A company designs cardboard boxes that have the shape of a right rectangular prism. For optimal stacking, the length of...

1. TRANSLATE the problem constraints

• Given information:
  • Box is a right rectangular prism with width w
  • Length is twice the width: \(\mathrm{l = 2w}\)
  • Height is 3 inches shorter than width: \(\mathrm{h = w - 3}\)

2. INFER the approach

• We need the surface area formula for a rectangular prism
• Since we have all dimensions in terms of w, substitute them into the formula
• The surface area formula is \(\mathrm{S = 2(lw + lh + wh)}\)

3. SIMPLIFY by substituting and expanding

• Substitute \(\mathrm{l = 2w}\) and \(\mathrm{h = w - 3}\) into the formula:

\(\mathrm{S = 2((2w)(w) + (2w)(w-3) + w(w-3))}\)

• Expand each term inside the parentheses:
  • First term: \(\mathrm{(2w)(w) = 2w^2}\)
  • Second term: \(\mathrm{(2w)(w-3) = 2w^2 - 6w}\)
  • Third term: \(\mathrm{w(w-3) = w^2 - 3w}\)

4. SIMPLIFY by combining like terms

• \(\mathrm{S = 2(2w^2 + 2w^2 - 6w + w^2 - 3w)}\)
• Combine like terms: \(\mathrm{2w^2 + 2w^2 + w^2 = 5w^2}\) and \(\mathrm{-6w + (-3w) = -9w}\)
• \(\mathrm{S = 2(5w^2 - 9w)}\)

5. SIMPLIFY final distribution

• \(\mathrm{S = 2(5w^2) + 2(-9w) = 10w^2 - 18w}\)

Answer: (C) \(\mathrm{S = 10w^2 - 18w}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when expanding terms or combining like terms.

For example, they might incorrectly calculate \(\mathrm{2w^2 + 2w^2 + w^2 = 4w^2}\) instead of \(\mathrm{5w^2}\), or combine \(\mathrm{-6w - 3w = -8w}\) instead of \(\mathrm{-9w}\). These calculation errors lead to incorrect coefficients in the final expression.

This may lead them to select Choice (A) (\(\mathrm{S = 8w^2 - 12w}\)) or Choice (B) (\(\mathrm{S = 8w^2 - 18w}\))

Second Most Common Error:

Poor TRANSLATE reasoning: Students might incorrectly interpret "3 inches shorter than width" as \(\mathrm{h = 3 - w}\) instead of \(\mathrm{h = w - 3}\).

This reversal leads to different terms in the expansion, ultimately producing an expression with positive terms instead of the correct negative term, potentially leading them to select Choice (D) (\(\mathrm{S = 10w^2 + 18w}\))

The Bottom Line:

This problem tests whether students can accurately translate word constraints into mathematical expressions and then perform multi-step algebraic simplification without making computational errors. Success requires careful attention to both language interpretation and arithmetic precision.