Question

To the nearest hundredth, \(\pi = 3.14\) and \(\sqrt{10} = 3.16\)

Column A	Column B
\(\pi^2\)	10

 

Hint:

For quantitative comparison questions involving squares and square roots, it's often easier to compare the numbers before squaring them (or after taking the square root of both columns). This can help avoid complex calculations.

Answer 1

Step 1: Understanding the Concept:
This question asks for a comparison between the square of \(\pi\) and the number 10. We are given approximate values for \(\pi\) and \(\sqrt{10}\) to help with the comparison.
Step 2: Key Formula or Approach:
The most direct way to compare \(\pi^2\) and 10 is to compare \(\pi\) and \(\sqrt{10}\). If \(a\) and \(b\) are positive numbers, then \(a<b\) is equivalent to \(a^2<b^2\).
Step 3: Detailed Explanation:
We are given the approximations \(\pi \approx 3.14\) and \(\sqrt{10} \approx 3.16\).
Based on these values, we can see that \(\pi\) is less than \(\sqrt{10}\).
\[ 3.14<3.16 \] \[ \pi<\sqrt{10} \] Since both \(\pi\) and \(\sqrt{10}\) are positive numbers, we can square both sides of the inequality without changing its direction:
\[ \pi^2<(\sqrt{10})^2 \] \[ \pi^2<10 \] This shows that the quantity in Column A is less than the quantity in Column B.
Alternatively, we could square the given approximation for \(\pi\):
\[ \pi^2 \approx (3.14)^2 = 3.14 \times 3.14 = 9.8596 \] Since 9.8596 is less than 10, the quantity in Column A is smaller.
Step 4: Final Answer:
Both methods show that \(\pi^2\) is less than 10. Therefore, the quantity in Column B is greater.