Question 853677: A Norman window has the shape of a rectangle surmounted by a semicircle as in the figure below. If the perimeter of the window is 35 feet, express the area, A, as a function of the width, x, of the window.
Answer by harpazo(655)   (Show Source):
You can put this solution on YOUR website!
The diameter of the semicircle is equal to the width of the rectangle. You'd still use the height in the calculations along the way.
If the height of the rectangle part was h and the width was x, then the perimeter would be
(length of 3 rectangle sides) + (curved part from semicircle) =
(h+h+x) + ( (1/2) 2 π r )
(h+h+x) + ( π r )
(2h+x) + ( π (x/2) ) = 35
You can use this to get h in terms of x, or vice versa. Then plug it into the expression for area to get things in terms of width.
2h+x = 35 - π(x/2)
2h = 35 - π(x/2) - w
h = 17.5 - π(x/4) - (x/2)
The area would be
(area of rectangle) + (area of semicircle) =
h*x + ((1/2)π r^2)
h*x + ((1/2)π (x/2)^2)
h*x + (π x^2)/8
Take the expression for h and plug it in here to get everything in terms of x.
[17.5 - π(x/4) - (x/2)]x + (π x^2)/8
A(x) = 17.5x-π(x^2/4)-(x^2/2) +
(π x^2)/8
:)
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