A right circular cylinder has a base circumference of 24pi centimeters and a height of 5 centimeters. What is the...

1. TRANSLATE the problem information

• Given information:
  • Right circular cylinder
  • Base circumference = 24π cm
  • Height = 5 cm
  • Need to find volume in cubic cm

2. INFER the approach needed

• To find volume, we need \(\mathrm{V = \pi r^2h}\)
• We have height (5 cm) but need radius
• We can get radius from the given circumference using \(\mathrm{C = 2\pi r}\)

3. SIMPLIFY to find the radius

• Start with circumference formula: \(\mathrm{C = 2\pi r}\)
• Substitute known values: \(\mathrm{24\pi = 2\pi r}\)
• Divide both sides by 2π: \(\mathrm{r = \frac{24\pi}{2\pi} = 12}\) cm

4. SIMPLIFY to calculate the volume

• Use volume formula: \(\mathrm{V = \pi r^2h}\)
• Substitute values: \(\mathrm{V = \pi(12)^2(5)}\)
• Calculate: \(\mathrm{V = \pi(144)(5) = 720\pi}\) cubic cm

Answer: B (720π)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students correctly find \(\mathrm{r = 12}\) but forget to square the radius in the volume formula.

They calculate \(\mathrm{V = \pi(12)(5) = 60\pi}\) instead of \(\mathrm{V = \pi(12)^2(5) = 720\pi}\). The volume formula requires \(\mathrm{r^2}\), but they use just \(\mathrm{r}\), treating it like finding the area of a rectangle instead of the area of a circle times height.

This leads them to select Choice A (60π).

Second Most Common Error:

Poor INFER reasoning: Students recognize they need the volume formula but incorrectly think they can use the diameter directly since circumference relates to the full width.

Since \(\mathrm{C = 24\pi}\) and they know \(\mathrm{C = 2\pi r}\), they might think 'diameter = 24' and use that in \(\mathrm{V = \pi(diameter)^2h = \pi(24)^2(5) = 2880\pi}\). They're conceptually confused about whether the volume formula uses radius or diameter.

This leads them to select Choice D (2880π).

The Bottom Line:

This problem tests whether students can chain together multiple formulas (circumference → radius → volume) and execute each step accurately. The key insight is recognizing that you must find radius as an intermediate step before applying the volume formula.