A structural engineer designs triangular support frames where stability requirements dictate that the sum of any two beam lengths must...

1. TRANSLATE the problem information

• Given information:
  • Two beam lengths: 5 feet and 14 feet
  • Third beam length: b feet (unknown)
  • Constraint: sum of any two beams ≥ third beam + 4 feet

• What this tells us: We need to apply this constraint in all possible combinations of beams

2. INFER the approach

• Since we have three beams total, there are three ways to pick "any two beams"
• We must write three separate inequalities, one for each pair
• All three inequalities must be true simultaneously

3. TRANSLATE each constraint into mathematical form

• Pair 1: (5 + 14) vs. b → \(5 + 14 \geq b + 4\)
• Pair 2: (5 + b) vs. 14 → \(5 + b \geq 14 + 4\)
• Pair 3: (14 + b) vs. 5 → \(14 + b \geq 5 + 4\)

4. SIMPLIFY each inequality

• From constraint 1: \(19 \geq b + 4\) → \(b \leq 15\)
• From constraint 2: \(5 + b \geq 18\) → \(b \geq 13\)
• From constraint 3: \(14 + b \geq 9\) → \(b \geq -5\)

5. APPLY CONSTRAINTS to find the final answer

• The third constraint (\(b \geq -5\)) is automatically satisfied since beam lengths are positive
• We need both \(b \leq 15\) AND \(b \geq 13\)
• Combined: \(13 \leq b \leq 15\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students apply the constraint in only one direction, typically just ensuring \(5 + 14 \geq b + 4\), which gives \(b \leq 15\). They miss that the constraint must work in all three directions and forget to check what happens when b is one of the "two beams" being added together.

This incomplete analysis may lead them to select Choice A (\(b \leq 15\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse this with the standard triangle inequality (where the sum just needs to be greater than, not greater than plus 4). They set up \(5 + 14 \gt b\), \(5 + b \gt 14\), and \(14 + b \gt 5\), getting \(b \lt 19\), \(b \gt 9\), and \(b \gt -9\), leading to \(9 \lt b \lt 19\).

This may lead them to select Choice D (\(9 \leq b \leq 19\)).

The Bottom Line:

This problem requires careful attention to the modified constraint (adding 4 feet) and systematic application in all directions. Students who rush through the setup or apply constraints incompletely will miss the correct range.