To determine where \(\sqrt{105}\) lies on a number line with respect to whole numbers and rational numbers to the nearest half, we first need to calculate the approximate value of \(\sqrt{105}\).
We know:
- \(\sqrt{100} = 10\)
- \(\sqrt{121} = 11\)
Since \(105\) is between \(100\) and \(121\), it follows that:
\[ 10 < \sqrt{105} < 11 \]
Next, we can refine our estimate of \(\sqrt{105}\). By squaring some numbers between \(10\) and \(11\):
- \(10.5^2 = 110.25\) (too high)
- \(10.2^2 = 104.04\) (too low)
- \(10.3^2 = 106.09\) (too high)
So we can narrow it down further:
- \(10.2 < \sqrt{105} < 10.3\)
Now let’s check \(10.25\):
\[ 10.25^2 = 105.0625 \quad (too high) \]
Now we try \(10.23\):
\[ 10.23^2 = 104.5929 \quad (too low) \]
Thus, we conclude that:
\[ 10.23 < \sqrt{105} < 10.25 \]
This means that \(\sqrt{105}\) is between \(10.2\) and \(10.3\). When rounding to the nearest half, the approximate value of \(\sqrt{105}\) is \(10.5\).
So, \(\sqrt{105}\) is actually between \(10\) and \(10.5\), which corresponds to option C: Between 10 and 10.5.