Question

Identify the approximate location of each of the numbers on a number line.
A. π^2
B. √180
C. 3√38
Places on the number line: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20.

Answer 1

To place the numbers \(A. \pi^2\), \(B. \sqrt{180}\), and \(C. 3\sqrt{38}\) on a number line with specific reference points, let's first calculate each of them to get approximate values.

### A. \(\pi^2\)

We know \(\pi \approx 3.14\), so:

\[
\pi^2 \approx (3.14)^2 \approx 9.8596
\]

Thus, \(\pi^2\) is approximately \(9.86\), which is between 9 and 10 on the number line.

### B. \(\sqrt{180}\)

To simplify \(\sqrt{180}\):

\[
\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}
\]

Now, we will approximate \(\sqrt{5}\). It's known that \(\sqrt{5} \approx 2.236\):

\[
\sqrt{180} \approx 6 \times 2.236 \approx 13.416
\]

Thus, \(\sqrt{180}\) is approximately \(13.42\), which is between 13 and 14 on the number line.

### C. \(3\sqrt{38}\)

To find \(3\sqrt{38}\):

First, approximate \(\sqrt{38}\). We know that:

\[
6^2 = 36 \quad \text{and} \quad 7^2 = 49 \quad \Rightarrow \quad 6 < \sqrt{38} < 7
\]

To estimate more precisely, we can know that \(\sqrt{36} = 6\) and use \(6.1^2 = 37.21\) and \(6.2^2 = 38.44\), leading us to:

\[
\sqrt{38} \approx 6.2
\]

Thus:

\[
3\sqrt{38} \approx 3 \times 6.2 = 18.6
\]

So, \(3\sqrt{38}\) is approximately \(18.6\), which is between 18 and 19 on the number line.

### Summary of Locations

- \(A. \pi^2 \approx 9.86\) (between 9 and 10)
- \(B. \sqrt{180} \approx 13.42\) (between 13 and 14)
- \(C. 3\sqrt{38} \approx 18.6\) (between 18 and 19)

You can place them on the number line roughly as follows:
- \(π^2\) near 10,
- \(\sqrt{180}\) near 13.5,
- \(3\sqrt{38}\) near 18.5.