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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.

Using the formula given, so far I've figured out how many years it will take (see working below) but how can I narrow it down to the nearest month?

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    $\begingroup$ Have you learned anything about logarithms yet? $\endgroup$ – Brian M. Scott Jul 10 '13 at 11:19
  • $\begingroup$ Nope, not up to that yet. $\endgroup$ – jaykirby Jul 10 '13 at 11:20
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    $\begingroup$ You could use the same reasoning with $e^{12r}=1.115$ instead, and $x$ is the time in months. But this is quite tedious, and I would recommend you to read up on the natural logarithm instead, because you would get an exact solution much faster. $\endgroup$ – zuggg Jul 10 '13 at 11:28
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Let $a=1.115^{1/12}=\sqrt[12]{1.115}$, the twelfth root of $1.115$. Then

$$1.115^x=(a^{12})^x=a^{12x}\;,$$

and $12x$ is the number of months that have gone by. Thus, if you can solve $a^y=1.5$, $y$ will be the desired number of months. Without logarithms the best that you’ll be able to do is find the smallest integer $y$ such that $a^y\ge 1.5$.

By my calculation $a\approx1.009112468437$. You could start with $a^{36}$ and work up until you find the desired $y$.

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  • $\begingroup$ Nice explanation. I get 3 years and 9 months, is that right? $\endgroup$ – jaykirby Jul 10 '13 at 11:56
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    $\begingroup$ @jaykirby: I believe that you’re one month short: for $3$ years, $8$ months I get about $1.4905$, while for $3$ years, $9$ months I get about $1.5041$. $\endgroup$ – Brian M. Scott Jul 10 '13 at 11:59
  • $\begingroup$ Noted. I amended my comment after realising that. $\endgroup$ – jaykirby Jul 10 '13 at 12:01
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    $\begingroup$ @jaykirby: Looks good, then. $\endgroup$ – Brian M. Scott Jul 10 '13 at 12:01
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You can apply the same principle as in my previous solution. What you need is the growth rate in a month. Call the rate per month $e^s$, then

$$(e^s)^{12}=e^r=1.115$$

because after 12 months you of course have 1 year. You can solve this to give you

$$e^s=\sqrt[12]{1.115}=1.009112$$

You can work it out from here.

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  • $\begingroup$ Got it. So if the monthly growth rate is 1.009112, what is the next step to figure out how many months total to reach 3000 people? $\endgroup$ – jaykirby Jul 10 '13 at 11:36
  • $\begingroup$ Follow the same steps of the previous exercise. $\endgroup$ – Raskolnikov Jul 10 '13 at 11:53

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