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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 $12 per linear foot to install and the farmer is not willing to spend more than $3000, find the dimensions for the plot that would enclose the most area.

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trccater54

If the northern side has length x and the east and west sides have length y, then the cost of the fence is 
C = 12x + 12y + 6y 
and the enclosed area is A = xy. 

12x + 18y = 6000 
x = 500 - 1.5y 
A = 500y - 1.5y^2 

A is maximized when y = 500/3. In which case x = 250. 

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