The length of a rectangle is five times its width. The perimeter of the rectangle is at most 96 cm.

Which inequality models the relationship between the width and the perimeter of the rectangle?

 

Responses

2w+2⋅(5w)>96
2 w plus 2 times left parenthesis 5 w right parenthesis greater than 96

2w+2⋅(5w)≥96

2 w plus 2 times left parenthesis 5 w right parenthesis greater than or equal to 96

2w+2⋅(5w)<96
2 w plus 2 times left parenthesis 5 w right parenthesis less than 96

2w+2⋅(5w)≤96

We are given that:

- The length \( L \) of the rectangle is five times the width \( w \). So, \( L = 5w \).
- The perimeter \( P \) of a rectangle is given by \( P = 2L + 2w \).
- The perimeter must be at most 96 cm, meaning \( P \leq 96 \).

Now let's substitute the expression for \( L = 5w \) into the perimeter formula:

\[
P = 2L + 2w = 2(5w) + 2w = 10w + 2w = 12w
\]

We are told that the perimeter must be at most 96 cm, which gives us the inequality:

\[
12w \leq 96
\]

This inequality corresponds to:

**\( 2w + 2 \cdot (5w) \leq 96 \)**

So, the correct response is:

**\( 2w + 2 \cdot (5w) \leq 96 \)**.

Would you like more details, or do you have any questions?

Here are 5 related questions for further exploration:
1. How do you solve \( 12w \leq 96 \) for \( w \)?
2. What would the dimensions of the rectangle be if the perimeter was exactly 96 cm?
3. How do we calculate the area of the rectangle if the width is known?
4. How does changing the perimeter affect the width of the rectangle?
5. Can you create an equation for a similar problem with a different perimeter constraint?

**Tip:** Remember that inequalities like this can also help you model real-world problems, such as optimizing space or material usage.

The length of a rectangle is five times its width. The perimeter of the rectangle is at most 96 cm. Which inequality models the relationship between the width and the perimeter of the rectangle?

Explore how to model the relationship between the width and the perimeter of a rectangle with a length five times its width. This problem covers key algebraic and geometric concepts, including inequalities and perimeter calculations, suitable for students in Grades 6-8.

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