The population \(\mathrm{P(t)}\) of a small town, in thousands of people, t years after 2020 is given by the function...

1. TRANSLATE the problem information

• Given information:
  • Population function: \(\mathrm{P(t) = kt + 12}\) (in thousands)
  • t represents years after 2020
  • In 2025, population was 17,000 people
• TRANSLATE the timeline: 2025 is 5 years after 2020, so when \(\mathrm{t = 5}\), we have \(\mathrm{P(5) = 17}\) thousand

2. INFER the approach needed

• We have one unknown constant (k) and one data point
• Strategy: Use the 2025 data to find k, then predict 2040

3. SIMPLIFY to find the constant k

• Substitute the known values: \(\mathrm{P(5) = 17}\)
• \(\mathrm{k(5) + 12 = 17}\)
• \(\mathrm{5k = 5}\)
• \(\mathrm{k = 1}\)

4. TRANSLATE and calculate the target year

• TRANSLATE: 2040 is 20 years after 2020, so we need \(\mathrm{P(20)}\)
• \(\mathrm{P(20) = 1(20) + 12 = 32}\)

Answer: 32

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students miscount the years after 2020, thinking 2025 is 4 years after (2024, 2023, 2022, 2021) or getting confused about which year to start from.

This leads to using the wrong t-value when setting up the equation to find k, resulting in an incorrect constant and wrong final answer.

Second Most Common Error:

Missing unit awareness: Students forget that the function gives population in thousands, so they use \(\mathrm{P(5) = 17,000}\) instead of \(\mathrm{P(5) = 17}\).

This creates the equation \(\mathrm{5k + 12 = 17,000}\), giving \(\mathrm{k = 3,397.6}\), and ultimately a wildly incorrect prediction. This leads to confusion and abandoning the systematic solution.

The Bottom Line:

This problem tests careful reading and unit tracking more than complex math. The linear function work is straightforward, but students must pay attention to timeline conversion and unit consistency throughout.