Question

Using a number line, what whole number and rational number to the nearest half is √105 between?

answers

between 9.5 and 10

between 11 and 11.5

between 10.5 and 11

between 10 and 10.5

Answer 1

you sure

Answer 2

To determine where \( \sqrt{105} \) falls, we can first calculate its approximate value.

We know that \( 10^2 = 100 \) and \( 11^2 = 121 \), which means

\[
10 < \sqrt{105} < 11
\]

Now let's find a more precise value for \( \sqrt{105} \).

Using a calculator, \( \sqrt{105} \approx 10.247 \).

Next, we need to identify which of the provided options is correct by estimating \( \sqrt{105} \) to the nearest half:

- The nearest half values to \( 10.247 \) are 10 and 10.5.
- Since \( 10.247 \) is greater than 10 but less than 10.5, it falls in the interval:

\[
\text{between } 10 \text{ and } 10.5
\]

Thus, the correct answer is:

**between 10 and 10.5**.

Answer 3

Let's verify the calculation and reasoning step by step:

1. We know that \( \sqrt{105} \) lies between \( 10 \) and \( 11 \) because:
- \( 10^2 = 100 \) (which is less than 105)
- \( 11^2 = 121 \) (which is more than 105)

2. Let's estimate \( \sqrt{105} \) further:
- Trying \( 10.2 \):
\[
10.2^2 = 104.04 \quad (\text{so } \sqrt{105} > 10.2)
\]
- Trying \( 10.3 \):
\[
10.3^2 = 106.09 \quad (\text{so } \sqrt{105} < 10.3)
\]
- Thus, we have \( 10.2 < \sqrt{105} < 10.3 \).

3. To find out to the nearest half:
- The nearest halves around \( 10.247 \) (which is approximately \( \sqrt{105} \)) are \( 10 \) and \( 10.5 \).
- As \( 10.247 \) is greater than \( 10 \) but less than \( 10.5 \), it confirms that \( \sqrt{105} \) is between \( 10 \) and \( 10.5 \).

So, my original conclusion stands correct:

**Between 10 and 10.5** is indeed the accurate range for \( \sqrt{105} \).