A wildlife biologist is monitoring the population of a certain species of frog in a protected wetlands area. The table...
GMAT Advanced Math : (Adv_Math) Questions
A wildlife biologist is monitoring the population of a certain species of frog in a protected wetlands area. The table below shows the estimated population, \(\mathrm{P}\), for \(\mathrm{t}\) years after the monitoring began.
| Time \(\mathrm{t}\) (years) | Population \(\mathrm{P}\) |
|---|---|
| 0 | 1200 |
| 1 | 1140 |
| 2 | 1083 |
The population is decreasing exponentially. Which of the following equations best models the relationship between \(\mathrm{t}\) and \(\mathrm{P}\)?
1. TRANSLATE the problem information
- Given information:
- Initial population (\(\mathrm{t=0}\)): \(\mathrm{P = 1200}\)
- After 1 year (\(\mathrm{t=1}\)): \(\mathrm{P = 1140}\)
- After 2 years (\(\mathrm{t=2}\)): \(\mathrm{P = 1083}\)
- The relationship is exponential decay
2. INFER the key difference between model types
- Exponential models have constant ratios between consecutive terms
- Linear models have constant differences between consecutive terms
- Since the problem states "exponential," I need to find the constant ratio
3. SIMPLIFY to find the decay factor
- Calculate ratio from year 0 to year 1: \(\mathrm{1140 ÷ 1200 = 0.95}\)
- Calculate ratio from year 1 to year 2: \(\mathrm{1083 ÷ 1140 = 0.95}\)
- The constant ratio confirms exponential decay with factor 0.95
4. INFER the correct model form
- Exponential decay model: \(\mathrm{P = P_0(decay\ factor)^t}\)
- With \(\mathrm{P_0 = 1200}\) and decay factor = 0.95: \(\mathrm{P = 1200(0.95)^t}\)
- This matches choice (B)
5. SIMPLIFY to verify with all data points
- \(\mathrm{t=0}\): \(\mathrm{P = 1200(0.95)^0 = 1200(1) = 1200}\) ✓
- \(\mathrm{t=1}\): \(\mathrm{P = 1200(0.95)^1 = 1140}\) ✓
- \(\mathrm{t=2}\): \(\mathrm{P = 1200(0.95)^2 = 1200(0.9025) = 1083}\) ✓
Answer: B
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Not understanding the fundamental difference between exponential and linear models. Students might calculate the differences between consecutive terms (\(\mathrm{1200-1140=60}\), \(\mathrm{1140-1083=57}\)) and think these are "close enough" to be constant, leading them to choose a linear model.
This may lead them to select Choice A (\(\mathrm{P = 1200 - 60t}\)) which gives the right answer for \(\mathrm{t=1}\) but fails for \(\mathrm{t=2}\).
Second Most Common Error:
Poor TRANSLATE reasoning: Misreading "decreasing exponentially" and selecting an exponential function with a growth factor instead of decay factor.
This may lead them to select Choice C (\(\mathrm{P = 1200(1.05)^t}\)) because they recognize the exponential form but miss that \(\mathrm{1.05 \gt 1}\) represents growth, not decay.
The Bottom Line:
The key insight is recognizing that exponential relationships maintain constant ratios between consecutive values, while linear relationships maintain constant differences. Students who try to force linear thinking onto exponential data will consistently get wrong predictions for future time periods.