alleryA village grows at a rate proportional to its population. It was 20,000 in 1999 and 25,000 in 2004. Its population in 2009 is:
- 30,000
- 28,750
- 32,500
- 31,250
Show answer and solution
Correct answer: D. 31,250
1. The model is $\dfrac{dy}{dt}=ky$, whose solution is $\ln y=kt+c$.
2. At $t=0$, $y=20{,}000$, so $c=\ln 20{,}000$.
3. At $t=5$, $y=25{,}000$, giving $k=\dfrac15\ln\dfrac54$.
4. At $t=10$: $\ln y=\dfrac{10}{5}\ln\dfrac54+\ln 20{,}000$, so $y=20{,}000\times\left(\dfrac54\right)^{2}$.
5. That is $20{,}000\times 1.5625=31{,}250$. Answering $30{,}000$ assumes a constant increase of 5,000 per five years, which is linear rather than exponential growth.
_Source: NECTA ACSEE 2023 Advanced Mathematics 142/2, Question 7: Differential Equations_