Question 268809
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if the U.S grows at an annual rate of 7.5%, how long will take to reach a population of 400 000 000?
(the population, now, is 300 000 000)
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        The solution in the post is incorrect conceptually.
        It is incorrect, since it uses a linear model, while an exponential model must be used.

        See my correct solution below.



<pre>
Use exponential function for the population

    P(t) = {{{300000000*(1+0.075)^t}}} = {{{300000000*1.075^t}}}.


Then you have this equation to find 't', the time from "now' in years

    400000000 = {{{300000000*1.075^t}}}.


Divide both sides by 300000000

    {{{4/3}}} = {{{1.075^t}}}.


Take logarithm of both sides

    {{{log((4/3))}}} = t*log((1.075)}}}


Express 't' and calculate

    t = {{{log((4/3))/log(1.075))}}} = 3.977868374.


Rounding, you may say that the process will take about 3.98 years (or 4 years).


In reality, the average percentage growing population US in the last 10-15 years (2010 - 2025) 
is about 1% (or below it).
</pre>

Solved correctly.