A right circular cone has a height of 12 meters (m) and a base with a radius of 6 m....

1. TRANSLATE the problem information

• Given information:
  • Right circular cone with height \(\mathrm{h = 12\ m}\)
  • Base radius \(\mathrm{r = 6\ m}\)
  • Need to find: Volume in m³

2. INFER the approach

• This is a cone volume problem, so we need the cone volume formula
• The formula \(\mathrm{V = \frac{1}{3}\pi r^2h}\) requires radius and height, which we have

3. SIMPLIFY through the calculation

• Apply the formula: \(\mathrm{V = \frac{1}{3}\pi r^2h}\)
• Substitute values: \(\mathrm{V = \frac{1}{3}\pi(6)^2(12)}\)
• Calculate the exponent: \(\mathrm{V = \frac{1}{3}\pi(36)(12)}\)
• Multiply inside: \(\mathrm{V = \frac{1}{3}\pi(432)}\)
• Final multiplication: \(\mathrm{V = 144\pi\ m^3}\)

Answer: \(\mathrm{(C)\ 144\pi}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about cone vs cylinder formulas: Students might use the cylinder volume formula \(\mathrm{V = \pi r^2h}\) instead of the cone formula \(\mathrm{V = \frac{1}{3}\pi r^2h}\), forgetting that cones have the 1/3 factor.

Using \(\mathrm{V = \pi(6^2)(12) = \pi(36)(12) = 432\pi}\)

This leads them to select Choice \(\mathrm{(D)\ (432\pi)}\)

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors, particularly in calculating \(\mathrm{6^2 = 36}\). Some might incorrectly compute \(\mathrm{6^2}\) as \(\mathrm{12}\) (confusing it with \(\mathrm{6 \times 2}\)), or make multiplication errors in the sequence.

These calculation mistakes can lead to various incorrect values, potentially causing them to select Choice \(\mathrm{(A)\ (24\pi)}\) or Choice \(\mathrm{(B)\ (72\pi)}\) depending on the specific error.

The Bottom Line:

This problem tests whether students can distinguish between cone and cylinder volume formulas and execute multi-step arithmetic accurately. The presence of 432π as an answer choice is particularly tricky since it's what you get with the cylinder formula.