To find a rational number between \(\sqrt{10}\) and \(\sqrt{11}\), we need to first approximate the values of \(\sqrt{10}\) and \(\sqrt{11}\).
- \(\sqrt{10} \approx 3.162\)
- \(\sqrt{11} \approx 3.317\)
Now, we can look for rational numbers between these two values.
From the choices provided:
- \(3.1\)
- \(3.1165\)
- \(\frac{16}{5} = 3.2\)
- \(10\)
- \(\frac{10}{3} \approx 3.333\)
- \(3.4\)
The numbers that fall between \(3.162\) and \(3.317\) are \(3.2\) and \(3.1165\).
So, both \(3.1165\) and \(\frac{16}{5}\) (which equals \(3.2\)) are rational numbers between \(\sqrt{10}\) and \(\sqrt{11}\).
If you are asked to select just one, then \(3.2\) (or \(\frac{16}{5}\)) is a clear choice, as it is a simple rational representation.