Question

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

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
between 11 and 11.5

between 10 and 10.5

Answer 1

To determine where \( \sqrt{105} \) falls on a number line between whole numbers and rational numbers to the nearest half, we can first approximate the value of \( \sqrt{105} \).

1. We know that:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)

Therefore, \( \sqrt{105} \) is between \( 10 \) and \( 11 \).

2. To find a better approximation:
- \( 10.2^2 = 104.04 \) (which is less than 105)
- \( 10.3^2 = 106.09 \) (which is greater than 105)

So, \( \sqrt{105} \) is between \( 10.2 \) and \( 10.3 \).

3. To round to the nearest half:
- The nearest halves are \( 10 \) and \( 10.5 \).
- Since \( 10.2 \) is closer to \( 10 \), we can conclude that \( \sqrt{105} \) rounded to the nearest half is \( 10 \).

Thus, \( \sqrt{105} \) is between **10 and 10.5**. The correct response is:

**between 10 and 10.5.**