A cylinder has a lateral surface area of 216pi square centimeters. The circumference of the base is 12pi centimeters. What...

1. TRANSLATE the problem information

• Given information:
  • Lateral surface area = \(216\pi\) square centimeters
  • Circumference of base = \(12\pi\) centimeters
  • Need to find: height in centimeters

2. INFER the most efficient approach

• We have two cylinder measurements and need a third
• Key insight: Both formulas involve radius, so we can connect them
• Strategy: Use circumference to find radius, then use lateral surface area to find height

3. SIMPLIFY to find the radius

• From circumference formula: \(\mathrm{C} = 2\pi\mathrm{r}\)
• \(12\pi = 2\pi\mathrm{r}\)
• Divide both sides by \(2\pi\): \(\mathrm{r} = 6\) cm

4. SIMPLIFY to find the height

• From lateral surface area formula: \(\mathrm{L} = 2\pi\mathrm{rh}\)
• \(216\pi = 2\pi(6)\mathrm{h}\)
• \(216\pi = 12\pi\mathrm{h}\)
• Divide both sides by \(12\pi\): \(\mathrm{h} = 18\) cm

Answer: D (18)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to find the radius first as an intermediate step. They might try to work directly with the given measurements without seeing the connection between the formulas. This leads to confusion about how to combine \(216\pi\) and \(12\pi\) meaningfully, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the radius correctly (\(\mathrm{r} = 6\)) but make algebraic errors when solving \(2\pi(6)\mathrm{h} = 216\pi\). They might incorrectly simplify to get \(6\pi\mathrm{h} = 216\pi\), leading to \(\mathrm{h} = 36\). This may lead them to select Choice E (36).

The Bottom Line:

This problem requires students to see that cylinder formulas are interconnected through the radius. The key breakthrough is recognizing that you can use one measurement to find the missing piece needed for the other formula.