Question

Using rational approximations, what statement is true?(1 point)Responses3>9–√3 is greater than square root of 9π<9–√π<9–√3<9–√3<9–√π>9–√

Answer 1

To analyze the statement using rational approximations, let's evaluate the terms involved.

1. **Square root of 9, \(\sqrt{9}\)**: This equals 3.
2. **Square root of \(\pi\), \(\sqrt{\pi}\)**: The approximate value of \(\pi\) is about 3.14, so \(\sqrt{\pi} \approx 1.77\).
3. **Square root of 3, \(\sqrt{3}\)**: The approximate value of \(\sqrt{3} \approx 1.73\).

Now, substituting these approximations into the inequality:

- \(\sqrt{9} = 3\)
- \(\sqrt{3} \approx 1.73\)
- \(\sqrt{\pi} \approx 1.77\)

Now we can see how the various terms relate:

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

Now let's analyze the proposed inequalities from the options given in the question:

The correct relationships derived:

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

Thus, this confirms that:

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

None of these directly correspond to properly written inequalities. However, from our simplification, we verify that:

- \(3 > \sqrt{9 - \sqrt{3}} > \sqrt{9 - \sqrt{\pi}}\) is one of the deductions made here.

Thus, the correct conclusion aligns with the separation of square roots and their numerical ranges.

So, **the true statement** is:

\[ 3 > 9 - \sqrt{3} > \sqrt{9 - \sqrt{\pi}}\]

This shows how the approximated terms work together based on rational comparisons.