Question

A Norman window is a window with a semi-circle on top of regular rectangular window. What should be the dimensions of the rectangular part of a Norman window to allow in as much light as possible, if there are only 12 ft of the frame material available?

Answer 1

Answer:

The dimensions of the rectangular part that allows as much light as possible are: 2.154 ft x 2.154 ft

Step-by-step explanation:

The dimensions of the Norman window are:

2x + 2y  +  pi x / 2  =   12 ft                               You should divide through by 2

x +  y  +   (1/4)pi x  =  6  ft                                   X is cleared from multiplication

(1 + (1/4)pi ) x  + y  =  6  ft                                    We clear Y

y  =  6  - (1 + 0.25pi)x

So the total Area , A, to ensure the greatest amount of light is:

A  = ( x * y  + pi  [(1/2)x]^2 ) / 2                           Substituting y, we have

A =  ( x * [ 6 - (1 + 0.25pi)x] + 0.25pix^2 ) / 2       Simplify

A = 6x - (1 + 0.25pi)*x^2 + 0.125pix^2

A = 6x - x^2 - 0.25pix^2 + 0.125pix^2

A = 6x - x^2 - 0.125pix^2

A= 6x -  ( 1 + 0.125pi)x^2

The value of x that maximizes the area ≈  2.154 ft

And y = [6 - (1 + 0.25pi)(2.154) ] ≈ 2.154 ft

 So  the rectangular part should be  =  2.154 ft x 2.154 ft

Answer 2

The dimensions of the rectangular portion of the window should be 4 by 4.

To maximize area, we make the dimensions as close to congruent as possible.  Since there will be 3 sides on the rectangular portion with a frame around them, we divide 12 by 3:
12/3 = 4

This means it should be 4 by 4.