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 $20 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.)

The dimensions for the plot that would enclose the most area are as follows:

Width  = 116.67 ft

Length = 175 ft.

Step-by-step explanation:

Note: Kindly refer to the attached diagram where:

x = Length of the rectangular area

y = Width of the rectangular area.

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. This implies that it is $10 per foot. Otherwise, it will be $20 per foot. Since no fencing is needed along the barn, it implies that there is no fence along the northern length.

Since the farmer is not willing to spend more than $7000, we have:

10y + 20x + 20y = 7000

20x + 30y = 7000

Divide through by 20, we have:

x + 1.5y = 350

x = 350 - 1.5y …………………………. (1)

The fenced area is

A = x * y ………………………………….. (2)

Substituting equation (1) into (2), we have:

A = (350 - 1.5y)y

A = 350y - 1.5y^2 ……………. (3)

To maximize A, equation (3) is differentiated with respect to y and equated it to 0 as follows:

That is,

A'(y) = 350 - 3y = 0

Solving for y, we have:

y = 350/3

y = 116.67 ft

Substituting y = 116.67 for y in equation (1), we have:

x = 350 - (1.5 * 116.67)

x = 175ft

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

Width  = 116.67 ft

Length = 175 ft.

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