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The problem involves determining the dimensions of a rectangular plot of land, where an agricultural worker must enclose the land using 4,000 meters of wire divided into four parts to surround the land. The goal is to maximize the area of the rectangle.

### Problem Breakdown:

1. **Perimeter Constraint**:  
   The equation for the perimeter is given by:
   \[
   2x + 2y = 1000
   \]
   This simplifies to:
   \[
   x + y = 500
   \]
   where \(x\) represents the length (largo), and \(y\) represents the width (ancho) of the rectangle.

2. **Solving for \(y\)**:
   \[
   y = 500 - x
   \]

3. **Area Function**:
   The area \(A(x)\) of the rectangle is the product of length and width:
   \[
   A(x) = x \cdot y = x(500 - x) = 500x - x^2
   \]
   This is a quadratic equation of the form:
   \[
   A(x) = -x^2 + 500x
   \]

4. **Maximizing the Area**:
   To find the maximum area, we use the formula for the vertex of a parabola:
   \[
   x = \frac{-b}{2a}
   \]
   For the equation \(A(x) = -x^2 + 500x\), the vertex occurs at:
   \[
   x = \frac{-500}{2(-1)} = 250
   \]

5. **Calculating \(y\)**:
   Substituting \(x = 250\) into the perimeter equation:
   \[
   y = 500 - 250 = 250
   \]

Thus, the dimensions of the rectangle that maximizes the area are **250 meters by 250 meters**. This results in a square.

### Conclusion:
The rectangular plot will be a square with both sides measuring 250 meters, providing the maximum possible enclosed area.

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Would you like more details or have any further questions?

### Related Questions:
1. How is the perimeter equation derived from the total wire available?
2. Why does the vertex formula give the maximum area in this case?
3. Could the result be different if the available wire were used differently?
4. How do we know that a square provides the maximum area in such problems?
5. Can this technique be applied to non-rectangular shapes for maximizing areas?

### Tip:
In optimization problems involving a fixed perimeter, a square often maximizes the enclosed area for rectangles.

An agricultural worker needs to enclose a rectangular piece of land using 4,000 meters of barbed wire. What should the dimensions of the land be to maximize its area?

Learn how to determine the dimensions of a rectangular plot that maximizes the area when enclosed by 4,000 meters of barbed wire. The solution involves algebra, quadratic equations, and optimization techniques. Suitable for students in Grades 9-12.

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