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g a farmer has 24002400 feet of fencing and wants to fence off a rectangular field that boarders a straight river. he needs no fence along the river. what are the dimensions of the field that has the largest area? (list the smallest dimension first)

User Martey
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1 Answer

7 votes
7 votes

Answer:

600 ft by 1200 ft

Explanation:

You want the dimensions of the largest rectangular area that can be enclosed on three sides with 2400 feet of fencing.

Perimeter

Let x represent the dimension of the field that is parallel to the river, and y the dimension out from the river. The sum of the three side lengths is ...

x + 2y = 2400

Solving for y gives ...

y = (2400 -x)/2

Area

The area is the product of the two dimensions:

A = xy = x(2400 -x)/2

We observe that this is the factored form of a quadratic with zeros at x=0 and x=2400. Its leading coefficient is negative, so the graph opens downward, and the vertex is the point where area is a maximum.

The vertex is located on the line of symmetry at the x-value that is halfway between the zeros:

x = (0 +2400)/2 = 1200

y = (2400 -x)/2 = (2400 -1200)/2 = 600

The dimensions of the field that has the largest area are 600 ft by 1200 ft.

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Additional comment

As we observed here, the side parallel to the "river" will use 1/2 of the fencing, and the remaining two sides will each use 1/4 of the fencing. This is the general solution to such a fencing problem.

User Lxndr
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