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.)

Let:
- $ x $ be the width of the plot (parallel to the barn wall),
- $ y $ be the length of the plot (perpendicular to the barn wall).

### Step 1: Formulate the cost constraint
Since no fencing is needed along the barn wall, only three sides need fencing:
- The west side (shared with the neighbor) with length $ y $,
- The east side with length $ y $,
- The south side with length $ x $.

The cost for these sides is:
\[
14 \times (x + 2y) \leq 7000
\]

Dividing both sides by 14 gives:
\[
x + 2y \leq 500
\]

### Step 2: Express area as a function of one variable
The area $ A $ of the rectangular plot is:
\[
A = x \cdot y
\]

Using the cost constraint $ x + 2y = 500 $, solve for $ x $ in terms of $ y $:
\[
x = 500 - 2y
\]

Substitute this into the area formula:
\[
A = (500 - 2y) \cdot y = 500y - 2y^2
\]

### Step 3: Find the maximum area
To maximize the area, differentiate $ A $ with respect to $ y $ and set the derivative to zero:
\[
\frac{dA}{dy} = 500 - 4y = 0
\]

Solving for $ y $:
\[
4y = 500 \Rightarrow y = 125
\]

Now substitute $ y = 125 $ back into the equation $ x = 500 - 2y $ to find $ x $:
\[
x = 500 - 2(125) = 250
\]

### Conclusion
The dimensions of the plot that maximize the area are:
\[
\text{Width (} x \text{)} = 250 \text{ feet, Length (} y \text{)} = 125 \text{ feet}
\]

Thus, the dimensions are:
**250, 125**

Would you like more details or have any questions?

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### Related Questions
1. How would the solution change if the cost per foot of fencing increased?
2. What if the farmer wanted fencing along all four sides?
3. How does sharing the cost impact the total fencing required for maximum area?
4. How could we generalize this approach to different shapes of land plots?
5. What would the optimal dimensions be if the farmer could spend only $5000?

### Tip
When maximizing areas with budget constraints, isolating one variable in terms of the other helps reduce complexity and allows using calculus more effectively.

Find the optimal dimensions for a rectangular plot of land using algebraic and calculus-based optimization techniques. Learn how a farmer can maximize the enclosed area given a fencing cost constraint, with a step-by-step solution including cost equations and area maximization. Suitable for high school students in Grades 10-12.

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