Afarmer 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 $16 per linear foot to install and the farmer is not willing to spend more than $8000, find the dimensions for the plot that would enclose the most area.

*I'm sure there is an easier way to do this, but I haven't found it. In the meantime sorry for the extensive answer, but hope it helps! 

 
100 feet by 250 feet or 25,000 sq. feet

Explanation: 
Okie-dokie - here we go:
So, think about it, and maybe sketch this out as you are going. You have a barn, so make a little square on a paper with a big x on it. The top side is the north facing wall that you don't need any fence along. The side that will be on the left is the side that will be half the cost since your neighbor is splitting it with you. So lets think about the other two sides first:

Keep in mind that the barn wall will not grow with the size of the area you are making, so the "vertical" sides on the left and right will get longer but the "horizontal" fence at the top of the area will always be as wide as the barn. This isn't very helpful in finding the area or even the perimeter, but is helpful to keep in mind visually. 

You are the farmer. You have $8,000 , and you are a stickler, so no more than that. You can go under by a bit, but no more than that limit. So see how many linear feet of fence you can buy at $16 per foot. 
8000 ÷ 16 = 500. You can get 500 feet out of this amount of money. But you can get more on the fence you are sharing with your neighbor, since it is half the price. Don't think about that for now. 

Let's set a width for the barn. I'm going to say the barn is 100 feet wide, so the top part of the fence will be 100 feet long. That is 400 feet left to build. If you were paying full price for both sides, that would be 200 feet on each side. However, you have a discount on the left fence, so you can stretch each side out another 1/2 the distance because he is splitting the cost of that fence with you. In other words: 
two 200 foot sides = $6400 at full price 
one 200 foot side = $3200
Find how much the neighbor is paying for: 
3200 ÷ 2 = 1600. 
So you only paying for 1 and a half sides, or:
1600 + 3200 = $4800 for 300 feet. 
You now have an extra 1600 to spend, which is another 100 feet. Split this between your two sides, so now each side is 250 feet long. 
But your neighbor is paying for half of that 50 feet you just added, so:
50 • 16 = $800 
So he pays for $400, giving you another $400 to buy fence with. 
If you understand what is happening, I'm just going to keep going mathematically without explanations:
$400 ÷ $16 = 25 feet
25 feet ÷ 2 = 12.5 feet  (250 + 12.5 = 262.5 feet)
12.5 • $16 = $200 ÷ 2 = $100
$100 ÷ $16 = 6.25 feet  
6.25 ÷ 2 = 3.125  (262.5 + 3.125 = 265.625 feet) 
3.125 • $16 = $50 ÷ 2 = $25
$25 ÷ $16 = 1.5625 feet
1.5625 ÷ 2 = .78125 (265.625 + .78125 = 266.40625 feet) 
And so on, this could keep going until you run out of money, but you'd really be milking it. Plus, let's say that you can only buy full linear feet, and not fractions, then you would stop at 250 feet per side. Which is: 
100 feet + 250 feet + (250 feet – 125 feet) = 475 feet 
475 feet • $16 = $7600 Under the limit! But you do have $400, which you could keep adding to your sides, or maybe your width is longer. Either way, find the area of 100 feet by 250 feet: 
A = l • w
A = 250 • 100 = 25,000 sq. feet