To find the whole number and the rational number to the nearest half that the square root of 10 (\(\sqrt{10}\)) falls between, let's first calculate \(\sqrt{10}\):
\[ \sqrt{10} \approx 3.162 \]
Next, we need to identify the whole number that \(\sqrt{10}\) is closest to. The whole numbers surrounding \(3.162\) are \(3\) and \(4\). Therefore, \(\sqrt{10}\) is between \(3\) and \(4\).
Now, we will find the rational number to the nearest half. The nearest halves surrounding \(3.162\) are \(3.0\) and \(3.5\). Since \(3.162\) is closer to \(3.0\) than to \(3.5\), we can conclude that \(\sqrt{10}\) is between \(3.0\) and \(3.5\).
Putting this all together:
\(\sqrt{10}\) is between 3 (whole number) and 3.0 (rational number to the nearest half).