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JEE Main & Advanced Mathematics Differential Equations Question Bank Critical Thinking

  • question_answer
    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]

    A) 8 times the original                 

    B) 16 times the original

    C) 32 times the original               

    D) 64 times the original

    Correct Answer: C

    Solution :

    • Let \[{{P}_{0}}\] be the initial population and let the population after t years be P. Then \[\frac{dP}{dt}=kP\Rightarrow \frac{dP}{P}=k\,dt\]        
    • On integrating, we have \[\log P=kt+c\]        
    • At \[t=0,\] \[P={{P}_{0}}\]        
    • \[\therefore \] \[\log {{P}_{0}}=0+c\]; \[\therefore \] \[\log P=kt+\log {{P}_{0}}\]Þ \[\log \frac{P}{{{P}_{0}}}=kt\]        
    • When \[t=5\] hrs, \[P=2{{P}_{0}}\]        
    • \[\therefore \] \[\log \frac{2{{P}_{0}}}{{{P}_{0}}}=5k\] Þ \[K=\frac{\log 2}{5}\]; \[\therefore \] \[\log \frac{P}{{{P}_{0}}}=\frac{\log 2}{5}t\]        
    • When \[t=25\] hours, we have           
    • \[\log \frac{P}{{{P}_{0}}}=\frac{\log 2}{5}\times 25=5\log 2=\log 32\]; \[\therefore \] \[P=32{{P}_{0}}\].


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