# calculus

A Norman window is constructed by adjoining a semicircle to the top of a rectangular window . (The diameter of the semicircle is the same as the width of the rectangular) If the perimeter of the Norman window is 20 ft, find the dimensions that will allow the window to admit the most light.

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1. base = 2 r
height of rectangle = h

area is A

A = 2 r h + (1/2) pi r^2

perimeter = 2 r + 2 h + (1/2)(2 pi r)=20

(2 + pi)r + 2 h = 20
(1 + pi/2)r + h = 10
so
h = 10 - 2.57 r

A = 2 r (10-2.57r) + 1.57 r^2

A = 20 r - 5.14 r^2 + 1.57 r^2
or
A = 20 r - 3.57 r^2
dA/dr = 0 for max = 20 - 7.14 r
or
r = 2.8
all yours now

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