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.

Question

Philoid · Accepted 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.