Question

Using a number line, what whole number and rational number to the nearest half is √105 between? a. between 11 and 11.5 b. between 9.5 and 10 c. between 10 and 10.5 d. between10.5 and 11

Answer 1

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 \( \sqrt{105} \):

1. Check \( 10.5^2 = 110.25 \)
2. Check \( 10.25^2 = 105.0625 \)
3. 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.