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 $ax^2 + bx + c$.