Question

Identify the approximate location of each of the numbers on a number line.
A. π^2
B. √180
C. 3√38

Answer 1

To identify the approximate locations of the numbers \( A: \pi^2 \), \( B: \sqrt{180} \), and \( C: 3\sqrt{38} \) on a number line, we need to calculate their approximate values.

### A. \( \pi^2 \)
We know that \( \pi \approx 3.14 \).
So,
\[
\pi^2 \approx (3.14)^2 \approx 9.86
\]

### B. \( \sqrt{180} \)
We can simplify it:
\[
\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}
\]
Now, since \( \sqrt{5} \approx 2.236 \):
\[
6\sqrt{5} \approx 6 \times 2.236 \approx 13.416
\]

### C. \( 3\sqrt{38} \)
We approximate \( \sqrt{38} \). Since \( 6^2 = 36 \) and \( 7^2 = 49 \), we know \( \sqrt{38} \) is between 6 and 7. Using better approximations,
\[
\sqrt{38} \approx 6.16
\]
Thus,
\[
3\sqrt{38} \approx 3 \times 6.16 \approx 18.48
\]

### Summary of Approximations
- \( A: \pi^2 \approx 9.86 \)
- \( B: \sqrt{180} \approx 13.416 \)
- \( C: 3\sqrt{38} \approx 18.48 \)

### Placement on Number Line
Based on these approximations:
- \( A (\pi^2) \) is approximately at 9.86
- \( B (\sqrt{180}) \) is approximately at 13.416
- \( C (3\sqrt{38}) \) is approximately at 18.48

So the order on the number line is \( A < B < C \) (or \( 9.86 < 13.416 < 18.48 \)).