To find the whole number and the rational number to the nearest half that \(\sqrt{10}\) falls between, we first need to calculate \(\sqrt{10}\).
Calculating \(\sqrt{10}\): \[ \sqrt{10} \approx 3.162 \]
Now, we can identify the whole numbers that \(\sqrt{10}\) is between. The whole numbers around \(3.162\) are \(3\) and \(4\). So, we can say:
\[ \sqrt{10} \text{ is between } 3 \text{ and } 4. \]
Next, we need to find the rational number to the nearest half. The nearest halves around \(3.162\) are \(3.0\) and \(3.5\). Since \(3.162\) is closer to \(3.0\) than \(3.5\), we can say that:
\(\sqrt{10} \text{ is approximately } 3.0\) when rounded to the nearest half.
Thus, the final answer would be: \[ \sqrt{10} \text{ is between } 3 \text{ and } 4. \]