A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $/$5$ per foot, and $/$1,/!200$ altogether. To the nearest foot, find the length of the side parallel to the barn that will maximize the area of the pasture.

Let us list out the given information:

Cost per foot = $5

Total fencing cost = $1200

Let L be the length of the barn that will maximize the area of the pasture and W be the width of the rectangular cow pasture.

From the given information, we need to do the fencing for three sides covering 1 L and 2 W's because other L will be the fourth side is part of the side of a barn. We need to write the equation based on the cost information.

If cost/foot is $5, then cost of length ' L ' will be 5L dollars and cost of '2W' width equals 10W dollars. Total cost is $1200. We can set up second equation as

5L + 10W = 1200. This can be simplified by dividing throughout the equation by 5 to get L + 2W = 240. Solving for L we get L = 240 - 2W

Area of a rectangle is given by A = L * W.

Substituting 240 - 2W for L we get the area function as ...

A = (240 - 2W) * W = 240W-2W²

A = - 2W²+240W.

When we have a function of the form f (x) = ax^2 + bx + c, then maximum value of the function can be obtained for x = - b/2a. Using that idea we can find the maximum area.

Here in the area function a = - 2 and b = 240

For W = - b/2a = - 240/2 (-2) = - 240/-4 = 60 feet, we get maximum area.

The question is asking us to find Length. We can substitute W = 60 in the equation L = 240 - 2W to find the Length.

L = 240 - 2 (60) = 240 - 120 = 120 feet.

Conclusion: The length of the side parallel to the barn that will maximize the area of the pasture is 120 feet.

Note: Here the information 400 feet in the sentence "the fourth side is part of the side of a barn that is 400 feet long" given to distract us, we might not require that information to find the Length.