The length of a rectangle is three times its width. The perimeter of the rectangle is at most 112 cm. Which inequality models the relationship between the width and the perimeter of the rectangle?

A. [tex]2w + 2 \cdot (3w) \geq 112[/tex]

B. [tex]2w + 2 \cdot (3w) \textless 112[/tex]

C. [tex]2w + 2 \cdot (3w) \textgreater 112[/tex]

D. [tex]2w + 2 \cdot (3w) \leq 112[/tex]

Sure! Let's solve the problem step by step.

1. Identify the given information:
- The length ([tex]$ L $[/tex]) of the rectangle is three times its width ([tex]$ W $[/tex]).
- The perimeter of the rectangle is at most [tex]$ 112 $[/tex] cm.

2. Express the length in terms of the width:
[tex]\[
L = 3W
\][/tex]

3. Write the formula for the perimeter of a rectangle:
The perimeter ([tex]$ P $[/tex]) of a rectangle is given by:
[tex]\[
P = 2 \times (\text{length}) + 2 \times (\text{width})
\][/tex]

4. Substitute the given length into the perimeter formula:
Since the length [tex]$ L $[/tex] is [tex]$ 3W $[/tex], the formula becomes:
[tex]\[
P = 2 \times (3W) + 2 \times W
\][/tex]
Simplifying this, we get:
[tex]\[
P = 6W + 2W
\][/tex]
[tex]\[
P = 8W
\][/tex]

5. Write the inequality for the perimeter:
The problem states that the perimeter is at most [tex]$ 112 $[/tex] cm. This means:
[tex]\[
8W \leq 112
\][/tex]

6. Rewrite the inequality using the perimeter formula:
[tex]\[
2W + 2(3W) \leq 112
\][/tex]

Thus, the inequality that models the relationship between the width and the perimeter of the rectangle is:
[tex]\[
2W + 2(3W) \leq 112
\][/tex]

So, the correct option is the fourth one:
[tex]\[
2W + 2(3W) \leq 112
\][/tex]