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Question

In a culture the bacteria count is 100000. The number is increased by 10% in 2 hours. In how many hours will the count reach 200000, if the rate of growth of bacteria is proportional to the number present.

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Solution

Let at any time the bacteria count be N.Given: dNdtα NdNdt=λN1NdN=λdtIntegrating both sides, we get1NdN=λdtln N=λt+ln C ...(1)Given:at t=0, N=100000therefore, ln C=ln 100000Putting the value in (1) we get,ln N=λt+ln 100000Also, at t=2N=110000Putting the values of t and N in (1), we getln 110000=2λ+ ln 10000012ln 1110=λSubstituting the values of ln C and λ in (1), we getln N=12ln 1110t+ ln 100000 ....(2)When N=200000, let t=T.Substituting these values in (2), we get ln 200000=T2ln 1110+ln 100000ln 2=T2ln 1110T=2ln 2ln 1110Therefore, in 2ln 2ln 1110 hours, the count will reach 200000.

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