An airline limits checked baggage so that the sum of the length, width, and height must not exceed 62 inches....

1. TRANSLATE the problem information

• Given information:
  • Airline rule: \(\mathrm{length + width + height \leq 62}\) inches
  • Height = 14 inches
  • Length = 3 times the width
  • Width = x inches

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

2. INFER the approach

• We need to substitute all dimensions into the airline constraint
• Since we're looking for "allowable values," we're solving an inequality
• Physical dimensions must be positive, so \(\mathrm{x \gt 0}\)

3. TRANSLATE dimensions in terms of x

• Width = x
• Length = 3x (since "length is 3 times its width")
• Height = 14

4. SIMPLIFY by setting up and solving the inequality

• Substitute into the constraint: \(\mathrm{3x + x + 14 \leq 62}\)
• Combine like terms: \(\mathrm{4x + 14 \leq 62}\)
• Subtract 14 from both sides: \(\mathrm{4x \leq 48}\)
• Divide by 4: \(\mathrm{x \leq 12}\)

5. APPLY CONSTRAINTS for the final answer

• Since width must be positive: \(\mathrm{x \gt 0}\)
• Combined with our solution: \(\mathrm{0 \lt x \leq 12}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly set up the dimensions, such as thinking length = \(\mathrm{3 + x}\) instead of length = \(\mathrm{3x}\), or forget to include all three dimensions in their inequality setup.

This leads to wrong inequalities like \(\mathrm{3 + x + x + 14 \leq 62}\), giving \(\mathrm{2x \leq 45}\) or \(\mathrm{x \leq 22.5}\), which doesn't match any answer choice and causes confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{4x + 14 \leq 62}\) but make arithmetic errors, such as incorrectly subtracting 14 (getting \(\mathrm{4x \leq 50}\) instead of \(\mathrm{4x \leq 48}\)) or dividing incorrectly.

This may lead them to select Choice D (\(\mathrm{0 \lt x \leq 14}\)) if they get \(\mathrm{x \leq 12.5}\) and round up, or other incorrect choices based on their arithmetic mistakes.

The Bottom Line:

This problem requires careful translation of relationships ("3 times the width") and systematic algebraic manipulation. The key insight is recognizing that all dimensions must be expressed in terms of the single variable x before applying the constraint.