https://file.aiquickdraw.com/mathbot/file/1729778172511pyzt0eal.jpg

The problem asks to find the dimensions (length and width) of a rectangle that has a perimeter of 64 meters and a maximum area.

### Steps:
1. **Perimeter Formula**: The perimeter \( P \) of a rectangle is given by:
   \[
   P = 2L + 2W
   \]
   where \( L \) is the length and \( W \) is the width.
   
   Given that the perimeter is 64 meters:
   \[
   64 = 2L + 2W \quad \text{or} \quad L + W = 32
   \]
   
2. **Maximizing Area**: The area \( A \) of a rectangle is given by:
   \[
   A = L \times W
   \]
   To maximize the area, we use the relationship \( L + W = 32 \). Solve for one variable in terms of the other. Let’s solve for \( L \):
   \[
   L = 32 - W
   \]
   
   Substitute this into the area equation:
   \[
   A = (32 - W) \times W = 32W - W^2
   \]
   This is a quadratic equation in terms of \( W \), and it represents a parabola that opens downward. The maximum area occurs at the vertex of the parabola, which is found by the formula:
   \[
   W = \frac{-b}{2a}
   \]
   where \( a = -1 \) and \( b = 32 \). So:
   \[
   W = \frac{-32}{2(-1)} = 16
   \]
   
3. **Finding \( L \)**: Since \( L + W = 32 \) and \( W = 16 \):
   \[
   L = 32 - 16 = 16
   \]
   
### Conclusion:
The length and width that give a maximum area for the rectangle are both 16 meters, meaning the rectangle is actually a **square**.

Would you like further details or have any other questions?

### Related Questions:
1. How do you find the perimeter of a rectangle given the area?
2. What are the steps for solving optimization problems in calculus?
3. Why is the area maximized when the rectangle is a square?
4. How can you verify the maximum area using derivatives?
5. What is the difference between maximizing and minimizing functions in geometry?

### Tip:
For rectangles, a square always gives the maximum area when the perimeter is fixed!

Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 64 meters.

Learn how to find the dimensions of a rectangle with a perimeter of 64 meters that maximizes area. This problem involves geometry, optimization, and quadratic equations. Suitable for students in Grades 8-10.

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