To determine where \( \sqrt{105} \) lies on a number line, we first need to calculate or estimate its value.
We know:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)
Since \( 100 < 105 < 121 \), it follows that \( 10 < \sqrt{105} < 11 \).
Next, let's find a more precise estimate by calculating \( \sqrt{105} \):
- Check \( 10.5^2 = 110.25 \)
- Check \( 10.25^2 = 105.0625 \)
- Check \( 10.2^2 = 104.04 \)
Thus, we can see:
- \( 10.2^2 < 105 < 10.25^2 \)
This shows that \( 10.2 < \sqrt{105} < 10.25 \), which also indicates that \( \sqrt{105} \) is greater than 10 but less than 10.5.
Next, to check the options given:
- a. between 11 and 11.5: Not correct
- b. between 9.5 and 10: Not correct
- c. between 10 and 10.5: Correct
- d. between 10.5 and 11: Not correct
Thus, the correct answer is (c) between 10 and 10.5.