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Question

Suppose that a rancher is build pens along a river so no fencing is needed along the river but wants 3 sections as shown. Find the length y that will maximize the area if the rancher has 6,548 feet of fencing to use. (round-off to 1 decimals place.)

Answer 1

The rancher has 6,548 feet of fencing to use with 4 equal vertical lines and one horizontal line to create the pen

We can set up on equation where x is the long horizontal side and y is the 4 equal vertical parts

x + 4y = 6,548

solve for x:

x = 6,548 - 4y

Plug in x into area equation

Area = x * y

A = (6,548 - 4y) * y

A = 6,548y - 4y²

Find the derivative of area

A' = 6,548 - 8y

Set A = 0 to find maximum value of y

0 = 6,548 - 8y

8y = 6,548

y = 818.5 feet

The length of y that will maximize the area is y = 818.5 feet