Using rational approximations, what statement is true?

Question

A. square root 16 > square root 4
B. square root 12 > pi
C. square root 16 < 4
D. square root 12 < pi

Answer 1

To evaluate the given statements, let's calculate the values of the square roots and $\pi$ as needed:

1. Statement A: $\sqrt{16}>\sqrt{4}$

   • $\sqrt{16}=4$
   • $\sqrt{4}=2$
   • Therefore, $4>2$ is true.

2. Statement B: $\sqrt{12}>\pi$

   • $\sqrt{12}=\sqrt{4\times 3}=2\sqrt{3}$
   • The approximate value of $\sqrt{3}\approx 1.732$, hence $2\sqrt{3}\approx 3.464$.
   • The approximate value of $\pi \approx 3.14159$.
   • Therefore, $3.464>3.14159$ is true.

3. Statement C: $\sqrt{16}<4$

   • We already established that $\sqrt{16}=4$.
   • Thus, $4<4$ is false.

4. Statement D: $\sqrt{12}<\pi$

   • We found that $\sqrt{12}\approx 3.464$ and $\pi \approx 3.14159$.
   • Therefore, $3.464<3.14159$ is false.

From this evaluation:

• Statements A and B are true.
• Statements C and D are false.

The true statements are A: $\sqrt{16}>\sqrt{4}$ and B: $\sqrt{12}>\pi$.