The dimension that would give the maximum area is 20.8569
How to solve for the maximum area
Let the shorter side be = x
Perimeter of the semi-circle is πx
Twice the Length of the longer side
[tex][70-(\pi )x -x][/tex]
Length = [tex][70-(1+\pi )x]/2[/tex]
Total area =
area of rectangle + area of the semi-circle.
Total area =
[tex]x[[70-(1+\pi )x]/2] + [(\pi )(x/2)^2]/2[/tex]
When we square it we would have
[tex]70x +[(\pi /4)-(1+\pi)]x^2[/tex]
This gives
[tex]70x - [3.3562]x^2[/tex]
From here we divide by 2
[tex]35x - 1.6781x^2[/tex]
The maximum side would be at
[tex]x = 35/2*1.6781[/tex]
This gives us 20.8569
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