There were no jackrabbits in Australia before 1788 when 24 jackrabbits were introduced. By 1920 the population of jackrabbits had...
GMAT Advanced Math : (Adv_Math) Questions
There were no jackrabbits in Australia before \(1788\) when \(24\) jackrabbits were introduced. By \(1920\) the population of jackrabbits had reached \(10\) billion. If the population had grown exponentially, this would correspond to a \(16.2\%\) increase, on average, in the population each year. Which of the following functions best models the population of jackrabbits \(\mathrm{t}\) years after \(1788\)?
1. TRANSLATE the problem information
- Given information:
- Initial population in 1788: 24 jackrabbits
- Average yearly increase: 16.2%
- Need function for population t years after 1788
- This describes exponential growth over time
2. INFER the appropriate model structure
- For exponential growth, we use: \(\mathrm{p(t) = A(1 + r)^t}\)
- \(\mathrm{A}\) = initial amount
- \(\mathrm{r}\) = growth rate (as decimal)
- \(\mathrm{(1 + r)}\) becomes the base of our exponential
3. TRANSLATE the specific values
- \(\mathrm{A = 24}\) (initial population)
- \(\mathrm{r = 16.2\% = 0.162}\) (convert percentage to decimal)
- So \(\mathrm{(1 + r) = 1 + 0.162 = 1.162}\)
4. INFER the complete model
- Substituting: \(\mathrm{p(t) = 24(1.162)^t}\)
- This matches choice C exactly
Answer: C
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students misunderstand how to incorporate the growth rate into the exponential model structure.
They might think the 16.2% growth rate should be used directly as a coefficient (like \(\mathrm{1.162 \times 24^t}\)) or confuse where each piece belongs in the formula. This structural confusion about exponential growth models leads them to select Choice A (\(\mathrm{1.162(24)^t}\)) where they've essentially swapped the roles of the initial value and growth factor.
Second Most Common Error:
Conceptual confusion about exponential function structure: Students incorrectly combine the initial population and growth rate through multiplication rather than understanding that the growth rate modifies the base.
They calculate \(\mathrm{24 \times 1.162 = 27.888}\) and think this should be the base raised to the t power, leading them to select Choice D (\(\mathrm{(24 \cdot 1.162)^t}\)).
The Bottom Line:
Success requires understanding that in exponential growth, the initial value stays as a coefficient while the growth rate (converted to decimal and added to 1) becomes the base of the exponential expression.