Question

Using a number line, what whole number and rational number to the nearest half is square root of 10 between

Answer 1

the whole number was right the rational number was wrong

Answer 2

To find the range within which \(\sqrt{10}\) lies, we can first identify two perfect squares between which 10 is situated. The perfect squares are:

- \(3^2 = 9\)
- \(4^2 = 16\)

Since \(9 < 10 < 16\), we conclude that:

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

Next, we can approximate \(\sqrt{10}\) more precisely. The square roots of 9 and 16 give us the following approximate values:

- \(\sqrt{9} = 3\)
- \(\sqrt{16} = 4\)

Now, we can calculate a more precise approximation of \(\sqrt{10}\) to find it to the nearest half. Testing values, we see that:

- \(3.1^2 = 9.61\) (too low)
- \(3.2^2 = 10.24\) (too high)

Since \(3.1 < \sqrt{10} < 3.2\), we can narrow it down further. Testing \(3.15\):

- \(3.15^2 = 9.9225\) (still too low)

Testing \(3.16\):

- \(3.16^2 = 10.0176\) (too high)

Thus, it’s clear that:

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

Rounding \(\sqrt{10}\) to the nearest half gives us \(3.0\) since \(3.15\) is closer to \(3.0\) than \(3.5\).

Now, let's summarize:

- The rational number to the nearest half that \(\sqrt{10}\) is between is \(3.0\) and \(4.0\).
- The whole number that \(\sqrt{10}\) is between is \(3\) and \(4\).

Thus, making it clear, the whole number is \(3\) and the rational number to the nearest half is \(3.0\).

Answer 3

I apologize for the oversight. Let's clarify the situation regarding the rational number.

We established that:

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

Calculating the approximate value, we found that \(\sqrt{10} \approx 3.162\).

Now, when rounding \(\sqrt{10}\) to the nearest half, we check the halves:

- **3.0**
- **3.5**
- **4.0**

Since \(3.162\) is closer to \(3.0\) than to \(3.5\), we round \(\sqrt{10}\) to:

- **The nearest half: 3.0**

Thank you for your patience, and I appreciate the correction. The whole number is **3** and the correct rational number to the nearest half is indeed **3.0**.