+41
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
+129899 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
+118687 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.
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