A model predicts that the population of Bergen was 15,000 in 2005. The model also predicts that each year for...

1. TRANSLATE the problem information

• Given information:
  • Population in 2005: 15,000
  • Each year population increases by 4% of previous year's population
  • \(\mathrm{x}\) = number of years after 2005, \(\mathrm{x \leq 5}\)
  • Need equation for population p

2. TRANSLATE the percentage increase

• "Increases by 4%" means: new amount = old amount × \(\mathrm{(100\% + 4\%)}\) = old amount × \(\mathrm{1.04}\)
• So each year: population becomes 1.04 times the previous year's population

3. INFER the pattern by building examples

• Let's see what happens year by year:
  • 2005 \(\mathrm{(x = 0)}\): \(\mathrm{p = 15{,}000}\)
  • 2006 \(\mathrm{(x = 1)}\): \(\mathrm{p = 15{,}000 \times 1.04 = 15{,}000(1.04)^1}\)
  • 2007 \(\mathrm{(x = 2)}\): \(\mathrm{p = 15{,}000 \times 1.04 \times 1.04 = 15{,}000(1.04)^2}\)
  • 2008 \(\mathrm{(x = 3)}\): \(\mathrm{p = 15{,}000(1.04)^3}\)

4. INFER the general form

• The pattern shows: \(\mathrm{p = 15{,}000(1.04)^x}\)
• This matches the exponential growth model where:
  • 15,000 is the initial value (when \(\mathrm{x = 0}\))
  • 1.04 is the growth factor (multiply by this each year)
  • x is the number of years after 2005

5. Match to answer choices

• Looking at the options, \(\mathrm{p = 15{,}000(1.04)^x}\) matches Choice D exactly

Answer: D. \(\mathrm{p = 15{,}000(1.04)^x}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse "increase by 4%" with "decrease by 4%" and think they should multiply by 0.96 instead of 1.04.

They reason: "4% means 0.04, so I multiply by that... wait, that makes the population tiny. Maybe it's 96%?" This confusion about whether to use 1.04 or 0.96 leads them to select Choice C (\(\mathrm{p = 15{,}000(0.96)^x}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly identify that 4% increase means multiply by 1.04, but they misunderstand the exponential structure and think the initial population should be the base of the exponent rather than the coefficient.

This leads them to select Choice B (\(\mathrm{p = 1.04(15{,}000)^x}\)), which has the exponential structure backwards.

The Bottom Line:

The key insight is recognizing that percentage increases create multiplicative patterns, and in exponential growth, the initial value stays as a coefficient while the growth factor becomes the base of the exponent.