To determine the whole number and the rational number to the nearest half that \(\sqrt{10}\) is between, we can start by finding perfect squares near 10.
- The perfect square just below 10 is \(3^2 = 9\).
- The perfect square just above 10 is \(4^2 = 16\).
Now, we know that:
\[ 3^2 < 10 < 4^2 \quad \Rightarrow \quad 3 < \sqrt{10} < 4 \]
So, the whole number that \(\sqrt{10}\) is between is 3 and 4.
Next, let's find \(\sqrt{10}\) to determine \( \sqrt{10} \) to the nearest half:
The approximate value of \(\sqrt{10}\) is around 3.162. To round this to the nearest half:
- The possible half numbers between 3 and 4 are 3, 3.5, and 4.
- Since 3.162 is closer to 3 than 3.5, we round it down to 3.
Thus, we can conclude that:
\(\sqrt{10}\) is between 3 and 4 and to the nearest half, it is 3.
To summarize:
\(\sqrt{10}\) is between 3 and 4.