Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 90 feet of fencing to go around it, what dimensions will give him the maximum area in the garden.

Let the rectangle dimensions be x and y. Assume that x represents the "short" side, so it is also the diameter of the semicircle. Then the perimeter is ...

P = x + 2y + πx/2 = 90

The area is the area of the rectangle added to the area of the semicircle:

A = xy + (1/2)π(x/2)^2 = x(π/8x +y)

Solving the perimeter equation for y, we get

y = (90 -x -π/2x)/2

and the area equation becomes ,..

A = x(π/8x +45 -x/2 -π/4x)

A = x(45 -x(1/2 +π/8))

The graph of A(x) is a parabola opening downward with zeros at x=0 and x=45/(1/2+π/8) = 360/(4+π).

The vertex (maximum) of the area curve will be halfway between the zeros, at ...

x = (0 +360/(4+π))/2 = 180/(4+π)

The value of y is then ...

y = (90 -x(1 +π/2))/2 = (1/2)(90 -(180/(4+π))(2+π)/2) = (1/2)(90)(1 -(2+π)/(4+π))

y = 45(4+π-2-π)/(4+π) = 90/(4+π)

The rectangle dimensions are 90/4+π) by 180/(4+π), with the semicircle diameter equal to 180/(4+π).

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That's about 25.2 ft by 12.6 ft with the semicircle attached to the long side.

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Please note that the overall dimensions of the garden are the same in both directions. It is effectively "square" with two corners rounded into quarter circles.