Question

The rate of increase of bacteria in a certain culture is proportional to the number present. If it double in 5 hours then in 25 hours, its number would be

• A. 8 times the original
• B. 16 times the original
• C. 32 times the original
• D. 64 times the original

Answer 1

The correct option is C 32 times the original

Let $P_0$ be the initial population and let the population after t years be P. Then $\frac{dP}{dt}=kP\Rightarrow \frac{dP}{P}=kdt$
On integrating, we have $logP=kt+c$
At $t=0,P=P_0$
$\therefore log{P}_0=0+c;\therefore logP=kt+log{P}_0\Rightarrow log\frac{P}{P_0}=kt$
When t =5 hrs, $P=2P_0$
$\therefore log\frac{2P_0}{P_0}=5k\Rightarrow K=\frac{log2}{5};\therefore log\frac{P}{P_0}=\frac{log2}{5}t$
When t =25 hours, we have
$log\frac{P}{P_0}=\frac{log2}{5}\times 25=5log2=log32;\therefore P=32P_0$