Say the population of a city is increasing at a constant rate of 11.5% per year. If the population is currently 2000, estimate how long it will take for the population to reach 3000.
How can this be solved using the formula below. I know how to solve when we input x number of years but don't know how to solve when the number of years is the unknown.
Best Answer
So, we have that after $x$ years, there are 3000 people where we started from 2000. Thus
$$3000=2000 e^{rx}$$
which can be rewritten as
$$e^{rx}=1.5$$
We also know that after 1 year, there is a 11.5% increase. Thus
$$e^r=1.115$$
From the conjugation of those two formulas, we have that
$$(1.115)^x=1.5$$
So all we have to do is see how many times we have to multiply 1.115 by itself to obtain 1.5. Since this will likely involve a non-integer number, we just take the integer solution such that we exceed 1.5. Which is 4. So somewhere in the fourth year, population exceeds 3000.