A printer produces pages at a constant rate, so the total number of pages P after t minutes is given...

1. TRANSLATE the problem information

• Given information:
  • Printer rate: \(\mathrm{P(t) = 12t}\) (pages after t minutes)
  • Need exactly 36 pages produced

• What this tells us: We need to find the time t when P(t) equals 36

2. INFER the solution strategy

• Since we know \(\mathrm{P(t) = 12t}\) and want \(\mathrm{P(t) = 36}\), we need to:
  • Set up the equation: \(\mathrm{P(t) = 36}\)
  • Substitute the given function: \(\mathrm{12t = 36}\)
  • Solve for t

3. SIMPLIFY the equation

• Starting with: \(\mathrm{12t = 36}\)
• Divide both sides by 12: \(\mathrm{t = 36 \div 12 = 3}\)
• Check our answer: \(\mathrm{P(3) = 12(3) = 36}\) ✓

Answer: B) 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they need to set up an equation \(\mathrm{P(t) = 36}\). Instead, they might try operations like \(\mathrm{36 - 12 = 24}\) or get confused about what the rate 12 represents in relation to the target of 36 pages.

This may lead them to select Choice D (24) through incorrect arithmetic manipulation.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret what "exactly 36 pages" means mathematically. They might think they need to add or multiply 36 with the rate 12, leading to calculations like \(\mathrm{36 + 12 = 48}\).

This may lead them to select Choice E (48) based on this incorrect relationship.

The Bottom Line:

This problem tests whether students can connect a real-world question to function evaluation. The key insight is recognizing that "exactly 36 pages" means finding the input value t that makes the output P(t) equal to 36.