The rate of growth of a population is proportional to the number present. If the population of a city doubled in the past 25 years, and the present population is 100000, when will the city have a population of 500000?
[Given loge 5 = 1.609, loge 2 = 0.6931]
Let the initial population be Po .
And the population after time t be P.
According to question,
⇒ 
⇒ 
⇒ 
Integrating both sides, we have
⇒ ∫
= k∫dt
⇒ log|P| = kt + c ……(1)
Given, we have P = Po when t = 0 sec
Putting the value in equation (1)
∴ log|P| = kt + c
⇒ log|Po| = 0 + c
⇒ c = log|Po| ……(2)
Putting the value of c in equation (1) we have,
log|P| = kt + log|Po|
⇒ log|P| – log|Po| = k t
⇒ (log |P| – log|Po|) = kt [
]
⇒ log (
= kt ……(3)
Now, the population doubled in 25 years.
Let P = 2Po at t = 25 years
∴ kt = log (
⇒ k×25 = log (
⇒ k = 
……(4)
Now, equation (3) becomes,
Now, let t1 be the time for the population to increase from 100000 to 500000
⇒ 
⇒ 
⇒ 
(log5 = 1.609 and log2 = 0.6931)
⇒ t1 = 58
∴ The time require for the population to be 500000 = 58 years.