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Ghella [55]
3 years ago
8

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the dimensions of a

Norman window of maximum area when the total perimeter is 16 feet.
Mathematics
1 answer:
Rzqust [24]3 years ago
6 0

Answer:

Step-by-step explanation:

Given Perimeter of window

P=16\ ft

Suppose x and y be the dimension of rectangular window

Perimeter of rectangular portion =x\cdot y

Perimeter of semicircle=\pi (0.5x)

so

2(x+y)+0.5\pi x=16

y=8-0.25\pi x-x

Area of the window is

A=xy+\frac{\pi x^2}{4}

A=x(8-0.25\pi x-x)+0.25\pi x^2

differentiate A w.r.t x to get maximum or minimum value

\frac{\mathrm{d} A}{\mathrm{d} x}=8-(0.5\pi +2)x+\frac{\pi x}{2}

put \frac{\mathrm{d} A}{\mathrm{d} x}=0 to get maxima/minima

thus x=4 ft

for x=4 , y=8-0.25\pi \cdot 4-4

y=0.858 ft

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