Anorman 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?

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