Question:A videographer books a single-day shoot that can last at most 6 hours. The videographer must spend 40 minutes setting...

1. TRANSLATE the problem information

• Given information:
  • Maximum total time: 6 hours
  • Setup time (required): 40 minutes
  • Time per segment: 17 minutes
  • Need: Greatest number of segments possible

• What this tells us: We have a fixed setup cost plus variable recording time that cannot exceed our time budget.

2. TRANSLATE time units consistently

• Convert the 6-hour limit to minutes: \(\mathrm{6 \times 60 = 360}\) minutes

• Now everything is in the same units (minutes)

3. INFER the mathematical relationship

• Total time used = Setup time + (Number of segments × Time per segment)

• Total time used = \(\mathrm{40 + 17n}\), where n = number of segments

• The constraint: \(\mathrm{Total\:time\:used \leq Time\:limit}\)

4. Set up and SIMPLIFY the inequality

• Write the constraint: \(\mathrm{40 + 17n \leq 360}\)

• Subtract 40 from both sides: \(\mathrm{17n \leq 320}\)

• Divide by 17: \(\mathrm{n \leq 320/17}\)

• Calculate the division (use calculator): \(\mathrm{n \leq 18.82}\)...

5. APPLY CONSTRAINTS to select final answer

• Since we cannot record a fraction of a segment, n must be a whole number

• Since we cannot exceed the time limit, we must round DOWN to 18

• Verify: \(\mathrm{40 + 17(18) = 346}\) minutes \(\mathrm{\leq 360}\) minutes ✓

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students correctly calculate \(\mathrm{n \leq 18.82}\) but then round UP to 19 segments, thinking "18.82 is closer to 19."

They fail to recognize that rounding up would violate the time constraint (\(\mathrm{40 + 17(19) = 363 \gt 360}\)). The inequality symbol \(\mathrm{\leq}\) means we cannot exceed the limit, so we must round down to stay within bounds.

This leads to the incorrect answer of 19 segments.

Second Most Common Error:

Poor TRANSLATE execution: Students forget to convert hours to minutes and work directly with 6 hours, setting up: \(\mathrm{40 + 17n \leq 6}\).

This creates an impossible situation since the setup time alone (40 minutes) already exceeds their perceived limit of 6 minutes, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students understand that "at most" constraints in real-world contexts require rounding down to whole numbers, not mathematical rounding to the nearest integer.