A small city had a population of 2,000 residents at the beginning of 2020. Three years later, the population had...

1. TRANSLATE the problem information

• Given information:
  • Initial population at \(\mathrm{t = 0}\): 2,000 residents
  • Population at \(\mathrm{t = 3}\): 54,000 residents
  • Model: \(\mathrm{P = A(3)^{kt}}\)
• We need to find the value of \(\mathrm{k}\)

2. TRANSLATE to find the constant A

• At the beginning (\(\mathrm{t = 0}\)): \(\mathrm{P = 2{,}000}\)
• Substitute:

\(\mathrm{2{,}000 = A(3)^{k×0}}\)
\(\mathrm{= A(3)^0}\)
\(\mathrm{= A × 1}\)
\(\mathrm{= A}\)

• Therefore: \(\mathrm{A = 2{,}000}\)

3. TRANSLATE the condition after 3 years

• At \(\mathrm{t = 3}\): \(\mathrm{P = 54{,}000}\)
• Substitute known values: \(\mathrm{54{,}000 = 2{,}000(3)^{3k}}\)

4. SIMPLIFY to isolate the exponential term

• Divide both sides by 2,000:

\(\mathrm{54{,}000 ÷ 2{,}000 = (3)^{3k}}\)

• This gives us: \(\mathrm{27 = (3)^{3k}}\)

5. INFER the relationship using exponent properties

• Recognize that \(\mathrm{27 = 3^3}\)
• So we have:

\(\mathrm{3^3 = 3^{3k}}\)

• Since the bases are equal, the exponents must be equal:

\(\mathrm{3 = 3k}\)

• Therefore: \(\mathrm{k = 1}\)

Answer: C (1)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may incorrectly substitute values or confuse which variable represents what in the exponential model.

For example, they might think \(\mathrm{A = 54{,}000}\) (the final population) instead of \(\mathrm{A = 2{,}000}\) (the initial population), or they might substitute \(\mathrm{t = 2020}\) instead of \(\mathrm{t = 0}\) for the initial condition. This leads to completely wrong equations and makes it impossible to solve correctly.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that \(\mathrm{27 = 3^3}\), or they may not know how to apply the exponent property that if \(\mathrm{b^m = b^n}\), then \(\mathrm{m = n}\).

Without recognizing that \(\mathrm{27 = 3^3}\), students get stuck with the equation \(\mathrm{27 = (3)^{3k}}\) and can't proceed to find \(\mathrm{k}\). They might try to guess or use trial and error with the answer choices.

This may lead them to select Choice B (1/3) by incorrectly thinking \(\mathrm{k = 1/3}\) because "3 years" appears in the problem.

The Bottom Line:

This problem requires careful attention to translating the word problem into the correct mathematical model, combined with recognizing exponential relationships and applying exponent properties systematically.