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?

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.