The circumference of the base of a right circular cylinder is 20pi meters, and the height of the cylinder is...

1. TRANSLATE the problem information

• Given information:
  • Circumference of base = 20π meters
  • Height = 6 meters
• Find: Volume in cubic meters

2. INFER what you need to solve the problem

• Volume formula requires radius: \(\mathrm{V = \pi r^2h}\)
• You have height (\(\mathrm{h = 6}\)) but need radius
• You can find radius from the given circumference

3. SIMPLIFY to find the radius

• Use circumference formula: \(\mathrm{C = 2\pi r}\)
• Substitute known values: \(\mathrm{20\pi = 2\pi r}\)
• Divide both sides by 2π: \(\mathrm{r = 20\pi \div 2\pi = 10}\) meters

4. SIMPLIFY to calculate volume

• Substitute into volume formula: \(\mathrm{V = \pi r^2h}\)
• \(\mathrm{V = \pi(10)^2(6)}\)
  \(\mathrm{= \pi(100)(6)}\)
  \(\mathrm{= 600\pi}\) cubic meters

Answer: C. 600π

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about radius vs diameter: Students may mistakenly use the circumference value as the diameter, leading to \(\mathrm{r = 20\pi \div 2 = 10\pi}\). Then calculating \(\mathrm{V = \pi(10\pi)^2(6)}\) results in an extremely large value that doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Missing the radius-squaring step: Weak SIMPLIFY execution where students forget to square the radius in the volume formula, calculating \(\mathrm{V = \pi rh}\) instead of \(\mathrm{V = \pi r^2h}\). This gives \(\mathrm{V = \pi(10)(6) = 60\pi}\), leading them to select Choice A (60π).

Third Most Common Error:

Using diameter instead of radius: Students correctly find \(\mathrm{r = 10}\) from circumference, but then mistakenly use the original circumference value (20π) as diameter in their head, treating 20 as the radius. This gives \(\mathrm{V = \pi(20)^2(6) = 2400\pi}\), leading them to select Choice D (2400π).

The Bottom Line:

This problem tests whether students can systematically work through connected formulas - circumference to find radius, then radius to find volume. The key insight is recognizing that volume requires radius, which must be extracted from the circumference first.