)
Again, Differentiating A' w.r.t x ,
=> A'' = -3 + √3/2
which is Less than 0 ,
Therefore , A is maximum if x = 16 / ( 8 - √3) and y = 24 - 8√3 / 8 - √3
Hence, For admitting the most light the area should be largest , dimensions of rectangle are x = 16 / ( 8 - √3) and y = 24 - 8√3 / 8 - √3
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