Question 1157957
Exponential functions are functions where the variable is in the exponent, such as  {{{y=2^x}}} .
Exponential functions of the form {{{y=A*e^kt}}} where {{{A}}} and {{{k}}} are constants
are used to model exponential growth and exponential decay.
Let's call our function {{{P}}} for  population.
Let {{{t}}} be the number of years after 2000.
The function is {{{P=A*e^kt}}} .
For the year 2000, {{{t=0}}} and {{{P=222030}}} , and
{{{222030=A*e^(k*0)}}} --> {{{222030=A*e^0}}} --> {{{222030=A*1}} -->{{{highlight(A=222030)}}}
For the year 2010, {{{t=10}}} , {{{P=259841}}} , and
{{{259841=222030*e^(k*10)}}} --> {{{259841/222030=e^(k*10)}}} --> {{{ln(259841/222030)=k*10}}} --> {{{highlight(k=ln(259841/222030)/10)}}}
That is the exact value.
{{{k=approximately}}}{{{0.0157257}}} or {{{k=approximately}}}{{{0.01573}}} .
 
We could write the growth function as
{{{P=222030*e^(("ln(259841/222030)/10")*t)}}} , or
{{{P=222030*e^(ln("259841/222030")*("t/10"))}}} , or
{{{P=222030*(e^ln("259841/222030"))^("t/10")}}} , or {{{P=222030*(259841/222030)^("t/10")}}} ,
or maybe use the approximate value for k, and write it as
{{{P=222030*e^(0.01573t)}}}
 
We can calculate the projected population in 2010 and 2020, using the equations found above.
For 2010, {{{t=10}}} . Substituting that value into {{{P=222030*e^(0.01573t)}}} , we get
{{{P=222030*e^(0.01573*10)=222030*e^0.1573=222030*1.17034667=259852}}}(rounded to whole number).
However using {{{P=222030*e^(0.0157257t)}}} , we get
{{{P=222030*e^(0.0157257*10)=222030*e^0.157257=222030*1.17029634=25984.08967}}}{{{"="}}}{{{259841}}}(rounded to whole number).
To match all 6 digits in the population for 2010, we need the exact value of {{{k}}} , or at least a better approximation and we need to carry more digits through the calculations.
For 2020 {{{t=20}}} , substituting it into {{{P=222030*e^(0.0157257t)}}} we get
{{{P=222030*e^"0.0157257*20/10"=222030*e^0.03146514=222030*1.3695943527=304090.85}}} .
Rounding to whole numbers, we get that the projected population in 2020 is {{{highlight(304091)}}} .
If we use the more accurate {{{P=222030*(259841/222030)^"t/10"}}} , we get
{{{P=222030*(259841/222030)^"20/10"=222030*(259841/222030)^2=259841^2/222030=304091.09}}} .
Rounding to whole numbers, we get that the projected population in 2020 is {{{highlight(304091)}}} .
 
Using the equation with the exact value of {{{k}}} , {{{P=222030*e^(("ln(259841/222030)/10")*t)}}} , we could set {{{P=2*222030=444060}}} and solve for {{{t}}} .
We get
{{{2=e^(("ln(259841/222030)/10")*t)}}}-->{{{ln(2)="ln(259841/222030)/10"*t}}}-->{{{t=10*ln(2)/"ln(259841/222030)"=44.077}}} (rounded to 3 decimal places).
Rounding to whole numbers, it takes {{{highlight(44)}}} years for the population to double at the rate observed between 2000 and 2010.