SOLUTION: Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 70 feet of fencing to go around it, what dimensions will give him the max

Question 86386: Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 70 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?
I know that eventually you may have to use the X=b/2a formula

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 70 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?
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Draw the picture.
Let the shorter side be "x"
Perimeter of the semi-circle is (pi)x
Twice the Length of the longer side is [70-(pi)x -x]
Length = [70-(1+pi)x]/2
Total area = area of rectangle + area of the semi-circle.
Total area = x[[70-(1+pi)x]/2] + [(pi)(x/2)^2]/2
2Total area = 70x -(1+pi)x^2 + [(pi)(x^2/4)]
2Total area = 70x +[(pi/4)-(1+pi)]x^2
2Total area = 70x - [3.3562]x^2
Total area = 35x - 1.6781x^2
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Max occurs at x = 35/2*1.6781= 20.8569(shorter side)
Longer side = [70-(1+pi)*10.4285]/2 = 13.4047 (longer side)
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Comment: The method here is correct; the arithmetic may be less
than the accuracy you require. I recommend you check it out.
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Cheers,
Stan H.

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