Question

Preston wants to create a garden that backs up to an existing fence in his yard. He won't need to enclose the garden with fencing and he only has 85 feet of fencing available. What was the largest dimensions of the garden be if you could encloses the three sides of the garden and use the existing fence as the fourth side? If necessary, round answer to the nearest tenth.

Answer 1

42.5 ft by 21.3 ft

Explanation:

The largest area is obtained when half the available fence is used parallel to the existing fence, and the other half is used to fence the two ends of the rectangle. Here, that means the dimension parallel to the existing fence is ...

(85 ft)/2 = 42.5 ft

and the ends of the rectangular garden are ...

(42.5 ft)/2 = 21.25 ft ≈ 21.3 ft

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You can figure this as follows:

Let x represent the length of fence parallel to the existing fence. Then the other dimension of the fenced area is (85 -x)/2 and the fenced area is the product of these dimensions.

area = x(85-x)/2

This expression describes a downward-opening parabola with zeros at x=0 and at x=85. The vertex (maximum) will be found where x is halfway between these values, at x = (0 +85)/2 = 42.5.

Area is maximized when 42.5 ft of fencing is used parallel to the existing fence, and the other half of the fencing is used for the other two sides of the enclosure.