A farmer plans to

Question

the farmer plans to make a pen against the side of a barn that is divided into three rectangular portions, each with an area of 120 square meters. Assuming that no fence needed along the side of the barn, what are the dimensions of each pen that minimize the amount of fence that must be used?

Answer 1

I will assume that the dividers are perpendicular to the side of the barn.

Let x = length of the side of the pen against the barn

y = length of a side perpendicular to the barn

Then xy = 120. So, y = 120/x.

A = amount of fencing = x + 4y = x + 480/x, x > 0

So, A' = 1 - 480/x² = (x² - 480) / x²

A' = 0 when x² = 480.

x = sqrt(480) = 4sqrt(30) = 21.9

When 0 < x < 4sqrt(30), A' < 0. So, A is decreasing.

When x > 4sqrt(30), A' > 0. So, A is increasing.

Therefore, A is minimized when x = 4sqrt(30) and y = sqrt(30).