A right circular cylinder has a lateral surface area of 48pi square inches. The height of the cylinder is 4...

1. TRANSLATE the problem information

• Given information:
  • Lateral surface area = \(48\pi\) square inches
  • Height = 4 inches
  • Need to find volume in cubic inches

2. INFER the solution strategy

• To find volume, I need the volume formula \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\)
• I have height (4), but I need radius
• I can find radius using the lateral surface area formula: \(\mathrm{A} = 2\pi\mathrm{rh}\)

3. TRANSLATE into equation and SIMPLIFY

• Set up the lateral surface area equation:
  \(48\pi = 2\pi\mathrm{r}(4)\)
  \(48\pi = 8\pi\mathrm{r}\)

• Solve for radius:
  \(\mathrm{r} = 48\pi \div 8\pi = 6\) inches

4. SIMPLIFY to find the volume

• Now substitute into volume formula:
  \(\mathrm{V} = \pi(6)^2(4)\)
  \(\mathrm{V} = \pi(36)(4)\)
  \(\mathrm{V} = 144\pi\) cubic inches

Answer: C (\(144\pi\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the multi-step approach needed. They see "lateral surface area" and "volume" and try to find a direct connection, not realizing they need to find radius first as an intermediate step.

This leads to confusion and guessing, or attempts to use incorrect formulas that don't apply to the given information.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equations correctly but make algebraic errors, such as:

• Incorrectly dividing: \(\mathrm{r} = 48\pi \div 8\pi = 6\pi\) (keeping \(\pi\) in denominator)
• Calculation errors: \((6)^2 = 12\) instead of 36
• Final arithmetic: \(\pi(36)(4) = 144\) instead of \(144\pi\)

This may lead them to select Choice A (\(24\pi\)) or Choice B (\(72\pi\)) depending on where the error occurred.

The Bottom Line:

This problem tests whether students can chain formulas together strategically. The key insight is recognizing that lateral surface area is the bridge to finding radius, which then unlocks the volume calculation.