The population of a city increases at a rate proportional to the population at that time. If the population of the city increase from 20 lakhs to 40 lakhs in 30 years, then after another 15 years the population is

We have \(\frac{\mathrm{dP}}{\mathrm{dt}} \propto \mathrm{P} \Rightarrow \frac{\mathrm{dP}}{\mathrm{dt}}=\mathrm{kP}\)
\(
\therefore \int \frac{\mathrm{dP}}{\mathrm{dt}}=\int \mathrm{k} d t \Rightarrow \log \mathrm{P}=\mathrm{kt}+\mathrm{c}
\)
From given data, we write
\(\log 20=\mathrm{k}(0)+\mathrm{c} \Rightarrow \mathrm{c}=\log 20 \)
\( \therefore \log \mathrm{P}=\mathrm{kt}+\log 20 \)
\( \text { Also } \log 40=30 \mathrm{k}+\log 20 \)
\( \therefore \log 40-\log 20=30 \mathrm{k} \Rightarrow \mathrm{k}=\frac{1}{30} \log 2 \)
\( \therefore \log \mathrm{P}=\left(\frac{\log 2}{30}\right) \mathrm{t}+\log 20 \)
\( \text { When } \mathrm{t}=30+15=45 \)
\( \therefore \log \mathrm{P}=\left(\frac{\log 2}{30}\right)(45)+\log 20=(\log 2)\left(\frac{3}{2}\right)\) \(+\log 20 \)
\( =\log (2)^{\frac{3}{2}}+\log 20=\log (2 \sqrt{2} \times 20) \)
\( \therefore \mathrm{P}=40 \sqrt{2} \text { lakhs}\)