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