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 $14 per linear foot to install and the farmer is not willing to spend more than $7000, find the dimensions for the plot that would enclose the most area. (Enter the dimensions as a comma separated list.)

Since, cost of the fence on the western side is shared with a neighbor, we can say it is $7 per foot. Otherwise, it is $14 per foot. The northern length is not fenced.

From the question, we know that the cost is not to exceed $7000, therefore we can expressed it as

7y + 14x + 14y = 7000

14x + 21y = 7000

Divide through by 14, we have

(x + 1.5) = 500 ------------- eqn (*)

But the area of the fenced

= A = (x×y) -------------- eqn (**)

Substitute (*) into (**) we have

A = (500 - 1.5y)*y

= 500y - 1.5y²

In other to maximize the Area, we can find the differential of A

A'(y) = 500 - 3y

At A'(y) = 0.

500-3y= 0

3y=500

y= 166.67 ft

From eqn(*)

(x+ 1.5) = 500

x = 500 - 1.5y

500-1.5(166.67)

= 250ft

Therefore, the dimensions for the plot that would enclose the most area are

Width of 166.67 ft and Length = 250 ft.