Question

The population of a certain country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is $20000$, estimate the number of people initially living in the country.

• A. $6071$
• B. $7071$
• C. $7061$
• D. $6077$

Answer 1

Answer: (B)

$7071$

Given that the rate of increase of the population of a country is proportional to the number of people living presently in the country.

i.e., $\frac{dP}{dt}=kP$,where $P$ is the population.

$k$ is the proportionalitiy constant.

$\int \frac{1}{P}dP=\int kdt$

$⟹\ln P=kt+c$

Let $P_i$ be the initial population of the country,

$\therefore$ $t=0⟹P=P_i$.

$\therefore P=P_i{e}^{kt}$

Given at $t=2years$ population is doubled which gives,

$2=e^{2k}$

$⟹k=\ln (\sqrt{2})$

Given at $t=3years$ population of the country is $20000$,

$⟹20000=P_i{e}^{3\ln \sqrt{2}}$

$\therefore P_i=\frac{20000}{2.8284}$

$⟹P_i=7071$

$\therefore$ the initial population of the town is $7071$.