A manufacturer produces rectangular storage containers where the length is twice the width and the height is 3 more than...

1. TRANSLATE the problem relationships

• Given information:
  • Width = \(\mathrm{w}\) inches
  • Length = twice the width → \(\mathrm{2w}\) inches
  • Height = 3 more than the width → \(\mathrm{w + 3}\) inches

2. INFER the approach needed

• For any rectangular container, volume equals \(\mathrm{length \times width \times height}\)
• We have all three dimensions in terms of \(\mathrm{w}\), so we can write a volume expression

3. Set up the volume formula

• Volume = \(\mathrm{length \times width \times height}\)
• \(\mathrm{V = (2w) \times w \times (w + 3)}\)

4. SIMPLIFY through algebraic expansion

• First multiply the first two terms: \(\mathrm{V = 2w^2 \times (w + 3)}\)
• Apply distributive property:
  \(\mathrm{V = 2w^2 \times w + 2w^2 \times 3}\)
• Simplify each term:
  \(\mathrm{V = 2w^3 + 6w^2}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "3 more than the width" as \(\mathrm{3w}\) instead of \(\mathrm{w + 3}\), or confuse which dimension is twice the other.

If they write height = \(\mathrm{3w}\), their volume becomes \(\mathrm{V = (2w) \times w \times (3w) = 6w^3}\), leading them toward Choice D (\(\mathrm{6w^3 + 2w^2}\)) after attempting some expansion.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{V = 2w^2(w + 3)}\) but make algebraic errors when expanding.

A common mistake is distributing incorrectly: writing \(\mathrm{2w^2(w + 3) = 2w^3 + 3w^2}\) instead of \(\mathrm{2w^3 + 6w^2}\). This leads them to select Choice A (\(\mathrm{2w^3 + 3w^2}\)).

The Bottom Line:

This problem tests whether students can accurately translate word relationships into algebraic expressions and then perform multi-step algebraic expansion without computational errors.