The population of a certain 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 20,000 estimates the number of people initially living in the country.

Explanation

Let N denote the number of people living in the country at any time t, and let `N_(0)` denote the number of people initially living in the country.
Then, from `(dN)/(dt) propto N, (dN)/(dt)-kN=0`
which has the solution, `N=ce^(kt)`…………………..(1)
At `t=0, N=N_(0)`, hence, equation (1) states that `N_(0)=ce^(k(0))`, or `c=N_(0)`.
Thus, `N=N_(0)e^(kt)`....................(2)
At `t=2, N=2N_(0)`.
Substituting these values into equation (2), we have
`2N_(0)=N_(0)e^(2k)`or `e^(k)=sqrt(2)`.
Substituting this value into equation (1) gives
`N=N_(0)(sqrt(2))^(k)` ...............(3)
At `t=3, N=20,000`.
Substituting these values into equation (3), we obtain `N_(0)=20,000//2sqrt(2)=7071`