Question 600561
First off, let's use the fact that the population is growing at an annual rate of 7.7% to find the value of k.


If the initial population is say 1000 people, then in one year, we add 7.7% of that 1000 to the initial population to get: 1000 + 0.077*1000 = 1000 + 77 = 1077



So I = 1000 and N = 1077. This all happens in one year, so t = 1. Use these three variables to find k



{{{N = Ie^(kt)}}}


{{{1077 = 1000*e^(k*1)}}}


{{{1077 = 1000*e^(k)}}}


{{{1077/1000 = e^(k)}}}


{{{1.077 = e^(k)}}}


{{{ln(1.077) = ln(e^(k))}}}


{{{ln(1.077) = k*ln(e)}}}


{{{ln(1.077) = k*(1)}}}


{{{ln(1.077) = k}}}


{{{0.0741793981742515 = k}}}


{{{k = 0.0741793981742515}}}



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Now that you know the value of k, use it to find the value of t when I = 1000, N = 2000 (double the population of the initial 1000)



{{{N = Ie^(kt)}}}


{{{2000 = 1000*e^(0.0741793981742515t)}}}


{{{2000/1000 = e^(0.0741793981742515t)}}}


{{{2 = e^(0.0741793981742515t)}}}


{{{ln(2) = ln(e^(0.0741793981742515t))}}}


{{{ln(2) = 0.0741793981742515t*ln(e)}}}


{{{ln(2) = 0.0741793981742515t*(1)}}}


{{{ln(2) = 0.0741793981742515t}}}


{{{ln(2)/0.0741793981742515 = t}}}


{{{0.693147180559945/0.0741793981742515 = t}}}


{{{9.34420064896866 = t}}}


{{{t = 9.34420064896866}}}


So it will take roughly 9.3442 years for the population to double.