Let’s evaluate each of the statements using rational approximations.
-
\(\sqrt{16} < 4\):
- \(\sqrt{16} = 4\)
- Therefore, this statement is false.
-
\(\sqrt{12} < \pi\):
- Approximating \(\sqrt{12}\):
\(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\).
- \(\sqrt{3} \approx 1.732\)
- Thus, \(\sqrt{12} \approx 2 \times 1.732 \approx 3.464\).
- Now, \(\pi \approx 3.14\).
- Therefore, this statement is true as \(3.464 > 3.14\).
- Approximating \(\sqrt{12}\):
\(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\).
-
\(\sqrt{12} > \pi\):
- Since we already approximated \(\sqrt{12} \approx 3.464\) and \(\pi \approx 3.14\), this statement is also true.
-
\(\sqrt{16} > 4\):
- As mentioned before, \(\sqrt{16} = 4\), thus this is false.
So the statements we evaluated show that \(\sqrt{12} > \pi\) and \(\sqrt{12} < \pi\) cannot both be true at the same time; \(\sqrt{12} > \pi\) is the correct statement.
Thus, the correct statements are:
- \(\sqrt{12} > \pi\) is true.
- \(\sqrt{12} < \pi\) is false.
- \(\sqrt{16} < 4\) is false.
- \(\sqrt{16} > 4\) is false.
The overall conclusion is that the only true statement is \(\sqrt{12} > \pi\).