Page 36 - How To Solve Word Problems In Calculus

6. If the radius of the semicircle is r , the base of the window will be
2r . Let x represent the height of the rectangle.

r

x x

2r

The area of the window is the sum of the areas of the rectangle and
semicircle.

1 2
A = 2rx + πr
2

Area of Area of
rectangle semicircle
The top of the figure is a semicircle and has a length equal to half
the circumference of a full circle. Thus

2x + 2r + πr = 100
C = 2πr for a full circle.
2x = 100 − 2r − πr
π 1
x = 50 − r − r C = πr .
2 2
Substituting into the area equation,

π 1
A = 2r 50 − r − r + πr 2
2 2
1
2
2
A = 100r − 2r − πr + πr 2
2
1 2
2
A(r ) = 100r − 2r − πr
2

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