The owner of a garden supply store wants to construct a fence to enclose a recta

Question

The owner of a garden supply store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store, using part of the side of the store (which is 280 feet long) for part of one of the sides. (See the figure below.) There are 470 feet of fencing available to complete the job. Find the length 3f the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure. Note: you can click on the image to get a enlarged view. Length of parallel side(s) = Length of perpendicular sides =

Explanation / Answer

Maintaining the aspect of the drawing, let x by the length of the sides parallel to the store and y be the length of the sides perpendicular.

The area of the extension is xy, but the new perimeter is only 3 sides because the existing wall of the store forms the fourth.

x + 2y = 470 feet.

Solve the perimeter expression for either variable and substitute into the area expression.

x = 470 - 2y

A = xy = (470 - 2y)y

Simplify and differentiate area with respect to y.

A = 470y - 2y2

dA/dy = 470 - 4y = 0

470 = 4y

y = 117.5 ft <-- length of perpendicular sides

Go back and solve for x.

x = 470 - 2y = 470 - 235

x = 235 ft <-- length of parallel sides

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