The population P of a certain city y years after the last census is modeled by the equation below, where...

1. TRANSLATE the problem information

• Given information:
  • Population model: \(\mathrm{P = P_0(1 + r)^y}\)
  • Population decreases by a fixed percent each year
  • Need to find constraints on r

• What this tells us: We need to determine what values of r make this model represent population decrease.

2. INFER what population decrease means mathematically

• In the exponential model \(\mathrm{P = P_0(1 + r)^y}\), the term \(\mathrm{(1 + r)}\) is the multiplier applied each year
• For population to decrease each year, this multiplier must be less than 1
• For population to remain positive (realistic), this multiplier must be greater than 0
• Therefore: \(\mathrm{0 \lt (1 + r) \lt 1}\)

3. SIMPLIFY the compound inequality to find r

• Starting with: \(\mathrm{0 \lt (1 + r) \lt 1}\)
• Subtract 1 from all three parts: \(\mathrm{0 - 1 \lt (1 + r) - 1 \lt 1 - 1}\)
• This gives us: \(\mathrm{-1 \lt r \lt 0}\)

4. APPLY CONSTRAINTS to verify our answer makes sense

• Check: If \(\mathrm{r = -0.3}\), then \(\mathrm{(1 + r) = 0.7}\)
• Each year population becomes 70% of previous year (30% decrease) ✓
• Population stays positive ✓
• This confirms our constraint \(\mathrm{-1 \lt r \lt 0}\) is correct

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not connect "population decreases by fixed percent" to the mathematical requirement that \(\mathrm{(1 + r) \lt 1}\). Instead, they might think decreasing population simply means \(\mathrm{r \lt 0}\), without considering the constraint that \(\mathrm{(1 + r)}\) must stay positive. This reasoning would lead them to select Choice A (\(\mathrm{r \lt -1}\)), not realizing this would make \(\mathrm{(1 + r)}\) negative and create unrealistic population values.

Second Most Common Error:

Conceptual confusion about exponential models: Students might incorrectly think that any positive value of r represents growth and any negative r represents decay, missing the crucial insight that it's the value of \(\mathrm{(1 + r)}\), not just r, that determines growth or decay behavior. This leads to confusion about why the answer isn't simply "\(\mathrm{r \lt 0}\)" and may cause them to get stuck and guess.

The Bottom Line:

This problem requires understanding that in exponential models, it's the base \(\mathrm{(1 + r)}\) that directly controls growth or decay behavior, not just the parameter r itself. Success depends on translating "decreasing population" into precise mathematical constraints on this base.