Q. 575.0( 1 Vote )

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.

Answer :

Let y be the number of bacteria at any instant t


It is given that the rate of growth of the bacteria is proportional to the number present.


=


= (where k is a constant)


=


Integrating both sides, we get


=


= log y = kt + C


Let y0 be the number of bacteria at t = 0.


= log y0 = C


Substitute the value of C in, we get


log y = kt + log y0


= log y - log y0 = kt


=


Also, it is given that the number of bacteria increased by 10% in 2 hours.


=


=


Substituting the value,


=


= k =


Therefore,


=


=


Now, the time when the number of bacteria increases from 100000 to 200000 be t1.


= at t = t1


Now, =


Hence, in hours the number if bacteria increases from 100000 to 200000.


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