To determine how many cups of lemonade Dalton used, we start with the amount that Alvin used. Alvin used \( 3 \frac{1}{6} \) cups of lemonade.
First, we convert \( 3 \frac{1}{6} \) to an improper fraction:
\[ 3 \frac{1}{6} = \frac{3 \times 6 + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6} \]
Dalton used 2 times as much lemonade as Alvin. Therefore, we need to calculate:
\[ 2 \times \frac{19}{6} = \frac{2 \times 19}{6} = \frac{38}{6} \]
Now, we simplify \( \frac{38}{6} \):
\[ \frac{38}{6} = \frac{19}{3} \]
Next, we can convert \( \frac{19}{3} \) to a mixed number:
\[ \frac{19}{3} = 6 \frac{1}{3} \]
Therefore, the amount of lemonade Dalton used is
\[ \boxed{6 \frac{1}{3}} \text{ cups.} \]
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