Applications Of Derivatives Question 66
Question: The rate of increase of bacteria in a certain culture is proportional to the number present. If it doubles in 5 hours then in 25 hours, its number would be
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:
[c] Let N be the no. of bacteria at time t. Let $ N_0 $ be the initial original no. of bacteria. Then, $ \frac{d}{dt}N\infty N\frac{d}{dt}N=kN\Rightarrow \frac{dN}{N}=kdt $
$ \Rightarrow \int{\frac{dN}{N}=k}\int{dt\Rightarrow \log N=kt+c} $ At $ t=0,N=N_0\Rightarrow \log N_0=0+c\Rightarrow c=\log N_0 $
$ \therefore \log N=kt+\log N_0\Rightarrow \log \frac{N}{N_0}=kt $ When t = 5 hrs. $ N=2N_0\therefore \log ( \frac{2N_0}{N_0} )=5k $
$ \Rightarrow k=\frac{\log 2}{5}\therefore \log \frac{N}{N_0}=\frac{\log 2}{5}t $ When $ t=25hrs,\log \frac{N}{N_0}=\frac{\log 2}{5}\times 25 $
$ \Rightarrow \log \frac{N}{N_0}=5\log 2\log \frac{N}{N_0}=\log 2^{5} $
$ \Rightarrow \frac{N}{N_0}=32\therefore N=32N_0 $ . Therefore, 32 times the original.
BETA
