Question

Using rational approximations, what statement is true?
48−−√<36−−√ start root 48 end root less than start root 36 end root 49−−√<7 start root 49 end root less than 7 48−−√>36−−√ start root 48 end root greater than start root 36 end root 49−−√>7

Answer 1

To analyze the inequalities given, we first need to determine the approximate values of the square roots involved.

1. **Calculate \(\sqrt{48}\)**:
\[
\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}
\]
Using a rational approximation for \(\sqrt{3}\) (approximately 1.732):
\[
\sqrt{48} \approx 4 \times 1.732 \approx 6.928
\]

2. **Calculate \(\sqrt{36}\)**:
\[
\sqrt{36} = 6
\]

3. **Calculate \(\sqrt{49}\)**:
\[
\sqrt{49} = 7
\]

Now, we can evaluate the inequalities:

- \( \sqrt{48} < \sqrt{36} \) translates to approximately \( 6.928 < 6 \), which is **false**.
- \( \sqrt{48} < \sqrt{49} \) translates to approximately \( 6.928 < 7 \), which is **true**.
- \( \sqrt{36} < \sqrt{49} \) translates to \( 6 < 7 \), which is also **true**.

Next, let's summarize the evaluations:
- We have established that \( \sqrt{48} \) is greater than \( \sqrt{36} \) and less than \( \sqrt{49} \).
- This implies that \( \sqrt{48} \) is indeed between \( \sqrt{36} \) and \( \sqrt{49} \).

Therefore, the true statements derived from the approximations are:
- \( \sqrt{36} < \sqrt{48} < \sqrt{49} \)

We can also state:
- \( \sqrt{48} > \sqrt{36} \)
- \( \sqrt{49} > 7 \)

The logical conclusion based on the calculated approximate values is:
\[
\sqrt{48} > \sqrt{36} \quad \text{and} \quad \sqrt{48} < \sqrt{49}
\]

Thus, the valid statements regarding these square roots would be:
- **\(\sqrt{48} > \sqrt{36}\)**

And the relevant part of your options suggests:
- \(48^{0.5} > 36^{0.5}\) and \(49^{0.5} > 7\)

So the statement "48−−√>36−−√" is true, and "49−−√>7" is also true.