We have an exponential function for the population A(t) and we have to find the growth rate.
[tex]A=6e^{0.003t}[/tex]This can be expressed as:
[tex]\frac{A(t+1)-A(t)}{A(t)}=\frac{A(t+1)}{A(t)}-1[/tex]For the exponential formula we get:
[tex]\begin{gathered} \frac{A(t+1)}{A(t)}-1 \\ \frac{6e^{0.003(t+1)}}{6e^{0.003t}}-1 \\ e^{0.003(t+1-t)}-1 \\ e^{0.003\cdot1}-1 \\ 1.003-1 \\ k=0.003 \end{gathered}[/tex]The growth rate is 0.003 or 0.3%.
We can calculate the time needed to double the population dividing ln(2) by the growth rate:
[tex]t=\frac{\ln (2)}{k}\approx\frac{0.693}{0.003}\approx231[/tex]Answer:
a) The growth rate is 0.003 or 0.3%
b) The number of years to duplicate the population is 231 years.
