Question

Question
Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point)
Responses

between 10.5 and 11
between 10.5 and 11

between 11 and 11.5
between 11 and 11.5

between 9.5 and 10
between 9.5 and 10

between 10 and 10.5

Answer 1

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:

1. **between 10.5 and 11**: No
2. **between 11 and 11.5**: No
3. **between 9.5 and 10**: No
4. **between 10 and 10.5**: Yes

Therefore, \( \sqrt{105} \) falls between \( 10 \) and \( 10.5 \).

The correct response is:

**between 10 and 10.5**.