To determine where \( \sqrt{105} \) falls on a number line between whole numbers and rational numbers to the nearest half, we can first approximate the value of \( \sqrt{105} \).
-
We know that:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Therefore, \( \sqrt{105} \) is between \( 10 \) and \( 11 \).
-
To find a better approximation:
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)
So, \( \sqrt{105} \) is between \( 10.2 \) and \( 10.3 \).
-
To round to the nearest half:
- The nearest halves are \( 10 \) and \( 10.5 \).
- Since \( 10.2 \) is closer to \( 10 \), we can conclude that \( \sqrt{105} \) rounded to the nearest half is \( 10 \).
Thus, \( \sqrt{105} \) is between 10 and 10.5. The correct response is:
between 10 and 10.5.