Question 998640
That is what we cal exponential growth.
We like to express it as
{{{P(t)=P[0]*e^kt}}} based on
{{{P[0]}}}= the initial population,
{{{k}}}= a rate constant that we will find, and
{{{e}}}= and irrational number.
We could use any other base, as in
{{{P(t)=P[0]*10^Kt}}} , with a power of base {{{10}}} ,
and a different constant {{{K}}} .
However, we really like to use the number {{{e}}} as a base,
because it makes the growth rate
{{{dP/dt=k*P(t)}}} ,
making {{{k}}} the ratio of the growth rate to the population size.


(a) So, with {{{P[0]=500}}} and {{{P(3)=1500}}} ,
{{{500*e^(3k)=1500}}}-->{{{e^(3k)=1500/500}}}-->{{{e^(3k)=3}}}-->{{{3k=ln(3)}}}-->{{{k=ln(3)/3}}} , and
{{{highlight(P(t)=500*e^("ln(3)t/3"))}}}<-->{{{highlight(P(t)=500*3^("t/3"))}}}<-->{{{highlight(P(t)=500*root(3,3^t))}}}
(b) {{{P(5)=500*e^("5ln(3)/3")=highlight(3120)}}}(rounded)
(c) {{{500*e^("ln(3)t/3")=2130}}}-->{{{e^("ln(3)t/3")=2130/500}}}-->{{{e^("ln(3)t/3")=4.26}}}-->{{{ln(3)t/3=ln(4.26)}}}-->{{{t/3=ln(4.26)/ln(3)}}}-->{{{t=3ln(4.26)/ln(3)}}}-->{{{highlight(t=3.96)}}}(rounded)