Question

Using rational approximations, what statement is true?(1 point) Responses 3>9–√ 3 greater than start root 9 end root π>9–√ pi greater than start root 9 end root π<9–√ pi less than start root 9 end root 3<9–√

Answer 1

To evaluate the given statements, we can first calculate the approximate values of the expressions involved.

1. **Calculate \(9 - \sqrt{3}\)**:
- \(\sqrt{3} \approx 1.732\)
- Therefore, \(9 - \sqrt{3} \approx 9 - 1.732 \approx 7.268\).

2. **Calculate \(\sqrt{9}\)**:
- \(\sqrt{9} = 3\).

3. **Calculate \(\pi\)**:
- \(\pi \approx 3.14159\).

4. **Calculate \(\sqrt{\pi}\)**:
- Since \(\pi \approx 3.14159\), then \(\sqrt{\pi} \approx \sqrt{3.14159} \approx 1.772\).

Now we can compare the values:

- **Comparing \( 9 - \sqrt{3} \) and \( \sqrt{9} \)**:
- \( 9 - \sqrt{3} \approx 7.268\)
- \( \sqrt{9} = 3\)
- So, \( 9 - \sqrt{3} > \sqrt{9} \).

- **Comparing \(\sqrt{\pi}\) and \(\sqrt{9}\)**:
- \( \sqrt{\pi} \approx 1.772\)
- Since \(1.772 < 3\), we have \(\sqrt{\pi} < \sqrt{9}\).

- **Comparing \(\pi\) and \(\sqrt{9}\)**:
- Since \(\pi \approx 3.14159\) and \(\sqrt{9} = 3\), we have \(\pi > 3\).

Given these comparisons, let's summarize the estimated inequalities:

1. \(9 - \sqrt{3} > \sqrt{9}\)
2. \(\sqrt{\pi} < \sqrt{9}\)
3. \(\pi > \sqrt{9}\)

Now, we can summarize the valid statement from your options:

- \( \pi > 9 - \sqrt{3} \) is the true statement.

With this analysis, the correct answer is:

**\( \pi > 9 - \sqrt{3} \)**.