Question

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. no fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. if the fencing costs $10 per linear foot to install and the farmer is not willing to spend more than $4000, find the dimensions for the plot that would enclose the most area.

Answer 1

Let the length of the northern side be x and length of the west sides be y. The cost of fence is:
C=10x+10y+5y

and the area enclosed be A=xy
10x+15y=4000
x=400-1.5y
A=400y-1.5y^2
A is maximized when y=400/3 and x=200