To find the value of \( \sqrt{105} \) and determine its position on a number line, we will first calculate \( \sqrt{105} \).
The perfect squares near 105 are:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Since \( 100 < 105 < 121 \), it follows that:
\[ 10 < \sqrt{105} < 11 \]
To find a more precise estimate, we can try squaring numbers between 10 and 11:
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)
Thus, we can conclude that:
\[ 10.2 < \sqrt{105} < 10.3 \]
So, we can state that \( \sqrt{105} \) is approximately \( 10.24 \).
Now, rounding 10.24 to the nearest half, we notice that:
- The nearest halves are 10.0 and 10.5.
- Since 10.24 is closer to 10.0 than to 10.5, it rounds to 10.
This leads us to conclude that:
- The whole number \( \sqrt{105} \) is between 10 and 11.
- The rational number to the nearest half is approximately 10.0.
Based on this analysis, the correct response is:
between 10 and 10.5.