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Answer :
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]
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]
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