Question:\(\mathrm{d(t) = 50 + 3t}\)The function d models the distance, in kilometers, a car has traveled t hours after starting...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{d(t) = 50 + 3t}\) models distance in kilometers
  • \(\mathrm{t}\) represents hours after starting the trip
  • Need to find the predicted speed in km/hour

• What this tells us: We have a linear function relating distance to time

2. INFER the mathematical meaning

• In any linear function \(\mathrm{f(x) = mx + b}\), the coefficient \(\mathrm{m}\) represents the slope
• In our function \(\mathrm{d(t) = 50 + 3t}\), the coefficient of \(\mathrm{t}\) is 3
• This means the slope is 3

3. INFER the real-world interpretation

• Slope in a distance-time function represents rate of change of distance with respect to time
• Rate of change of distance with respect to time is exactly what speed means
• Therefore: speed = 3 kilometers per hour

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about function components: Students focus on the constant term (50) thinking it represents speed, or they evaluate the function at a specific time point.

They might think: "The car starts 50 km away, so maybe that's the speed?" or "Let me find \(\mathrm{d(1) = 50 + 3(1) = 53}\), so the speed is 53."

This may lead them to select Choice C (50) or Choice D (53)

Second Most Common Error:

Weak INFER skills: Students recognize this is about rate but don't connect slope to speed properly. They might try averaging or combining the numbers in some incorrect way.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand that in a linear distance-time function, the coefficient of the time variable directly gives you the speed. The key insight is recognizing slope as rate of change in the specific context of motion.