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.
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Answers
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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