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A semicircle is placed on one side of a square so that it's diameter coincides with a side of the square. Find the side length of the square if the total area of the square plus the semicircle is 200 square inches.
Let x = one side & diamter of the semicircle
Square area + semicircle area = 200
Radius = .5x
x^2 + (pi(.5x)^2)/2 = 200
x^2 + (pi*.25x^2)/2 = 200
2x^2 + pi*.25x^2 = 400; multiplied eq by 2
2x^2 + .785x^2 = 400
2.785x^2 = 400
x^2 = 400/2.785
x^2 = 143.6
x = SqRt[143.6]
x = 11.98 ~ 12 inches:
Check: r = 6 inches; 144 + (pi*36)/2 = 200.55 ~ 200