To analyze the statements provided using rational approximations, let's evaluate the values of the square roots:
-
√48: We know that \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \). Approximating \( \sqrt{3} \approx 1.732 \), we have: \[ \sqrt{48} \approx 4 \times 1.732 = 6.928 \]
-
√36: This is straightforward, as \( \sqrt{36} = 6 \).
-
√49: This is also straightforward, as \( \sqrt{49} = 7 \).
Now let's evaluate the options:
-
√48 > √36
- We have \( \sqrt{48} \approx 6.928 \) and \( \sqrt{36} = 6 \). So, this statement is true.
-
√49 < 7
- Since \( \sqrt{49} = 7 \), this statement is false.
-
√49 > 7
- Also false, because \( \sqrt{49} = 7 \).
-
√48 < √36
- False, as we determined \( \sqrt{48} \approx 6.928 > 6 \).
The only true statement is:
√48 > √36.