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This problem is about optimizing the dimensions of a rectangular chicken pen that uses 500 feet of fencing, with one side along a barn wall (meaning only three sides require fencing). The goal is to find the dimensions that yield the largest possible area.

### Step 1: Define the variables
- Let \( x \) represent the length of the side parallel to the barn.
- Let \( y \) represent the length of the two sides perpendicular to the barn.

### Step 2: Set up the perimeter constraint
Since only three sides of the pen need fencing, the total amount of fencing used is:
\[
x + 2y = 500
\]
This equation represents the total length of fencing available.

### Step 3: Write the area formula
The area \( A \) of the rectangle is:
\[
A = x \cdot y
\]

### Step 4: Express \( y \) in terms of \( x \)
Using the perimeter constraint \( x + 2y = 500 \), solve for \( y \):
\[
2y = 500 - x \implies y = \frac{500 - x}{2}
\]

### Step 5: Write the area formula in terms of \( x \)
Substitute \( y = \frac{500 - x}{2} \) into the area formula:
\[
A(x) = x \cdot \frac{500 - x}{2}
\]
Simplify:
\[
A(x) = \frac{500x - x^2}{2}
\]

### Final equation to be optimized:
\[
A(x) = \frac{500x - x^2}{2}
\]
This is the equation for the area in terms of one variable \( x \), and this equation needs to be optimized to find the maximum area.

Would you like further details or help with optimizing this equation? 

### 5 Related Questions:
1. How do you find the critical points of the area function to maximize it?
2. What are the dimensions of the pen if you solve this equation?
3. How does the shape of the pen (length vs. width) affect the maximum area?
4. What would happen if there was no barn and all four sides needed fencing?
5. How does changing the total amount of fencing available affect the optimal dimensions?

### Tip:
Always look for a way to reduce the number of variables in optimization problems, typically using constraints to express one variable in terms of another.

A rectangular chicken pen using 500 feet of fencing is being built next to a barn wall. Only three sides of fencing will be needed. What dimensions should the farm owner use to construct the chicken pen with the largest possible area?

Learn how to optimize the dimensions of a rectangular chicken pen with 500 feet of fencing next to a barn wall, using algebraic and quadratic concepts to find the maximum possible area.

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