Question

Using rational approximations what statement is true
√16<4
√12<π
√12>π
√16>4

Answer 1

Let’s evaluate each of the statements using rational approximations.

1. **\(\sqrt{16} < 4\)**:
- \(\sqrt{16} = 4\)
- Therefore, this statement is **false**.

2. **\(\sqrt{12} < \pi\)**:
- 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 \(3.464 > 3.14\).

3. **\(\sqrt{12} > \pi\)**:
- Since we already approximated \(\sqrt{12} \approx 3.464\) and \(\pi \approx 3.14\), this statement is also **true**.

4. **\(\sqrt{16} > 4\)**:
- 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 **\(\sqrt{12} > \pi\)**.