Question

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

Answer 1

To determine where \( \sqrt{105} \) lies among the given intervals, we need to first find its approximate value.

Calculating \( \sqrt{105} \):

1. Since \( 10^2 = 100 \) and \( 11^2 = 121 \), we know \( 10 < \sqrt{105} < 11 \).
2. To narrow it down further, we can calculate \( 10.2^2 \) and \( 10.3^2 \):
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)

So, \( 10.2 < \sqrt{105} < 10.3 \).

This means \( \sqrt{105} \) is definitely between 10.2 and 10.3. Now, let's identify which of the listed intervals contains \( \sqrt{105} \):

- **Between 9.5 and 10:** No, \( \sqrt{105} > 10 \).
- **Between 10 and 10.5:** Yes, \( 10 < \sqrt{105} < 10.3 \).
- **Between 10.5 and 11:** No, since \( \sqrt{105} < 10.5 \).
- **Between 11 and 11.5:** No, \( \sqrt{105} < 11 \).
- **Between 9.5 and 10:** No, already addressed.

Now, rounding \( \sqrt{105} \approx 10.247 \) to the nearest half, we get \( 10.5 \).

In summary:
- **Whole number:** \( 10 \)
- **Rational number to the nearest half:** \( 10.5 \)

Thus, \( \sqrt{105} \) is between 10 and 10.5.