A right circular cylinder has a circular base with circumference 8pi inches and a height of 12 inches. What is...

1. TRANSLATE the problem information

• Given information:
  • Right circular cylinder with circumference = \(8\pi\) inches
  • Height = 12 inches
  • Need to find: volume in cubic inches

2. INFER the solution strategy

• To find volume of a cylinder, we need the radius and height
• We have the height (12 inches) but need to find the radius
• We can use the given circumference to find the radius first

3. SIMPLIFY to find the radius

• Use the circumference formula: \(\mathrm{C} = 2\pi\mathrm{r}\)
• Substitute known values: \(8\pi = 2\pi\mathrm{r}\)
• Divide both sides by \(2\pi\): \(\mathrm{r} = \frac{8\pi}{2\pi} = 4\) inches

4. SIMPLIFY to calculate the volume

• Use cylinder volume formula: \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\)
• Substitute our values: \(\mathrm{V} = \pi(4)^2(12)\)
• Calculate: \(\mathrm{V} = \pi(16)(12) = 192\pi\) cubic inches

Answer: C (\(192\pi\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about radius vs diameter: Students might incorrectly think the circumference represents the diameter, leading them to use \(\mathrm{r} = 8\) instead of \(\mathrm{r} = 4\).

This would give \(\mathrm{V} = \pi(8)^2(12) = \pi(64)(12) = 768\pi\), causing them to select Choice D (\(768\pi\)).

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly find \(\mathrm{r} = 4\) but forget to square the radius in the volume formula, calculating \(\mathrm{V} = \pi(4)(12)\) instead of \(\mathrm{V} = \pi(4)^2(12)\).

This gives \(\mathrm{V} = 48\pi\), leading them to select Choice A (\(48\pi\)).

The Bottom Line:

This problem tests whether students can work backwards from circumference to radius, then forwards to volume. The key insight is recognizing that circumference gives you the radius through \(\mathrm{C} = 2\pi\mathrm{r}\), not that circumference equals diameter.