<span>Let the short side be W</span>
<span>
The half circumference of the semi-circle = .5 * pi *W</span>
<span>
2Lengths + width + semicircle = 20 ft</span>
<span>
2L + W + (.5*pi*W) = 20</span>
<span>
2L + W + 1.57W = 20</span>
<span>
Combine like terms</span>
<span>
</span>
<span>
2L + 2.57W = 20</span>
<span>
Simplify divide by 2</span>
<span>
L + 1.285W = 10</span>
<span>
L = (10 - 1.285W); for substitution:</span>
<span>
Get the area equation:</span>
<span>
Rectangle area + semicircle area</span>
<span>
A = L * W + (.5*pi*(.5W) ^2)</span>
<span>
A = LW + (1.57*.25W^2)</span>
<span>
A = LW + .3925W^2</span>
<span>
Replace L with (10-1.285W)</span>
<span>
A = W(10-1.285W) + .3925W^2</span>
<span>
A = -1.285W^2 + .3925W^2 + 10W</span>
<span>
A = -.8925W^2 + 10W
Find W by finding the axis of symmetry of this equation a=-.8925, b=10</span>
<span>
W = -10 / 2 * (-.8925)</span>
<span>
</span>
<span>
W = -10 / -1.785</span>
<span>
W = 5.60 is the width for max area</span>
<span>
then we solve for the length</span>
<span>
</span>
<span>L = 10 - 1.285(5.60)</span>
<span>
L = 10 – 7.20 = 2.8 ft is the length
</span>
<span>Then, Check the perimeter</span>
<span>
2(2.8) + 5.60 + 1.57(5.60) = 20ft</span>
<span>
</span>
5.6 + 5.6 + 8.736 = 20ft
<span>
Rectangle: 2.8 by 5.60 has semicircle circumference 8.736 has a maximum area: (2.8*5.60)
+ (.5 * pi * 2.8 ^ 2)</span>
<span>
</span>
=15.68 + 12.31
=28 sq/ft
