A right circular cylinder has a base with circumference 10pi centimeters and height 9 centimeters. A right rectangular prism has...

1. TRANSLATE the problem information

• Given information:
  • Cylinder: circumference = \(\mathrm{10\pi}\) cm, height = \(\mathrm{9}\) cm
  • Rectangular prism: length = \(\mathrm{15}\) cm, width = \(\mathrm{5}\) cm, same volume as cylinder
  • Find: height of rectangular prism

2. INFER the solution approach

• To find the cylinder's volume, we need its radius, not circumference
• Strategy: Find radius → calculate cylinder volume → set equal to prism volume → solve for prism height

3. TRANSLATE circumference to radius

• Circumference formula: \(\mathrm{C = 2\pi r}\)
• Given: \(\mathrm{10\pi = 2\pi r}\)
• Therefore: \(\mathrm{r = 5}\) cm

4. SIMPLIFY to find cylinder volume

• Volume of cylinder: \(\mathrm{V = \pi r^2h}\)
• \(\mathrm{V = \pi(5)^2(9)}\)
  \(\mathrm{= \pi(25)(9)}\)
  \(\mathrm{= 225\pi}\) cubic cm

5. INFER the equal volume relationship

• Since volumes are equal:
• Cylinder volume = Rectangular prism volume
• \(\mathrm{225\pi = length \times width \times height}\)
• \(\mathrm{225\pi = 15 \times 5 \times h = 75h}\)

6. SIMPLIFY to solve for height

• \(\mathrm{75h = 225\pi}\)
• \(\mathrm{h = \frac{225\pi}{75} = 3\pi}\) cm

Answer: B. 3π

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge of circumference formula: Students may not remember that \(\mathrm{C = 2\pi r}\), or they might try to use circumference directly in the volume formula without finding radius first.

Without the radius, they cannot calculate the cylinder's volume correctly, leading to confusion about how to proceed. This causes them to get stuck and randomly select an answer.

Second Most Common Error:

Weak SIMPLIFY execution: Students correctly set up \(\mathrm{75h = 225\pi}\) but make arithmetic errors when reducing the fraction.

Common mistakes include:

• Forgetting to divide both 225 and 75 by their common factor (75)
• Incorrectly calculating \(\mathrm{225 \div 75 = 4}\) instead of 3
• Dropping the π from the final answer

This may lead them to select Choice D (15π) or Choice C (9π).

The Bottom Line:

This problem tests whether students can work backwards from circumference to radius, then forward to volume comparisons. The key insight is recognizing that you need radius, not circumference, to calculate cylinder volume.