The growth of population is proportional to the number present. If the population of a colony doubles is 50 years, then the population will become triple in _____ years

Let \(\mathrm{P}_{0}=\) Initial population
Given \(\frac{\mathrm{dP}}{\mathrm{dt}} \propto \mathrm{P} \Rightarrow \frac{\mathrm{dP}}{\mathrm{dt}}=\lambda \mathrm{P} \Rightarrow \int \frac{\mathrm{dP}}{\mathrm{P}}=\int \lambda \mathrm{dt}\)
\(\log |\mathrm{P}|=\lambda \mathrm{t}+\mathrm{C}\ldots(1)\)
At \(t=0, \quad P=P_{0}\) we get \(\log P_{0}=0+C \Rightarrow C=\log P_{0}\)
\(\log P=\lambda t+\log P_{0} \Rightarrow \log \left(\frac{P}{P_{0}}\right)=\lambda t\)
When \(P=2 P_{0}, t=50 \Rightarrow \log \left(\frac{2 P_{0}}{P_{0}}\right)=50 \lambda\)
\(\therefore \log 2=50 \lambda \Rightarrow \lambda=\frac{1}{50} \log 2\)
\(\therefore \log \left(\frac{\mathrm{P}}{\mathrm{P}_{0}}\right)=\frac{\mathrm{t}}{50}(\log 2)\)
When \(\mathrm{P}=3 \mathrm{P}_{0}\) we get
\(\log 3=\frac{t}{50}(\log 2) \Rightarrow t=50\left(\frac{\log 3}{\log 2}\right)\) yrs.