Let's denote the length of the rectangle by $x$ cm and the width by $y$ cm.

We are given two pieces of information:

1. The area of the rectangle is $108 \text{ cm}^2$.
2. The perimeter of the rectangle is $48 \text{ cm}$.

Step 1: Write the equations

1. Area equation: $x \times y = 108$ This equation comes from the formula for the area of a rectangle, which is length times width.

2. Perimeter equation: $2x + 2y = 48$ This equation comes from the formula for the perimeter of a rectangle, which is twice the sum of the length and the width.

Step 2: Simplify the perimeter equation

Let's simplify the perimeter equation by dividing all terms by 2:

$x + y = 24$

Step 3: Express $y$ in terms of $x$

We can solve the simplified perimeter equation for $y$:

$y = 24 - x$

Step 4: Write the area equation using $x$

Substitute $y = 24 - x$ into the area equation:

$x \times (24 - x) = 108$

This equation represents the relationship between the length $x$ and the width $y$ based on the area of the rectangle.

Would you like to proceed with solving this equation, or do you have any questions?

Here are 5 related questions you might find useful:

1. How do you solve the quadratic equation formed to find the dimensions of the rectangle?
2. What happens if you set up the problem with different variable names?
3. Can you find the dimensions using a graph instead of algebra?
4. What is the significance of the solutions to this quadratic equation in the context of the problem?
5. How would the problem change if the perimeter were different?

Tip: In geometry problems involving both area and perimeter, setting up a system of equations is a powerful method to find unknown dimensions.