SOLUTION: A norman window is one that consists of a rectangle with a curved top to it. the curved top is actually a semicircle that sits on top of the rectangle. if the perimeter around the

Question 194404: A norman window is one that consists of a rectangle with a curved top to it. the curved top is actually a semicircle that sits on top of the rectangle. if the perimeter around the norman window is 20 feet, what radius of the semicircle will provide the greatest window area?
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A norman window is one that consists of a rectangle with a curved top to it.
the curved top is actually a semicircle that sits on top of the rectangle.
if the perimeter around the norman window is 20 feet, what radius of the
semicircle will provide the greatest window area?
:
Let H = the height of the rectangular portion of the window
:
Let x = the radius of the semi circle on top
:
Half the circumference = pi*x
:
Width of the window = 2x
:
The perimeter:
2H + 2x + pix = 20
Solve for H
2H = 20 - 2x - pix
H =
:
Area formula: A = (H*2x) + .5pi*x^2
Replace H with
A = 2x()+
Cancel the 2
A = +
A =
Leaving us with
A =
.5*pi = 1.57
A =
A =
A quadratic equation, find the vertex for max area; a=-3.57, b=20
x =
x =
x = 2.8 ft radius for max area
:
Check solution by finding the perimeter with this radius
Find 2H
2H = 20 - 2(2.8) - 2.8pi
2H = 5.6 ft
Find perimeter:
2H + 2x + pix = 20
5.6 + 2(2.8) + 2.8*pi =
5.6 + 5.6 + 8.8 = 20
;
note that max area is when the rectangular portion, is a square, 5.6 by 5.6

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