A carpenter builds rectangular storage boxes for a workshop. Each box has a width of 9 inches. The height of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{Width = 9\text{ inches}}\)
  • \(\mathrm{Height = h\text{ inches}}\)
  • Length = "3 inches less than twice the height"

• TRANSLATE the length description:
  • "Twice the height" = \(\mathrm{2h}\)
  • "3 inches less than twice the height" = \(\mathrm{2h - 3}\)

2. INFER the approach

• For any rectangular box, we need: \(\mathrm{Volume = length \times width \times height}\)
• We have all three dimensions, so we can write the volume function
• The volume will be expressed in terms of h since that's what the problem asks for

3. SIMPLIFY to create the volume function

• \(\mathrm{V(h) = length \times width \times height}\)
• \(\mathrm{V(h) = (2h - 3) \times 9 \times h}\)
• \(\mathrm{V(h) = 9h(2h - 3)}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students struggle to convert "3 inches less than twice the height" into the correct mathematical expression. They might write \(\mathrm{2h + 3}\) instead of \(\mathrm{2h - 3}\), confusing "less than" with "more than."

This reasoning leads them to select Choice E (\(\mathrm{9h(2h + 3)}\)) because they have the correct structure but the wrong sign.

Second Most Common Error:

Incomplete INFER reasoning: Students correctly translate the length but forget to include all three dimensions in the volume calculation. They might only multiply length and height, forgetting the constant width of 9 inches.

This causes them to select Choice A (\(\mathrm{h(2h - 3)}\)) because they're missing the width factor of 9.

The Bottom Line:

This problem tests whether students can accurately translate word descriptions into mathematical expressions and then systematically apply the volume formula. Success requires careful attention to the phrase "less than" and remembering that volume always involves all three dimensions.