To find the whole number and rational number to the nearest half between which \(\sqrt{10}\) lies, we first calculate the decimal approximation of \(\sqrt{10}\).
Calculating it gives: \[ \sqrt{10} \approx 3.162 \]
Now, we need to determine the whole number and the nearest half around this value.
The whole number less than \(3.162\) is \(3\), and the whole number greater than \(3.162\) is \(4\).
Next, we will check for the nearest halves:
- Halfway between \(3\) and \(4\) is \(3.5\).
Thus, we can see that: \[ 3 < \sqrt{10} < 3.5 \]
Therefore, we conclude that \(\sqrt{10}\) lies between the whole number \(3\) and the rational number \(3.5\).
In summary: \[ \sqrt{10} \text{ is between } 3 \text{ and } 3.5. \]