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.
There cannot be decimal points. Only whole real numbers, only the options I put.

Answer 1

To locate each of the numbers on a number line from 0 to 20, we'll calculate the approximate values of \( π^2 \), \( \sqrt{180} \), and \( 3\sqrt{38} \).

### A. \( π^2 \)

The value of \( π \) is approximately 3.14.

Calculating \( π^2 \):
\[
π^2 ≈ (3.14)^2 ≈ 9.8596
\]

Since we are looking for whole numbers, \( π^2 \) is approximately 10.

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

To simplify \( \sqrt{180} \):
\[
\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}
\]
Now let's find \( \sqrt{5} \) which is approximately 2.236. Therefore:
\[
6\sqrt{5} ≈ 6 \times 2.236 ≈ 13.416
\]
Rounding to the nearest whole number, \( \sqrt{180} \) is approximately 13.

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

First, we approximate \( \sqrt{38} \). Since \( 36 < 38 < 49 \), we know \( 6 < \sqrt{38} < 7 \). A better approximation gives us:
\[
\sqrt{38} ≈ 6.164
\]
Multiplying this by 3 gives:
\[
3\sqrt{38} ≈ 3 \times 6.164 ≈ 18.492
\]
Rounding to the nearest whole number, \( 3\sqrt{38} \) is approximately 18.

### Summary of Approximate Locations
- \( π^2 \) is approximately **10**
- \( \sqrt{180} \) is approximately **13**
- \( 3\sqrt{38} \) is approximately **18**

Thus, on the number line provided, we can place:
- **10** for \( π^2 \)
- **13** for \( \sqrt{180} \)
- **18** for \( 3\sqrt{38} \)