Question

1. A window is to be in the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 100 inches,
express its area as a function of its semicircular radius r.
ase dimension
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Answer 1

Given :  A window is to be in the shape of a rectangle surmounted by a semicircle

perimeter of the window is 100 inches,

To Find : area as a function of its semicircular radius r.

Solution:

semicircular radius r   inches

Diameter = 2r    inches  (one side of rectangle window)

Perimeter of semicircle =   πr

Other side of rectangle = b

Perimeter =   πr  + 2r  + 2b  = 100

=> πr/2 + r + b = 50

=> b  = 50 - ( πr/2 + r )

=> b = 50 - r( π/2  + 1)

Area of Rectangle = 2r.b  + (1/2)πr²

= 2r.(50 - r( π/2  + 1))  + (1/2)πr²

= 100r  - 2r² ( π/2  + 1) + (1/2)πr²

=  100r  -   (1/2)πr² - 2r²

area as a function of its semicircular radius r.  = 100r  -   (1/2)πr² - 2r²

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