Using rational approximations, what statement is true?

Question

A. √12 > pi
B.√16 > 4
C.√12 < pi
D.√16 < 4

Answer 1

To evaluate the statements, we will calculate the values of each expression.

1. For statement A: \( \sqrt{12} > \pi \)
   We can approximate \( \sqrt{12} \) as follows:
   \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \).
   \( \sqrt{3} \) is approximately \( 1.732 \), so:
   \( 2\sqrt{3} \approx 2 \times 1.732 \approx 3.464 \).
   We know that \( \pi \) is approximately \( 3.14159 \).
   Hence, \( 3.464 > 3.14159 \) → \( \sqrt{12} > \pi \) is true.

2. For statement B: \( \sqrt{16} > 4 \)
   Here, \( \sqrt{16} = 4 \).
   So, \( 4 > 4 \) is false.

3. For statement C: \( \sqrt{12} < \pi \)
   Since we determined that \( \sqrt{12} > \pi \) is true, this statement is false.

4. For statement D: \( \sqrt{16} < 4 \)
   Again, since \( \sqrt{16} = 4 \), the statement \( 4 < 4 \) is false.

Thus, the only statement that is true is A: \( \sqrt{12} > \pi \).