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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?

 

https://study.com/cimages/multimages/16/graph33-resizeimage6659431018903054284.jpg

 May 19, 2019
 #1
avatar+103097 
+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

 

 

 

cool cool cool

 May 19, 2019
edited by CPhill  May 19, 2019
edited by CPhill  May 22, 2019
 #2
avatar+45 
+1

thanks so much, Cphil

DungeyDabs  May 19, 2019
 #3
avatar+45 
-1

however the answer is wrong

DungeyDabs  May 19, 2019
 #4
avatar+103097 
0

OK....let me see if I can spot my error....

 

 

cool cool cool

CPhill  May 19, 2019
 #5
avatar+103097 
0

 

 

cool cool cool

CPhill  May 19, 2019
edited by CPhill  May 19, 2019
 #6
avatar+103688 
0

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

 

For instance,

 

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

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