Question

The rate of increase of bacteria in a culture is proportional to the number of bacteria present and it is found that the number doubles in 6 hours. Prove that the bacteria becomes 8 times at the end of 18 hours.

Answer 1

Let the original count of bacteria be N and the count of bacteria at any time t be P.
Given: $\frac{dP}{dt}\alpha P$
$$\begin{gathered} \Rightarrow \frac{dP}{dt}=aP \\ \Rightarrow \frac{dP}{P}=adt \\ \Rightarrow \log \left|P\right|=at+C.....\left(1\right) \\ \mathrm{Now}, \\ P=N\mathrm{at}t=0 \\ \mathrm{Putting}P=N\mathrm{and}t=0\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|N\right|=C \\ \mathrm{Putting}C=\log \left|N\right|\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ \log \left|P\right|=at+\log \left|N\right| \\ \Rightarrow \log \left|\frac{P}{N}\right|=at.....\left(2\right) \\ \mathrm{According}\mathrm{to}\mathrm{the}\mathrm{question}, \\ \log \left|\frac{2N}{N}\right|=6a \\ \Rightarrow a=\frac{1}{6}\log \left|2\right| \\ \mathrm{Putting}a=\frac{1}{6}\log \left|2\right|\mathrm{in}\left(2\right),\mathrm{we}\mathrm{get} \\ \log \left|\frac{P}{N}\right|=\frac{t}{6}\log \left|2\right|.....\left(3\right) \\ \mathrm{Putting}t=18\mathrm{in}\left(3\right)\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{bacteria}\mathrm{after}18\mathrm{hours},\mathrm{we}\mathrm{get} \\ \log \left|\frac{P}{N}\right|=\frac{18}{6}\log \left|2\right| \\ \Rightarrow \log \left|\frac{P}{N}\right|=3\log \left|2\right| \\ \Rightarrow \log \left|\frac{P}{N}\right|=\log \left|8\right| \\ \Rightarrow \frac{P}{N}=8 \\ \Rightarrow P=8N \end{gathered}$$