A cylinder has a height of 8 inches and a volume of 200pi cubic inches. What is the radius, in...

1. TRANSLATE the problem information

• Given information:
  • Height: \(\mathrm{h = 8}\) inches
  • Volume: \(\mathrm{V = 200\pi}\) cubic inches
• What we need to find: radius r

2. INFER the approach

• We need the volume formula for a cylinder to connect these quantities
• Since we have volume and height, we can solve for radius directly
• Strategy: Substitute known values into \(\mathrm{V = \pi r^2h}\) and solve for r

3. SIMPLIFY through algebraic steps

• Start with the volume formula: \(\mathrm{V = \pi r^2h}\)
• Substitute our values: \(\mathrm{200\pi = \pi r^2(8)}\)
• This gives us: \(\mathrm{200\pi = 8\pi r^2}\)
• Divide both sides by π: \(\mathrm{200 = 8r^2}\)
• Divide both sides by 8: \(\mathrm{25 = r^2}\)
• Take the square root: \(\mathrm{r = 5}\) inches

Answer: B (5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing conceptual knowledge: Not remembering the volume formula for a cylinder, or confusing it with other geometric formulas (like surface area or volume of other shapes).

Without the correct formula, students may try to guess relationships between the given numbers or use incorrect formulas, leading them to calculate values that don't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Making algebraic errors during the multi-step solution process.

Common computational mistakes include: forgetting to divide by π, incorrectly dividing 200 by 8, or forgetting to take the square root of 25. These errors can lead students to select Choice A (2.5) if they get \(\mathrm{r^2 = 6.25}\) instead of 25, or other incorrect values.

The Bottom Line:

This problem tests whether students can recall a key geometric formula and then execute a straightforward algebraic solution. Success depends on having the volume formula memorized and being systematic with algebraic manipulations.