A right circular cylinder has a base circumference of 12pi centimeters. The volume of the cylinder is 2,880pi cubic centimeters....

1. TRANSLATE the problem information

• Given information:
  • Base circumference = \(12\pi\text{ cm}\)
  • Volume = \(2,880\pi\text{ cubic cm}\)
  • Need to find: height h

2. INFER the solution strategy

• To use the volume formula \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\), we need the radius
• We can find radius from the given circumference first
• Then substitute into volume formula to solve for height

3. TRANSLATE circumference into radius equation

• Circumference formula: \(\mathrm{C} = 2\pi\mathrm{r}\)
• Set up equation: \(2\pi\mathrm{r} = 12\pi\)

4. SIMPLIFY to find radius

• Divide both sides by \(2\pi\): \(\mathrm{r} = \frac{12\pi}{2\pi} = 6\text{ cm}\)

5. TRANSLATE volume information using found radius

• Volume formula: \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\)
• Substitute known values: \(\pi(6)^2\mathrm{h} = 2,880\pi\)

6. SIMPLIFY to find height

• Calculate: \(\pi(36)\mathrm{h} = 2,880\pi\)
• Divide both sides by \(36\pi\): \(\mathrm{h} = \frac{2,880\pi}{36\pi} = \frac{2,880}{36} = 80\)

Answer: D (80)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor INFER reasoning: Students might try to use the volume formula immediately without recognizing they need the radius first. Some students see the circumference and think it can be directly substituted into the volume formula, leading to confusion about what to do next.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Arithmetic errors during SIMPLIFY: Students might make calculation errors when dividing 2,880 by 36, potentially getting 60 or other incorrect values.

This may lead them to select Choice C (60) if they incorrectly calculate \(2,880 ÷ 36\).

The Bottom Line:

This problem requires students to recognize the logical sequence: circumference → radius → volume → height. Students who jump ahead without finding the radius first, or who make arithmetic errors in the final division, will struggle to reach the correct answer.