Question

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 11 and 11.5
between 11 and 11.5

between 10.5 and 11
between 10.5 and 11

between 9.5 and 10

Answer 1

To find which whole number and rational number to the nearest half \( \sqrt{105} \) is between, we first need to calculate \( \sqrt{105} \).

1. **Estimate \( \sqrt{105} \)**:
We know that:
- \( 10^2 = 100 \)
- \( 11^2 = 121 \)

Therefore, \( \sqrt{105} \) is between 10 and 11. To narrow it down further, we can check the squares of numbers between 10 and 11.

- \( 10.5^2 = 110.25 \) (too high)
- \( 10.4^2 = 108.16 \) (still too high)
- \( 10.3^2 = 106.09 \) (still too high)
- \( 10.2^2 = 104.04 \) (too low)

Therefore, we have \( 10.2 < \sqrt{105} < 10.3 \).

2. **Finding the rational number to the nearest half**:
Since \( \sqrt{105} \) is approximately between 10.2 and 10.3, we round to the nearest half. The nearest halves around this interval are:
- 10.0
- 10.5

In this case, since it is closer to 10.2, \( \sqrt{105} \) rounds to 10.0 when considering halves.

Thus, the conclusion is that \( \sqrt{105} \) is between **10 and 10.5**.

The answer is:
**between 10 and 10.5**.