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 is proportional to the number of inhabitants?
[Given: $\log 2=0.6912$ and $\log 3=1.0986$.]

Answer 1

Let $P$ be the number of population

$\frac{dP}{dt}=kP$

$\int \frac{dP}{P}=\int kdt$

$\Rightarrow \ln P=kt+C$……….. (Genral Solution)

When $t=0,P=P_0$

$\ln P_0=k\left(0\right)+C$

$\Rightarrow C=\ln P_0$

When $t=60$ years, $P=2P_0$

$\ln 2P_0=k\left(60\right)+\ln P_0$

$\Rightarrow k=0.01152$

When $t=?,P=3P_0$

$\ln 3P_0=0.01152\left(t\right)+\ln P_0$

$\Rightarrow t=95$ Years.