\(\mathrm{P(t) = 280(1.04)^{(t/18)}}\)The function P models the population, in thousands, of a certain city t years after 2005. According to...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{P(t) = 280(1.04)^{(t/18)}}\)
  • Models population t years after 2005
  • Population increases by 4% every 18 months
• The question asks for the value of n

2. INFER what n represents

• Since the problem explicitly states the population "increases by 4% every 18 months"
• And we're asked to find n
• The most logical interpretation is that n represents this percentage increase value

3. TRANSLATE percentage increase to the answer

• A 4% increase every 18 months means n = 4
• Looking at the answer choices, this corresponds to Choice C

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the growth factor \(\mathrm{(1.04)}\) with the percentage increase \(\mathrm{(4\%)}\)

They see \(\mathrm{1.04}\) in the function and think this must be the answer, leading them to select Choice B (1.04). However, \(\mathrm{1.04}\) represents the growth factor \(\mathrm{(100\% + 4\% = 104\% = 1.04)}\), while the percentage increase is just \(\mathrm{4\%}\).

Second Most Common Error:

Poor INFER reasoning: Students get confused about what n represents in the context

Since n doesn't appear explicitly in the given function, some students may overthink the problem and try complex algebraic manipulations instead of recognizing that the problem statement directly tells them about the 4% increase. This leads to confusion and potentially guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can distinguish between growth factors and percentage increases, and whether they can identify what the question is actually asking for when the variable n isn't explicitly shown in the given function.