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 10 and 10.5 between 10 and 10.5 between 9.5 and 10 between 9.5 and 10 between 11 and 11.5

Answer 1

are you sure??

Answer 2

To find where \( \sqrt{105} \) falls on a number line:

1. First, calculate \( 10.5^2 \) and \( 11^2 \):
\[
10.5^2 = 110.25
\]
\[
11^2 = 121
\]
Since \( 105 < 110.25 \) and \( 105 > 100 \) (because \( 10^2 = 100 \)), this indicates that \( \sqrt{105} \) is between 10 and 10.5.

2. Next, we can narrow it down further. To find the square of \( 10.25 \):
\[
10.25^2 = 105.0625
\]
Because \( 105 < 105.0625 \), this shows that \( \sqrt{105} < 10.25 \).

Thus, we can conclude that:
\[
10 < \sqrt{105} < 10.25
\]

Now, rounding \( \sqrt{105} \) to the nearest half, it is approximately \( 10.25 \), which is closer to \( 10.5 \) than \( 10 \).

Thus, \( \sqrt{105} \) is between **10 and 10.5**, and the rational number to the nearest half is **10.5**. The correct response from the options given is:

**between 10 and 10.5**.

Answer 3

Let’s carefully reassess \( \sqrt{105} \).

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

2. **Narrowing down the interval further**:
- Check \( 10.5 \):
\[
10.5^2 = 110.25
\]
- Since \( 105 < 110.25\), that confirms \( \sqrt{105} < 10.5 \).

3. **Check \( 10.25 \)**:
\[
10.25^2 = 105.0625
\]
- Since \( 105 < 105.0625\), that confirms \( \sqrt{105} < 10.25 \).

Thus, we have:
\[
10 < \sqrt{105} < 10.25
\]

4. **Estimate more accurately**: We can estimate \( \sqrt{105} \) more precisely using a calculator:
\[
\sqrt{105} \approx 10.247
\]

5. **Find the nearest half**:
- Rounding \( 10.247 \) to the nearest half gives us \( 10.5 \) (since \( 10.247 \) is closer to \( 10.5 \) than \( 10 \)).

Thus, \( \sqrt{105} \) is indeed between **10 and 10.5**, but it is closer to **10.5** when considering rational numbers rounded to the nearest half.

So the conclusions still stand. \( \sqrt{105} \) is between **10 and 10.5**, confirming the previous assertion. If there are any specific points of uncertainty, please let me know!