\(\mathrm{P(t) = 450 - 12t}\) The function P models the distance, in meters, that a runner is behind a moving...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{P(t) = 450 - 12t}\) represents the distance (in meters) the runner is behind the checkpoint
  • Both the runner and checkpoint start moving at \(\mathrm{t = 0}\)
  • We need to find the runner's speed relative to the checkpoint

• What this tells us: We have a linear function where \(\mathrm{P(t)}\) shows how the gap between runner and checkpoint changes over time

2. INFER what the coefficient means

• In the linear function \(\mathrm{P(t) = 450 - 12t}\), the coefficient \(\mathrm{-12}\) tells us the rate of change
• Since the coefficient is negative, \(\mathrm{P(t)}\) decreases by 12 meters every second
• This means the distance between runner and checkpoint shrinks by 12 meters per second

3. INFER the connection to relative speed

• If the gap decreases by 12 meters per second, the runner must be catching up at that rate
• "Catching up at \(\mathrm{12\,m/s}\)" means the runner is moving \(\mathrm{12\,m/s}\) faster than the checkpoint
• Therefore, the runner's speed relative to the checkpoint is 12 meters per second

Answer: A (12)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see the number 450 in the function and think this might be the speed, or they focus on other values like \(\mathrm{450/12 = 37.5}\).

They miss that in a linear function, the coefficient of the variable (not the constant term) represents the rate of change. The 450 just tells us the initial distance behind, not anything about speed.

This may lead them to select Choice C (37.5) or Choice E (450).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "relative speed" means and think they need to find the runner's absolute speed or the checkpoint's speed separately.

They might think the problem is asking for something more complex than what the coefficient directly tells us, leading to confusion about what calculation to perform.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

This problem tests whether students recognize that in a linear function modeling change over time, the coefficient of t directly gives the rate of that change. The key insight is connecting "rate of change of distance behind" to "relative speed."