To find the value of \( \sqrt{105} \) and determine between which whole number and rational number it falls, we first calculate \( \sqrt{105} \).
Calculating \( \sqrt{105} \):
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Since \( 10^2 < 105 < 11^2 \), we know that \( \sqrt{105} \) is between 10 and 11.
To find a more precise value, we can calculate:
- \( 10.2^2 = 104.04 \)
- \( 10.3^2 = 106.09 \)
This tells us that \( 10.2 < \sqrt{105} < 10.3 \).
Next, to round \( \sqrt{105} \) to the nearest half, we consider 10.2:
- \( 10.2 \) is closer to \( 10.0 \) than to \( 10.5 \).
Thus, it seems \( \sqrt{105} \) is approximately 10.24. Given this rounding, it is nearest to:
- \( 10.0 \)
Now we can analyze the options:
- between 10.5 and 11: No
- between 11 and 11.5: No
- between 9.5 and 10: No
- between 10 and 10.5: Yes
Therefore, \( \sqrt{105} \) falls between \( 10 \) and \( 10.5 \).
The correct response is:
between 10 and 10.5.