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# In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

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#### Let the number of bacteria at a given times 't' be NThen,dNdt=kN where k is the proportionality constant.dNN=kdt∫NN0dNN=∫t0kdtln(NN0)=ktN=N0ekt ...(i)If N0=105 and t=2hours and N=(1+10100)×105=1.1×105Hence lnNN0=ktln(1.1×105105)=2kln(1.1)=2kk=12ln(1.1)hours−1=ln(√1.1)hours−1Now ln(NN0)=ktN is now 2×105Hence NN0=2×105105=2Thus ln(NN0)=kt implies ln2=ln√1.1tln2=ln(1.1)2t2ln2=ln(1.1)tt=ln4ln(1.1)=14.54 hours

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