A Norman window is a window with a semi-circle on top of regular rectangular window. (See the link.) 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?

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DungeyDabs May 19, 2019

#1**+2 **

We have the following perimeter

2x + 2y + pi x / 2 = 12 divide through by 2

x + y + (1/4)pi x = 6

(1 + (1/4)pi ) x + y = 6

y = 6 - (1 + .25pi)x

So....the total Area , A, to be maximized is

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

A = x * [ 6 - (1 + .25pi)x] + .25pix^2 /2 simplify

A = 6x - (1 + .25pi)x^2 + .125pix^2

A = 6x - x^2 - .25pix^2 + .125pix^2

A = 6x - x^2 - .125pix^2

A= 6x - ( 1 + .125pi)x^2

This might be most easily solved with a graph: https://www.desmos.com/calculator/brhi8ngjzf

The value of x that maximizes the area ≈ 2.154 ft

And y = [6 - (1 + .25pi)(2.154) ] ≈ 2.154 ft

So the rectangular part = 2.154 ft x 2.154 ft

Corrected Answer

CPhill May 19, 2019

#6**0 **

If you tell someone that their answer is wrong then you need to give your reason for thinking so:

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Thanks for the time you spent helping me CPhill but I think you answer is wrong because my teacher tells me that the answer is 42.

That is just an example of what you might write.

Melody
May 20, 2019