The volume of right circular cylinder A is 22 cubic centimeters. What is the volume, in cubic centimeters, of a...

1. TRANSLATE the problem information

• Given information:
  • Original cylinder A: volume = 22 cubic cm
  • New cylinder: radius is twice cylinder A's radius, height is half cylinder A's height

• What this tells us: If cylinder A has radius r and height h, the new cylinder has radius 2r and height h/2

2. INFER the approach

• We need to find how the volume changes when dimensions change
• Key insight: We don't need to find the actual values of r and h - we just need the ratio between the volumes
• Strategy: Express both volumes using the cylinder volume formula, then find the relationship

3. SIMPLIFY the volume expressions

• Original cylinder A: \(\mathrm{V_1 = πr^2h = 22}\)
• New cylinder: \(\mathrm{V_2 = π(2r)^2(h/2)}\)
• Expand the new volume: \(\mathrm{V_2 = π(4r^2)(h/2) = 2πr^2h}\)

4. INFER the final relationship

• Since \(\mathrm{πr^2h = 22}\), we can substitute:
• \(\mathrm{V_2 = 2πr^2h = 2(22) = 44}\) cubic centimeters

Answer: C. 44

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly evaluate \(\mathrm{(2r)^2}\) as \(\mathrm{2r^2}\) instead of \(\mathrm{4r^2}\)

When squaring "twice the radius," they mistakenly think \(\mathrm{(2r)^2 = 2r^2}\), leading to:
\(\mathrm{V_{new} = π(2r^2)(h/2) = πr^2h = 22}\)

This may lead them to select Choice B (22)

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly identify that radius doubles but miss that this gets squared in the volume formula

They might think: "radius doubles, height halves, so volume stays the same" without accounting for the \(\mathrm{r^2}\) term in the formula.

This may lead them to select Choice B (22)

The Bottom Line:

This problem tests whether students understand how changes in dimensions affect volume formulas, particularly remembering that radius gets squared in the cylinder volume formula.