The lateral surface area of a right circular cylinder is 120pi square meters, and the height of the cylinder is...

1. TRANSLATE the problem information

• Given information:
  • Lateral surface area = \(120\pi\) square meters
  • Height = \(6\) meters
  • Need to find: Volume in cubic meters
• What this tells us: We have LSA and height, but need radius to find volume

2. TRANSLATE the lateral surface area relationship

• The lateral surface area of a cylinder is the "wraparound" area: \(\mathrm{LSA} = 2\pi\mathrm{rh}\)
• Setting up our equation: \(2\pi\mathrm{rh} = 120\pi\)

3. SIMPLIFY to find the radius

• Substitute the known height: \(2\pi\mathrm{r}(6) = 120\pi\)
• This gives us: \(12\pi\mathrm{r} = 120\pi\)
• Divide both sides by \(12\pi\): \(\mathrm{r} = 120\pi \div 12\pi = 10\) meters

4. TRANSLATE and SIMPLIFY to find volume

• Use the volume formula: \(\mathrm{V} = \pi\mathrm{r}^2\mathrm{h}\)
• Substitute our values: \(\mathrm{V} = \pi(10)^2(6)\)
• Calculate: \(\mathrm{V} = \pi(100)(6) = 600\pi\) cubic meters

Answer: (C) \(600\pi\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse lateral surface area with total surface area and use the wrong formula \(\mathrm{LSA} = 2\pi\mathrm{r}(\mathrm{h} + \mathrm{r})\) or \(\mathrm{LSA} = 2\pi\mathrm{rh} + 2\pi\mathrm{r}^2\).

Using \(2\pi\mathrm{r}(\mathrm{h} + \mathrm{r}) = 120\pi\) with \(\mathrm{h} = 6\) gives:

\(2\pi\mathrm{r}(6 + \mathrm{r}) = 120\pi\)

leading to

\(12\pi\mathrm{r} + 2\pi\mathrm{r}^2 = 120\pi\)

This creates a quadratic equation that's much more complex than needed and typically leads to calculation errors or confusion.

This leads to confusion and abandoning systematic solution, causing students to guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up \(2\pi\mathrm{r}(6) = 120\pi\) but make arithmetic errors when solving for radius, such as getting \(\mathrm{r} = 20\) instead of \(\mathrm{r} = 10\).

With \(\mathrm{r} = 20\), they calculate:

\(\mathrm{V} = \pi(20)^2(6)\)

\(\mathrm{V} = \pi(400)(6)\)

\(\mathrm{V} = 2400\pi\)

This may lead them to select Choice (E) \(2{,}400\pi\).

The Bottom Line:

This problem requires precision in distinguishing lateral surface area (just the curved side) from total surface area. Students who rush often grab the wrong formula and create unnecessarily complex equations.