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.

Step-by-step explanation:

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.