A population grows at a rate proportional to its size, $\frac{dP}{dt} = kP$. It is 1,000 now and doubles in 5 years. What will it be after 15 years?

Explanation

1. 01

   $\frac{dP}{P} = k\,dt$ gives $P = P_0e^{kt}$.

2. 02

   Doubling in 5 years means $e^{5k} = 2$, so $P = 1000 \times 2^{t/5}$.

3. 03

   At $t = 15$: $1000 \times 2^3 = 8000$.