Question

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

between 10 and 10.5
between 10 and 10.5

between 9.5 and 10
between 9.5 and 10

between 10.5 and 11
between 10.5 and 11

between 11 and 11.5

Answer 1

To determine where \( \sqrt{105} \) lies on a number line, we can calculate its approximate value.

Calculating \( \sqrt{105} \):

1. We know that \( 10^2 = 100 \) and \( 11^2 = 121 \). So, \( \sqrt{105} \) must be between 10 and 11.
2. To narrow it down, we can try some values:
- \( 10.2^2 = 104.04 \) (too low)
- \( 10.3^2 = 106.09 \) (too high)

This tells us that \( \sqrt{105} \) is between 10.2 and 10.3.

To find the nearest half (0.5), we note that:
- 10.2 rounds to 10.0
- 10.3 rounds to 10.5

Thus, \( \sqrt{105} \) is between approximately 10.2 and 10.3, so it is:

**between 10 and 10.5.**

So the correct response is: **between 10 and 10.5.**