To find the length of the diagonal [tex]\( x \)[/tex] of a rectangular napkin where the length [tex]\( l \)[/tex] is twice the width [tex]\( w \)[/tex], we can use the Pythagorean theorem. Here’s the step-by-step solution:
1. Let the width be [tex]\( w \)[/tex].
2. The length [tex]\( l \)[/tex] is given as twice the width, so [tex]\( l = 2w \)[/tex].
3. According to the Pythagorean theorem, the square of the diagonal [tex]\( x \)[/tex] can be found using the lengths of the sides of the rectangle:
[tex]\[
x^2 = l^2 + w^2
\][/tex]
4. Substituting the values for [tex]\( l \)[/tex] and [tex]\( w \)[/tex]:
[tex]\[
x^2 = (2w)^2 + w^2
\][/tex]
5. Simplify the equation:
[tex]\[
x^2 = 4w^2 + w^2 = 5w^2
\][/tex]
6. Taking the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \sqrt{5w^2} = w\sqrt{5}
\][/tex]
7. To express [tex]\( x \)[/tex] in the form [tex]\( \frac{\sqrt{a}}{b} \)[/tex], note that:
[tex]\[
x = w\sqrt{5}
\][/tex]
This matches the form [tex]\( \frac{\sqrt{a}}{b} \)[/tex] if [tex]\( a = 5 \)[/tex] and [tex]\( b = 1 \)[/tex].
Therefore, the correct values are:
[tex]\( \boxed{1} \)[/tex]
