# 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.

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