The 34 feet perimeter and width, <em>x</em>, of the window which is in the shape of a rectangle surmounted by a semicircle is, <em>A </em>= 17•x - x²•(1/2 - π/8)
<h3>How can the area of the window be expressed as a function of <em>x</em>?</h3>
The shape of the window = A rectangle surmounted by a semicircle
Perimeter of the window, <em>P</em> = 34 feet
Width of the window = x
Required; The area, <em>A</em>, of the window as a function of <em>x</em>
Solution:
Diameter of the semicircle = x
Length of the semicircular arc = π•x/2
Let <em>y </em>represent the height of the window, we have;
P = 2•y + x + π•x/2 = 34
Therefore;
y = (34 - (x + π•x/2))/2 = 17 - x•(1 + π/2)/2
Area of the window, <em>A </em>= x × y + π•x²/8
Which gives;
A = x × (17 - x•(1 + π/2)/2) + π•x²/8 = 17•x - x²/2 - x²•π/8
A = 17•x - x²/2 - x²•π/8 = 17•x - x²•(1/2 - π/8)
Therefore;
Window area, <em>A</em> = 17•x - x²•(1/2 - π/8)
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