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 $24 per linear foot to install and the farmer is not willing to spend more than $6000, find the dimensions for the plot that would enclose the most area

The dimensions of the largest area are

W  = 83.33 ft

L = 125 ft.

Step-by-step explanation:

Let 'x' be the length of the rectangular area

and, 'y' be the width of the rectangular area.

Cost 'C'= 1/2(24x) + 24y + 24x =36x + 24y

To Maximize area

6000= 36x + 24y

y= 250-3x/2

Area'A' = xy= x( 250-3x/2) => 250x - 3/2x²

Taking derivative on both sides and then setting it to zero

A'= 250- 3x =0

x= 250/3 =>83.33ft

->y= 250-3(83.33)/2 =>125ft

A''= -3<0 , verifying by taking second derivative.

Therefore, the dimensions for the plot that would enclose the most area 83.33ft by 125ft

step-by-step explanation: