Question

Kelly has 300 ft. of garden edging he wants to use to make some planters in his back yard. He

wants to build the gardens in such a manner as to create two equal-sized rectangular
gardens sharing no sides within the yard against an existing wall requiring only three sides
each to be built.
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What are the dimensions that will produce the maximum enclosed area.
(Round your answer to 1 decimal place.)
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Answer 1

Dimensions of each garden:

• 75 ft parallel to the wall
• 37.5 ft out from the wall

Explanation:

You want the dimensions of two identical enclosures made from 300 ft of garden edging using an existing wall for one side of each. They share no sides.

Identical

Since the two enclosures are identical and share no sides, each will use half the edging: 150 ft.

Max area

The maximum area is obtained when half of the perimeter is parallel to the existing wall, and the other half is perpendicular. The dimensions of one enclosure are ...

• 75 ft parallel to the wall
• 37.5 ft perpendicular to the wall

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Additional comment

The general solution of the maximum area problem is that half the fence is used in each of the directions (parallel or perpendicular to the wall). If different costs are involved, the least cost is obtained when the costs in each direction are equal.

If x is the length parallel to the wall, then (p-x)/2 is the length perpendicular. The area is the quadratic x(p-x)/2, which has a maximum at x=p/2. Here, p=150 for each of the two enclosures. Then the two dimensions are x=150/2=75, and (150-75)/2=37.5.

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