The population of an area is $3500$ lacs with the annual growth rate of $2%$ at $t=0$. After this, the population stabilizes at $14000$ lacs. How much time will it take for the population to reach $7000$ lacs?
The options provided are:
A. $41.2$ Years
B. $35$ Years
C. $5$ Years
D. $17.9$ Years.
I have used the exponential growth formula i.e. $P = A.(1+r)^t$, taking $P = 7000$ and $A = 3500$ and got the answer $35$ Years. By using the rule of $70$ also, we will get the same answer as $35$ Years. However, the answer is given as $41.2$ Years, so am I doing it correctly or this initial point and the final point has got to do something with it?
Because of the limited growth, i.e. population stabilizes at 14000 lacs, you should consider the logistic growth model: the ODE is $$x'(t)=Cx(t)(14000-x(t))$$ with $x(0)=3500$ and $x'(0)/x(0)=0.02$.
Then $C=1/52500$ and, by separation of variables, the solution is $$x(t)=\frac{14000}{1+3\exp(-2x/75)}.$$ Finally, by solving $x(t)=7000$ we find $t=75\ln(3)/2\approx 41.2$