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:

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