Q. 35.0( 3 Votes )

# 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 log_{e} 5 = 1.609, log_{e} 2 = 0.6931]

Answer :

Let the initial population be P_{o} .

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 = P_{o} when t = 0 sec

Putting the value in equation (1)

∴ log|P| = kt + c

⇒ log|P_{o}| = 0 + c

⇒ c = log|P_{o}| ……(2)

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

log|P| = kt + log|P_{o}|

⇒ log|P| – log|P_{o}| = k t

⇒ (log |P| – log|P_{o}|) = kt []

⇒ log ( = kt ……(3)

Now, the population doubled in 25 years.

Let P = 2P_{o} 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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