Alan is building a garden shaped like a rectangle with a semicircle attached to one short side. If he has 80 feet of fencing to go around it, what dimensions will give him the maximum area in the garden?

To solve this problem, we need to use calculus and the formula for the perimeter of a rectangle and the circumference of a circle.

Step 1: Define the variables and equations Let's denote the length of the rectangle as 'x' and the width as 'y'. The semicircle has a radius of 'y'.

The total length of the fence, which is 80 feet, is equal to the perimeter of the rectangle plus the circumference of the semicircle. This gives us the equation: 2x + 2y + πy = 80

Step 2: Solve for one variable We can simplify this equation by solving for 'x': x = (80 - 2y - πy) / 2

Step 3: Define the area equation The total area of the garden, which we want to maximize, is the area of the rectangle plus the area of the semicircle. This gives us the equation: A = xy + 0.5πy^2

Step 4: Substitute 'x' into the area equation Substitute 'x' from step 2 into the area equation to get an equation for 'A' in terms of 'y' only: A = y(80 - 2y - πy) / 2 + 0.5πy^2

Step 5: Differentiate the area equation Differentiate the equation with respect to 'y' to find the maximum area. Set the derivative equal to zero and solve for 'y'.

Step 6: Second derivative test Take the second derivative of the area equation and substitute the 'y' value found in step 5. If the result is negative, then the area is maximized.

Step 7: Substitute 'y' back into the equation from step 2 to find 'x'.

These are the dimensions that will give Alan the maximum area for his garden.