What is the maximum possible area of a rectangle is square unit of length 16
units?
A. 128 B. 64 C. 48 D. 256

The maximum possible area of a rectangle given a perimeter of 16 units can be found by recognizing that, for a fixed perimeter, a square gives the maximum area.

### Step-by-step solution:

1. The formula for the perimeter of a rectangle is:
   \[
   P = 2 \times (L + W)
   \]
   where \(L\) is the length and \(W\) is the width. Given \(P = 16\), we have:
   \[
   2 \times (L + W) = 16
   \]
   Simplifying this:
   \[
   L + W = 8
   \]

2. The area of the rectangle is:
   \[
   A = L \times W
   \]
   To maximize the area, we express \(W\) in terms of \(L\) using \(L + W = 8\):
   \[
   W = 8 - L
   \]
   So, the area becomes:
   \[
   A = L \times (8 - L)
   \]
   Expanding this:
   \[
   A = 8L - L^2
   \]

3. To find the maximum area, take the derivative of the area function with respect to \(L\) and set it to zero:
   \[
   \frac{dA}{dL} = 8 - 2L = 0
   \]
   Solving for \(L\):
   \[
   2L = 8 \implies L = 4
   \]

4. Since \(L + W = 8\), if \(L = 4\), then \(W = 4\). Therefore, the rectangle is a square with side lengths of 4 units.

5. The area of this square is:
   \[
   A = 4 \times 4 = 16 \text{ square units}
   \]

Thus, the maximum possible area of a rectangle with a perimeter of 16 units is **16 square units**. None of the answer choices listed are correct.

Would you like further details or have any questions?

### Related questions:
1. What is the relationship between perimeter and area for different rectangles?
2. How does maximizing the area of a rectangle relate to optimization in calculus?
3. How do you calculate the perimeter of a square?
4. How does the area of a rectangle change if one side is doubled?
5. How would the maximum area change if the perimeter were increased?

**Tip**: For a fixed perimeter, the rectangle with the largest area is always a square.

What is the maximum possible area of a rectangle with a perimeter of 16 units?
A. 128 B. 64 C. 48 D. 256

Discover how to find the maximum possible area of a rectangle with a perimeter of 16 units. Learn the relationship between perimeter and area, and understand why a square gives the largest area. Suitable for students in Grades 7-10.

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