Q. 4

# 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 20000, if the rate of growth of bacteria is proportional to the number present?

Let the count of bacteria be C at any time t.

According to question,

where k is a constant

Integrating both sides, we have

= k∫dt

log|C| = kt + a ……(1)

Given, we have C = 100000 when t = 0 sec

Putting the value in equation (1)

log|C| = kt + a

log|100000| = 0 + a

a = log|100000| ……(2)

Putting the value of a in equation (1) we have,

log|C| = kt + log|100000|

log|C| – log|100000| = k t []

log ( = kt ……(3)

Also, at t = 2 years, = 110000

From equation(3),we have

kt = log (

k×2 = log (

k = ……(4)

Now, equation (3) becomes,

Now, let t1 be the time for the population to reach 200000

The time require for the population to be 200000 = hours

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