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yssk8322 yssk8322
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11 years ago
A window has the shape of a rectangle surmounted by a semicircle. if the perimeter of the window is 15 ft., find the dimensions that will allow the maximum amount of light to enter.
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#1
11 years ago
Hello

let sides of the rectangle = a and b, when b = base side.

Area of the window = a*b + 1/2*pi*(b/2)^2

Perimeter = 2a + b + pi*b/2 = 15
---> a = 7,5 - b/2 - pi*b/4--> plug in:

A = (7,5 - b/2 - pi*b/4)*b + 1/2pi(*b^2/4)

A = b^2(- pi/8 - 1/2) + 7,5b

A' = (- pi/4 - 1)*b + 7,5  Rightwards Arrow set = 0 to find the maximum area:

b = 4,2 ft

The maximum area is found  at b = 4,20074 ft
a = 7,5 - b/2 - pi*b/4
a = 2,1 ft

Dimensions of the rectangle: base side = 4,2 m, height = 2,1 m
The maximum area = 15,7528 ft^2


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