The area of a rectangle will be maximum for the given perimeter when the square is a

The area of a rectangle is maximized for a given perimeter when the rectangle is a square. This means that for any given perimeter, the rectangle with the largest possible area is a square.

### Explanation:

Let the length of the rectangle be \( l \) and the width be \( w \).

1. The perimeter \( P \) of the rectangle is given by:
   \[
   P = 2l + 2w
   \]

2. The area \( A \) of the rectangle is:
   \[
   A = l \times w
   \]

3. For a given perimeter, we can express the width in terms of the length:
   \[
   w = \frac{P}{2} - l
   \]

4. Substituting this into the area formula:
   \[
   A = l \times \left(\frac{P}{2} - l\right)
   \]
   \[
   A = \frac{P}{2}l - l^2
   \]

5. To maximize the area, we can take the derivative of \( A \) with respect to \( l \) and set it equal to zero:
   \[
   \frac{dA}{dl} = \frac{P}{2} - 2l = 0
   \]

6. Solving for \( l \), we find:
   \[
   l = \frac{P}{4}
   \]
   Since \( l = w \), the rectangle becomes a square.

Thus, when the length and width are equal, i.e., when the rectangle is a square, the area is maximized for a given perimeter.

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### Related Questions:
1. How do you derive the formula for the area of a rectangle?
2. Can the area of a rectangle be maximized in other ways besides being a square?
3. What is the maximum area of a square with a given perimeter?
4. How do you find the perimeter of a square if the area is known?
5. What happens to the area of a rectangle if the perimeter is doubled but the ratio of length to width remains the same?

### Tip:
Always remember that the maximum area of a rectangle for a given perimeter is achieved when the rectangle becomes a square.

Discover how to maximize the area of a rectangle for a given perimeter. Understand the relationship between perimeter, length, and width to achieve optimal area. Suitable for students and enthusiasts interested in geometry and optimization problems.

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