A right circular cylinder has a circular base whose circumference is 22pi centimeters. The height of the cylinder is 6...

1. TRANSLATE the problem information

• Given information:
  • Circumference of circular base: \(\mathrm{C = 22\pi}\) cm
  • Height of cylinder: \(\mathrm{h = 6}\) cm
  • Need to find: Volume in cubic centimeters

2. INFER the solution strategy

• To find volume of a cylinder, I need the volume formula \(\mathrm{V = \pi r^2h}\)
• I have height (6 cm) but need radius
• The circumference can give me the radius using \(\mathrm{C = 2\pi r}\)

3. SIMPLIFY to find the radius

• Start with circumference formula: \(\mathrm{C = 2\pi r}\)
• Substitute known value: \(\mathrm{22\pi = 2\pi r}\)
• Divide both sides by \(\mathrm{2\pi}\): \(\mathrm{r = \frac{22\pi}{2\pi} = 11}\) cm

4. SIMPLIFY to find the volume

• Apply volume formula: \(\mathrm{V = \pi r^2h}\)
• Substitute known values: \(\mathrm{V = \pi(11)^2(6)}\)
• Calculate: \(\mathrm{V = \pi(121)(6) = 726\pi}\) cubic centimeters

Answer: (C) 726π

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find radius first before applying the volume formula. They might try to directly use the circumference value (\(\mathrm{22\pi}\)) in place of radius in the volume formula, calculating something like \(\mathrm{V = \pi(22\pi)^2(6)}\), leading to a massively incorrect result that doesn't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Calculation error in SIMPLIFY: Students correctly identify the strategy but make arithmetic mistakes when computing \(\mathrm{r = \frac{22\pi}{2\pi}}\) or when calculating \(\mathrm{\pi(11)^2(6)}\). For example, they might incorrectly calculate \(\mathrm{(11)^2 = 111}\) instead of \(\mathrm{121}\), leading to \(\mathrm{V = \pi(111)(6) = 666\pi}\), which doesn't match any given choice. This may lead them to select Choice (A) (132π) as the "closest" answer.

The Bottom Line:

This problem tests whether students can work backwards from circumference to radius before applying a volume formula. The key insight is recognizing the two-step process: circumference → radius → volume.