Answer:
[tex]t \approx 49.860\,years[/tex]
Explanation:
The increase rate of population is described by the following ordinary differential equation is:
[tex]\frac{dP}{dt} = \frac{P}{\tau}[/tex]
Where [tex]\tau[/tex] is the time constant. The solution of the differential equation is:
[tex]P(t) = P_{o}\cdot e^{\frac{t}{\tau} }[/tex]
The time constant in years is found by substituting known variables:
[tex]\ln 1.014 = \frac{1}{\tau}[/tex]
[tex]\tau = \frac{1}{\ln 1.014}[/tex]
[tex]\tau \approx 71.927\,years[/tex]
The doubling time is:
[tex]\ln 2 = \frac{t}{71.927\,years}[/tex]
[tex]t = (71.927\,years)\cdot \ln 2[/tex]
[tex]t \approx 49.860\,years[/tex]
