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To solve this problem, let's break down the situation:

1. **Objective:** The farmer wants to enclose a rectangular area using the barn wall as one side and 64 feet of fencing for the other three sides.
2. **Variables:**
   - Let \( W \) be the width (the side perpendicular to the barn wall).
   - Let \( L \) be the length (the side along the barn wall).

3. **Fence Usage:** Since the barn wall forms one side of the rectangle, only three sides need to be fenced:
   - Two widths (\( 2W \)) and one length (\( L \)) will use the fence.
   - The total fencing required is therefore given by:
     \[
     2W + L = 64
     \]

Now, let's match this equation to the options given:

- **Option A:** \( 64 = 2W + L \)

This is indeed the correct answer, as it accurately represents the situation.

**Answer:** **A. \( 64 = 2W + L \)**

Would you like a further breakdown of how to maximize the area using this equation? Or do you have any specific questions about this setup?

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Here are five related questions for further understanding:

1. How would you maximize the area of this rectangle given this fencing constraint?
2. If 80 feet of fence were available instead, how would the equation change?
3. What would the dimensions of the rectangle be if we wanted to maximize the area with the 64 feet of fencing?
4. How would the equation change if the barn wall wasn't used, and all four sides needed fencing?
5. Could this problem be solved graphically to find the maximum area?

**Tip:** When solving these kinds of problems, carefully define your variables and translate word problems into mathematical equations step-by-step.

A farmer wants to fence a rectangular area by using the wall of a barn as one side of the rectangle and then enclosing the other three sides with 64 feet of fence. Find the dimensions of the rectangle that give the maximum area inside.
Which of the following equations describes the situation?

Explore how to solve a real-world optimization problem involving fencing a rectangular area with constraints. The problem uses algebra and geometry concepts to maximize area with a given fence length of 64 feet. Suitable for high school students in Grades 9-12.

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