Differential Equations Question 18
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
[Pb. CET 2004]
Options:
A) 8 times the original
B) 16 times the original
C) 32 times the original
D) 64 times the original
Show Answer
Answer:
Correct Answer: C
Solution:
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 $ \log P=kt+c $
At $ t=0, $
$ P=P_0 $
$ \therefore $ $ \log P_0=0+c $ ;
$ \therefore $ $ \log P=kt+\log P_0 $
Therefore $ \log \frac{P}{P_0}=kt $
When $ t=5 $ hrs, $ P=2P_0 $
$ \therefore $ $ \log \frac{2P_0}{P_0}=5k $
Therefore $ K=\frac{\log 2}{5} $ ;
$ \therefore $ $ \log \frac{P}{P_0}=\frac{\log 2}{5}t $
When $ t=25 $ hours, we have
$ \log \frac{P}{P_0}=\frac{\log 2}{5}\times 25=5\log 2=\log 32 $ ;
$ \therefore $ $ P=32P_0 $ .
BETA
