Question

Using rational approximations, what statement is true?(1 point)%0D%0AResponses%0D%0A%0D%0A48−−√<36−−√%0D%0Astart root 48 end root less than start root 36 end root%0D%0A%0D%0A48−−√>36−−√%0D%0Astart root 48 end root greater than start root 36 end root%0D%0A%0D%0A49−−√>7%0D%0Astart root 49 end root greater than 7%0D%0A%0D%0A49−−√<7

Answer 1

To determine the truth of each statement using rational approximations, let's evaluate them one by one.

1. \(\sqrt{48} < \sqrt{36}\)

   • We know that \(\sqrt{36} = 6\).
   • For \(\sqrt{48}\), we can approximate it: \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \), which is approximately \( 4 \times 1.732 \approx 6.928 \).
   • Therefore, \( \sqrt{48} \approx 6.928 \) and \( \sqrt{36} = 6 \).
   • This statement is false because \( 6.928 > 6 \).

2. \(\sqrt{48} > \sqrt{36}\)

   • As calculated, \( \sqrt{48} \approx 6.928 \) and \( \sqrt{36} = 6 \).
   • Therefore, this statement is true because \( 6.928 > 6 \).

3. \(\sqrt{49} > 7\)

   • We know that \(\sqrt{49} = 7\).
   • Therefore, this statement is false because \( 7 \) is not greater than \( 7 \).

4. \(\sqrt{49} < 7\)

   • Again, since \(\sqrt{49} = 7\), this statement is also false because \( 7 \) is not less than \( 7 \).

Conclusion: The true statement from the options is \(\sqrt{48} > \sqrt{36}\).