Question

The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?

Answer 1

Let the original population be N and the population at any time t be P.
Given: $\frac{dP}{dt}\alpha P$
$$\begin{gathered} \Rightarrow \frac{dP}{dt}=aP \\ \Rightarrow \frac{dP}{P}=adt \\ \Rightarrow \log \left|P\right|=at+C.....\left(1\right) \\ \mathrm{Now}, \\ P=N\mathrm{at}t=0 \\ \mathrm{Putting}P=N\mathrm{and}t=0\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|N\right|=C \\ \mathrm{Putting}C=\log \left|N\right|\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|P\right|=at+\log \left|N\right| \\ \Rightarrow \log \left|\frac{P}{N}\right|=at.....\left(2\right) \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \log \left|\frac{2N}{N}\right|=25a \\ \Rightarrow a=\frac{1}{25}\log \left|2\right|=\frac{1}{25}\times 0.6931=0.0277 \\ \mathrm{Putting}a=0.0277\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ \log \left|\frac{P}{N}\right|=0.0277t...\left(3\right) \\ \mathrm{For}P=500000\mathrm{and}N=100000: \\ \log \left|\frac{500000}{100000}\right|=0.0277t \\ \Rightarrow t=\frac{\log 5}{0.0277}=\frac{1.609}{0.0277}=58.08\mathrm{years}=\mathrm{Approximately}58\mathrm{years} \end{gathered}$$