Question

If the population of a country doubles in 60 years, in how many years will it be triple under
the assumption that the rate of increase in proportional to the number of inhabitants?
[Given : log 2 = 0.6912 and log 3 = 1.0986.]

Answer 1

Let P be the population of the country at time t.

Given `"dP"/dt prop P`

`therefore "dP"/dt =kP` (where k is a constant)

`therefore 1/PdP=kdt`

Integrating both the side w.r.t x

`int 1/Pdp=kint 1 dt+c`

`logP=kt+c`

`P=e^(kt+c)=e^(kt).e^c`

Let `e^c=alpha`

`therefore P=alpha.e^(kt)`

Let initial population at t = 0

`therefore N=alpha.e^0   thereforeN=alpha`

`P=N.e^(kt)`

Given P = 2N when t = 60 years,

`therefore 2N=Ne^(60k)`

`therefore 2=e^(60k)=>k=1/60 log 2`

`therefore P=N.e^(60k)`

Required t when P = 3N

`3=e^(kt)=>log3=kt`

`log3=(1/60log2).t`

`t=(60log3)/log2`

`=(60xx1.0986)/0.6912`

`=95.4 years(approx.)`

The population of the countr will triple approximately in 95.4 years.