The exact value of x is 4.48 ft so that greatest possible amount of light is admitted .
Norman window shap is rectangular surrounding by semi circle .
let x be the width and h be the height of window.
Therefore, radius of semi- circle is x/2 .
we have given that
perimeter of the window = 32 ft
the perimeter of semi-circle = π × x/2
the perimeter of rectangular = 2h + x
perimeter of window ( rectangular surrounding by semi circle) = 2h + x + πx/2 = 32 ---(1)
Area of rectangular part = x × h
Area of semi circle= π×(x/2)²
Area of window = π×(x²)/4 + xh ---(2)
Now, we must maximize the area of window so that greatest amount of light admitted.
using (1) , h = 16 - x/2 - πx/4
Area of window (A) = πx²/4 + 16x - x²/2 - πx²/4
A = 16x - x²----(3)
for maximize we take first order derivative of area and then put it equal to the zero for finding the critical value of x .
differentating equation (3) with respect to x we get, dA/dx = 16 - 2x
now, dA/dx = 0 so, 16 - 2x = 0 => x= 8
Hence , the exact value of x is 8 .
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Complete question:
Norman window has the shape of rectangle surmounted by semicircle: (Thus the diameter of the semicircle equal to the width of the rectangle labeled x. ) If the perimeter of the window Is 32 feet, find the exact value of x (In ft) so that the greatest possible amount of Iight Is admitted
