a) Find the dimensions of a Norman window of maximum area if the perimeter is 16

Question

                    a) Find the dimensions of a Norman window of maximum area if the perimeter is 16 feet. Give exact answers as well as decimal approximations.                 

                    
                

                    b) Verify that your answer yields maximum area.                 

                    
                

                    c) What is the maximum area? Give an exact answer as well as decimal approximations.                 

Find the dimensions of a Norman window of maximum area if the perimeter is 16 feet. Give exact answers as well as decimal approximations. Verify that your answer yields maximum area. What is the maximum area? Give an exact answer as well as decimal approximations.

Explanation / Answer

Let x denote half the width of the rectangle (so x is the radius of the semicircle),

and let h denote the height of the rectangle. Then the perimeter of the window is

2y + 2x +1/2(2?x) = 2y + (2 + ?)x.

Since the perimeter is 16, we have that

y =[16 ? (2 + ?)x]/2= 8-(2+?)x/2....................*

Therefore, the area of the window (which is proportional to the amount of light admitted),

is given by

A = (2x)y +1/2(?x^2) = 2xy +?x^2/2....................**

.

Substituting the above value for y yields

A(x) = 2x[8-(2+pi)x/2]+ pi x^2/2

and for maximizing we find A' and equate to zero

we get x=16/4+pi

x=2.24039

by equation * , y=2.24039

maximum area,by equation ** =17.9230

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