SOLUTION: A Norman Window is a rectangle with a semi-circle on top of it. Big Sky Window is designing a Norman window that will require 8m of trim around the perimeter of the window. What

Question 675817: A Norman Window is a rectangle with a semi-circle on top of it. Big Sky Window is designing a Norman window that will require 8m of trim around the perimeter of the window. What dimensions will allow the maximum amount of light to enter a house?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A Norman Window is a rectangle with a semi-circle on top of it.
Big Sky Window is designing a Norman window that will require 8m of trim around the perimeter of the window.
What dimensions will allow the maximum amount of light to enter a house?
:
This can be summed up in the statement:
"Find the max area for a perimeter of 8 meters"
:
Let W = the width of the rectangular portion and the diameter of semi-circle
.5W = the radius of the semicircle
let L = the length of the rectangular portion (height of the window)
:
Perimeter
2 vert lengths + 1 lower width + half the circumference
2L + W + .5W*pi = 8
2L + W + 1.57W = 8
2L + 2.57W = 8
Divide by 2
L + 1.285W = 4
L = (4-1.285W)
:
Area
A = L*W + [.5*pi*(.5W)^2]
A = L*W + [.5*pi*.25W^2
A = L*W + .3927W^2
Replace L with (4-1.285W)
A = W(4-1.285W) + .3927W^2
A = 4W-1.285W^2 + .3927W^2
A quadratic equation
A = -.8923W^2 + 4W
Find the axis of symmetry
W = -4/2(-.8923)
W = -4/-1.7846
W = 2.24 meters is the width of the window for max area
Find L
L = 4 - 1.285(2.24)
L = 4 - 2.8784
L = 1.12 meters is the height of the window for max area
:
:
:
Check this by finding the perimeter
p = 2(1.12 + 2.24 +(.5*pi*2.24)
p = 2.24 + 2.24 + 3.52
p = 8 meters
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