To find a rational number between \( \sqrt{10} \) and \( \sqrt{11} \), we first need to determine the approximate values of these square roots.
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Calculate \( \sqrt{10} \):
- \( \sqrt{10} \approx 3.162 \)
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Calculate \( \sqrt{11} \):
- \( \sqrt{11} \approx 3.317 \)
Now we look for a rational number between these two values. Among the options provided, we can analyze them:
- 3.1: This is less than 3.162, so it is not between \( \sqrt{10} \) and \( \sqrt{11} \).
- \( \frac{10}{3} \approx 3.333 \): This is greater than \( \sqrt{11} \), so it is not between the two.
- \( \frac{16}{5} = 3.2 \): This is between 3.162 and 3.317.
- 3.4: This is greater than \( \sqrt{11} \), so it is not between the two.
The only rational number from your list that lies between \( \sqrt{10} \) and \( \sqrt{11} \) is \( \frac{16}{5} \).
Thus, the answer is: \( \frac{16}{5} \).
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