A manufacturer determined that right cylindrical containers with a height that is 4 inches longer than the radius offer the...

1. TRANSLATE the problem information

• Given information:
  • Right cylindrical containers
  • Height is 4 inches longer than the radius
  • Need volume V in terms of radius r
• What this tells us: \(\mathrm{h = r + 4}\)

2. INFER the approach

• We need the volume formula for a cylinder: \(\mathrm{V = \pi r^2h}\)
• Since we know height in terms of radius, we can substitute directly
• This will give us volume entirely in terms of r

3. SIMPLIFY by substitution and distribution

• Start with: \(\mathrm{V = \pi r^2h}\)
• Substitute \(\mathrm{h = r + 4}\): \(\mathrm{V = \pi r^2(r + 4)}\)
• Distribute \(\mathrm{\pi r^2}\): \(\mathrm{V = \pi r^2 \cdot r + \pi r^2 \cdot 4}\)
• Final form: \(\mathrm{V = \pi r^3 + 4\pi r^2}\)

Answer: D. V = \(\mathrm{\pi r^3 + 4\pi r^2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "4 inches longer than the radius" as multiplication rather than addition, thinking height = 4r instead of r + 4.

When students make this error, they substitute \(\mathrm{h = 4r}\) into \(\mathrm{V = \pi r^2h}\), getting \(\mathrm{V = \pi r^2(4r) = 4\pi r^3}\). This leads them to select Choice A (\(\mathrm{4\pi r^3}\)).

Second Most Common Error:

Conceptual confusion about cylinder volume formula: Using \(\mathrm{V = \pi rh}\) instead of \(\mathrm{V = \pi r^2h}\), forgetting that the base area is \(\mathrm{\pi r^2}\) not \(\mathrm{\pi r}\).

Students who make this error get \(\mathrm{V = \pi r(r + 4) = \pi r^2 + 4\pi r}\), which looks similar to Choice C (\(\mathrm{\pi r^2 + 4\pi}\)) and may lead them to select it despite the missing 'r' in the second term.

The Bottom Line:

This problem tests whether students can accurately translate verbal relationships into algebraic expressions while correctly applying the cylinder volume formula. The key challenge is careful reading combined with systematic algebraic manipulation.