Question

Using a number line, what whole number and rational number to the nearest half is 10−−√
between?(1 point)
10−−√ is between (response area) and (response area).

Answer 1

To find where \( \sqrt{10} \) lies on the number line, we can first estimate its value.

Calculating:

\[
\sqrt{9} = 3 \quad \text{and} \quad \sqrt{16} = 4
\]

Since \( 9 < 10 < 16 \), we know:

\[
3 < \sqrt{10} < 4
\]

To get a better estimate, we can test the value \( 3.1 \):

\[
(3.1)^2 = 9.61 \quad \text{(too low)}
\]

Now let’s try \( 3.2 \):

\[
(3.2)^2 = 10.24 \quad \text{(too high)}
\]

So, \( \sqrt{10} \) is between \( 3.1 \) and \( 3.2 \).

Next, we can get a more precise estimate. Trying \( 3.15 \):

\[
(3.15)^2 = 9.9225 \quad \text{(too low)}
\]

Now try \( 3.16 \):

\[
(3.16)^2 = 10.0356 \quad \text{(too high)}
\]

So, we conclude that:

\[
3.15 < \sqrt{10} < 3.16
\]

Approximately, we can round \( \sqrt{10} \) to \( 3.16 \), which is slightly above \( 3.1 \) and below \( 3.2 \).

Now, if we want to express \( \sqrt{10} \) to the nearest half, we find that it is closest to \( 3.5 \).

However, strictly on whole numbers and the rational number\, to the nearest half:

The whole number range for \( \sqrt{10} \) is between \( 3 \) and \( 4 \), and the nearest half for rational approximation is \( 3.5 \).

Thus, we can say:

\(\sqrt{10}\) is between **3** and **3.5**.