Certainly! Let's solve the given problem step-by-step.
1. Determine the dimensions:
- The length of the napkin, [tex]\( l \)[/tex], is twice as long as the width, [tex]\( w \)[/tex].
- So if the width [tex]\( w = 1 \)[/tex] unit, then the length [tex]\( l = 2 \times w = 2 \)[/tex] units.
2. Calculate the length of the diagonal:
- The napkin forms a right triangle with the length and width.
- According to the Pythagorean theorem: [tex]\[ \text{diagonal}^2 = l^2 + w^2 \][/tex]
- Substituting the values: [tex]\[ \text{diagonal}^2 = (2 \times 1)^2 + (1)^2 \][/tex]
- Simplify: [tex]\[ \text{diagonal}^2 = 2^2 + 1^2 = 4 + 1 = 5 \][/tex]
- Therefore, [tex]\[ \text{diagonal} = \sqrt{5} \][/tex]
3. Replace [tex]\( a \)[/tex] and [tex]\( b \)[/tex] with appropriate values:
- Here, [tex]\( a \)[/tex] corresponds to the value under the square root in the expression [tex]\( \sqrt{a} \)[/tex].
- From the calculation above, we have [tex]\( \text{diagonal} = \sqrt{5} \)[/tex].
- Therefore, [tex]\( a = 5 \)[/tex].
- The expression given is: [tex]\[ x = \frac{\sqrt{a}}{b} \][/tex]
- Given the length [tex]\( l = 2 \)[/tex], it corresponds to [tex]\( b \)[/tex]=2.
Thus, the values we are looking for are:
[tex]\[ a = 5, \, b = 2 \][/tex]
Finally, substituting [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the expression:
[tex]\[ x = \frac{\sqrt{5}}{2} \][/tex]
This matches the result given in the problem. Here is your final boxed answer:
[tex]\[
a = 5, \, b = 2
\][/tex]
