To determine the length of the diagonal [tex]\( x \)[/tex] of a rectangular napkin where the length [tex]\( l \)[/tex] is twice as long as the width [tex]\( w \)[/tex], we start with the given relationships:
1. The length [tex]\( l \)[/tex] of the napkin is [tex]\( l = 2w \)[/tex].
Using the Pythagorean theorem, the diagonal [tex]\( x \)[/tex] is related to the length and the width by the equation:
[tex]\[ x^2 = l^2 + w^2 \][/tex]
Since [tex]\( l = 2w \)[/tex]:
[tex]\[ x^2 = (2w)^2 + w^2 \][/tex]
[tex]\[ x^2 = 4w^2 + w^2 \][/tex]
[tex]\[ x^2 = 5w^2 \][/tex]
Taking the square root of both sides:
[tex]\[ x = \sqrt{5w^2} \][/tex]
[tex]\[ x = \sqrt{5} \cdot w \][/tex]
Thus, the expression for [tex]\( x \)[/tex] can be written in the form [tex]\( x = \frac{\sqrt{a}}{b} \)[/tex] by identifying the corresponding terms:
[tex]\[ x = \frac{\sqrt{5}}{1} \][/tex]
where [tex]\( a = 5 \)[/tex] and [tex]\( b = 1 \)[/tex].
Therefore, we have:
[tex]\[ x = \frac{\sqrt{5}}{1} \][/tex]
So, the correct values to replace [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are [tex]\( a = 5 \)[/tex] and [tex]\( b = 1 \)[/tex]. Thus, the final answer is:
[tex]\[ x = \frac{\sqrt{a}}{b} = \frac{\sqrt{5}}{1} \][/tex]
