Farmer Jinzhou has  300 feet of fencing to enclose two identical rectangular pens that share a common fence.

If  x represents the length of the enclosure, what is the domain in the context of this situation?

To find the domain of the function \( x \), which represents the length of the enclosure, let's first understand the problem.

### Problem Breakdown:

1. **Two Identical Rectangular Pens Sharing a Common Fence:**
   - The pens are placed side by side, so they share one side of the fence.
   - Let \( x \) be the length of each pen.
   - Let \( y \) be the width of each pen.

2. **Total Fencing Available:**
   - The total amount of fencing available is 300 feet.

### Forming the Equation:

- The total fencing used can be broken down as:
  - 2 lengths (\( 2x \)) for the two pens.
  - 3 widths (\( 3y \)) because the pens share one side.

Thus, the equation relating \( x \) and \( y \) is:

\[
2x + 3y = 300
\]

### Solving for \( y \):

Solve for \( y \):

\[
3y = 300 - 2x
\]

\[
y = \frac{300 - 2x}{3}
\]

### Determining the Domain:

- \( y \) represents a length, so it must be non-negative. Therefore, we require:

\[
\frac{300 - 2x}{3} \geq 0
\]

Multiply both sides by 3:

\[
300 - 2x \geq 0
\]

Solving for \( x \):

\[
300 \geq 2x
\]

\[
x \leq 150
\]

- Also, \( x \) must be non-negative:

\[
x \geq 0
\]

### Domain of \( x \):

The domain of \( x \) in this context is:

\[
0 \leq x \leq 150
\]

This means \( x \) can take any value between 0 and 150 feet, inclusive.

---

Would you like more details or have any other questions?

Here are some related questions to explore:

1. What is the range of \( y \) given the domain of \( x \)?
2. How do we find the area of each pen in terms of \( x \)?
3. What would the maximum area be if Farmer Jinzhou wants to maximize the enclosed area?
4. How would the problem change if the pens were not identical?
5. What if there was no common fence? How would the domain of \( x \) change?

**Tip:** When determining the domain of a function in a real-world context, consider both the mathematical constraints and the physical or practical limitations.

Explore how to determine the domain of x in a problem involving the enclosure of two identical rectangular pens sharing a common fence. This mathematical problem involves geometry and linear equations, where you'll learn to solve for x given a total fencing constraint of 300 feet. Suitable for high school students.

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