a rectangle has a perimeter 16m. Express the area A (in m^2) of the rectangle as a function of the length, L, of one of its sides. A(L)= ?m^2

To express the area \(A\) of a rectangle as a function of the length \(L\) of one of its sides, given that the perimeter is \(16\) meters, we can follow these steps:

### Step-by-step Solution:

1. **Recall the Perimeter Formula for a Rectangle:**

   The perimeter \(P\) of a rectangle with length \(L\) and width \(W\) is given by:

   \[
   P = 2L + 2W
   \]

   Given that the perimeter is \(16\) meters:

   \[
   2L + 2W = 16
   \]

2. **Solve for the Width \(W\):**

   Rearrange the equation to solve for \(W\):

   \[
   2W = 16 - 2L
   \]

   \[
   W = \frac{16 - 2L}{2}
   \]

   Simplify the expression:

   \[
   W = 8 - L
   \]

3. **Express the Area \(A\) as a Function of \(L\):**

   The area \(A\) of a rectangle is given by:

   \[
   A = L \times W
   \]

   Substitute the expression for \(W\):

   \[
   A(L) = L \times (8 - L)
   \]

   \[
   A(L) = 8L - L^2
   \]

### Conclusion:

The area \(A\) of the rectangle as a function of the length \(L\) is:

\[
\boxed{A(L) = 8L - L^2 \text{ m}^2}
\]

Would you like further details or have any questions?

Here are 5 related questions that might interest you:

1. What is the maximum area of the rectangle, and what is the corresponding length \(L\)?
2. How would you express the width \(W\) as a function of the length \(L\)?
3. How would you find the derivative of the area function \(A(L)\) to determine critical points?
4. What are the possible dimensions of the rectangle if its area is \(12 \, \text{m}^2\)?
5. How would the area function change if the perimeter were different?

**Tip:** To maximize or minimize quadratic functions, use the vertex formula \(L = -\frac{b}{2a}\) when in standard form \(ax^2 + bx + c\).

Learn how to express the area of a rectangle as a function of its side length, given a perimeter of 16 meters. Step-by-step solution provided with formulas for perimeter and area calculations. Suitable for students in Grades 7-9.

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