A farmer has 360 feet of fencing to make three identical adjacent rectangular pens. What dimensions of each pen will maximize the total enclosed area?

Length = 45 ft and Width = 15 ft

Step-by-step explanation:

We have three pens, and they share a common side, which will be the bigger side, so we have the maximum benefit from the fencing.

If we call L the length and W the width of the pens, we have:

(See the figure attached for better understanding)

6W + 4L = 360

Total area = L * 3W

From the first equation:

3W + 2L = 180

W = (180 - 2L)/3

Using this value of W in the area equation:

Area = L * (180 - 2L) = 180L - 2L2

to find the maximum area, we need to find the vertix of the quadratic equation:

L = -b/2a (a and b are coefficients of the quadratic equation: a = -2 and b = 180)

L = -180/(-4) = 45 ft

Now we can find the width:

W = (180 - 3*45)/3 = 15 ft

So we have length = 45 ft and width = 15 ft

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