Farmer Ed has 3,000 meters of fencing and wants to enclose a rectangular plot that borders a river. If Farmer Ed does not fence the side along the river, what is the largest area that can be enclosed?

Question

29-12-2024
Mathematics

High School

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Farmer Ed has 3,000 meters of fencing and wants to enclose a rectangular plot that borders a river. If Farmer Ed does not fence the side along the river, what is the largest area that can be enclosed?

Answer :

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joymathew99608 joymathew99608 · Answer 1 · 2024-01-08T06:32:58-05:00

Farmer Ed can enclose the largest area by using 3,000 meters of fencing to create a rectangular plot with width of 750 meters and length of 1,500 meters along the river, yielding a maximum area of 1,125,000 square meters.

Farmer Ed is looking to enclose the largest area with a given 3,000 meters of fencing, not needing to fence one side because it borders a river. Since one side is the riverfront and does not need fencing, we can consider the fencing to be used for three sides of a rectangle. Let's call the sides that are perpendicular to the river width (W) and the side parallel to the river length (L). The total fencing used would then be for two widths and one length, which can be expressed as 2W + L = 3,000m. To find the maximum enclosed area, we rearrange the equation to express L in terms of W, and we get L = 3,000m - 2W.

The area A can be expressed as A = W imes L. Substituting L into the area equation, we get A = W imes (3,000m - 2W). To find the maximum A, we take the derivative of A with respect to W and set it to zero. Solving this, we find that W = 750m and thus L = 1,500m. The maximum area that Farmer Ed can enclose with 3,000 meters of fencing, alongside a river, is A = 1,125,000 m2 (or 750m imes 1,500m).