A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction...

1. TRANSLATE the problem information

• Given information:
  • Height = 60 inches
  • Length = 2.5 times the width
  • Width = x inches (what we're solving for)
  • Constraint: (perimeter of base) + height ≤ 130 inches

• What this tells us: We need to express everything in terms of x and set up an inequality.

2. TRANSLATE the perimeter calculation

• Since length = \(\mathrm{2.5x}\) and width = \(\mathrm{x}\):
  • Perimeter of base = \(\mathrm{2(length + width)}\)
  • \(\mathrm{= 2(2.5x + x)}\)
  • \(\mathrm{= 2(3.5x)}\)
  • \(\mathrm{= 7x}\)

• The constraint becomes: \(\mathrm{7x + 60 \leq 130}\)

3. SIMPLIFY the inequality

• Starting with: \(\mathrm{7x + 60 \leq 130}\)
• Subtract 60 from both sides: \(\mathrm{7x \leq 70}\)
• Divide both sides by 7: \(\mathrm{x \leq 10}\)

4. APPLY CONSTRAINTS to complete the solution

• Since x represents width (a physical dimension): \(\mathrm{x \gt 0}\)
• Combined with our solved inequality: \(\mathrm{0 \lt x \leq 10}\)

Answer: A. \(\mathrm{0 \lt x \leq 10}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly calculate the perimeter by forgetting the factor of 2, thinking perimeter = length + width instead of \(\mathrm{2(length + width)}\).

If they use perimeter = \(\mathrm{x + 2.5x = 3.5x}\), then their constraint becomes \(\mathrm{3.5x + 60 \leq 130}\), leading to \(\mathrm{3.5x \leq 70}\), so \(\mathrm{x \leq 20}\).

This may lead them to select Choice D (\(\mathrm{0 \lt x \leq 20}\)).

Second Most Common Error:

Inadequate APPLY CONSTRAINTS execution: Students correctly solve the inequality to get \(\mathrm{x \leq 10}\) but forget that physical dimensions must be positive.

They might think the answer is just "\(\mathrm{x \leq 10}\)" and get confused when this doesn't match any answer choice exactly, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students can accurately translate multi-step word problems into algebraic inequalities. The key challenge is correctly identifying that perimeter requires doubling the sum of length and width, not just adding them directly.