SOLUTION: Hey Guys, came across this problem in my maths homework and I can't seem to solve it. Can someone maybe help me out? A Norman window has the shape of a rectangle with a s

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Question 1057214: Hey Guys,
came across this problem in my maths homework and I can't seem to solve it.
Can someone maybe help me out?
A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions w * h of the Norman window whose perimeter is 500in. that has maximal area.
Answer in inches:
w=
h=
There is also a picture with it.
http://imgur.com/a/dr4ti
Thanks for the help!
Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
A Norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the width of the rectangle. Find the dimensions w * h of the Norman window whose perimeter is 500in. that has maximal area.
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Perimeter P = w + w*pi/2 + 2h = 500
w*(pi/2 + 1) + 2h = 500
h = 250 - w*(pi/4 + 1/2)
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Area = w*h + pi*(w/2)^2/2
Sub for h
Area = w*(250 - w*(pi/4 + 1/2)) + pi*w^2/8
Area = (pi/8)*w^2 -w^2*(pi/4 + 1/2) + 250w
Area = (-pi/8 - 1/2)*w^2 + 250w
dArea/dw = (-pi/4 - 1)w + 250 = 0
w = 250/(pi/4 + 1)
w =~ 140.025
h =~ 70.012