https://file.aiquickdraw.com/mathbot/file/1728331139220ybipoi2f.jpg

To solve this problem, let's analyze the information provided and use the Pythagorean theorem.

### Problem Breakdown:
1. The napkin is a rectangle with:
   - Length \( l \)
   - Width \( w \)
   - Diagonal \( x \)
2. The length \( l \) is twice as long as the width \( w \), so \( l = 2w \).
3. We need to express \( x \) in terms of \( l \).

### Solution:
The diagonal \( x \) of a rectangle can be found using the Pythagorean theorem:
\[
x = \sqrt{l^2 + w^2}
\]

Since \( l = 2w \), substitute \( w = \frac{l}{2} \) into the equation:
\[
x = \sqrt{l^2 + \left(\frac{l}{2}\right)^2}
\]
Expanding \( \left(\frac{l}{2}\right)^2 \):
\[
x = \sqrt{l^2 + \frac{l^2}{4}} = \sqrt{\frac{4l^2 + l^2}{4}} = \sqrt{\frac{5l^2}{4}}
\]
Now, simplifying further:
\[
x = \frac{\sqrt{5} \cdot l}{2}
\]

### Final Answer:
Replace \( a \) and \( b \) with the appropriate values:
\[
x = \frac{\sqrt{5}}{2} l
\]

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

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Here are some additional related questions to consider:
1. How would the formula change if \( l \) were three times \( w \) instead of two?
2. Can you calculate \( x \) if \( l \) is a specific length, say 10 cm?
3. What happens to the value of \( x \) as \( l \) increases? Does it increase proportionally?
4. How would you find \( w \) if you only know \( x \) and \( l \)?
5. If this were a square napkin, how would you calculate the diagonal \( x \)?

**Tip:** In geometry, remember that the Pythagorean theorem applies to any right triangle, including the ones formed by the diagonals of rectangles.

A restaurant uses rectangular napkins where the length, l, is twice as long as the width. The length of the napkin along the diagonal is x. What is x in terms of l? Replace a and b with the correct values.

Learn how to find the diagonal of a rectangle when the length is twice the width. This problem uses the Pythagorean theorem to express the diagonal in terms of the length. Suitable for students in Grades 8-10, with step-by-step guidance and formula breakdown.

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