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.
