Question

Consider the following problem: A farmer with 850 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

Answer 1

18,062.5 square feet

Explanation:

The largest area will be obtained when half the fence is used in each of the orthogonal directions. That is, the pen will have two parallel sides that total 425 feet, and 5 parallel sides and partitions that total 425 feet.

The long sides are 212.5 feet, and the short sides are 85 feet, so the overall area is ...

(212.5 ft)(85 ft) = 18,062.5 ft²

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Working Out

Suppose the long side is length x. Then the lengths of the 2 ends and 3 dividers are ...

short side = (850 -2x)/5

Then the overall area is the product of long side and short side:

A = x(850 -2x)/5 = (2/5)(x)(425 -x)

This equation is that of a parabola that opens downward. Its vertex (maximum) is at the value of x halfway between the zeros of 0 and 425. That is, area is a maximum when x=212.5.

That maximum area is ...

A = (2/5)(212.5)(425 -212.5) = 18,062.5 . . . square feet