# Exponential Growth Question

A city doubles its population in 25 years. If it is growing exponentially, when will it triple its population?

The above is a question in my maths textbook in the topic Exponential Growth & Decay.

I'm a bit confused as to how I should approach this question.

We have been taught to use the formula: $$Q=Ae^{kt}$$ Where $Q$ is the quantity, $A$ is the initial quantity, $k$ is the growth/decay constant and $t$ is the time.

In reference to the question, I don't think I need $A$ so here is the equation I ended up with: $$2Q=e^{25k}$$

Edit:

I found out that $$k=\frac{\ln2}{25}$$ I then let $Q=3A$ and the following is my working: $$3A=Ae^{25\frac{\ln2}{25}t}$$ $$3A=Ae^{\ln2t}$$ $$3=e^{\ln2t}$$ $$3=2^{t}$$ $$\ln{3}=t\ln{2}$$ $$t=\frac{\ln{3}}{\ln{2}}$$ $$t=1.6$$

I can't figure out what is wrong in my working out.

The provided answer is: 39.6 years

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You do need $A$; after 25 years, the population is twice $A$, not twice $Q$. And you want to know when the population will be three times $A$. –  Gerry Myerson Jun 25 '12 at 12:58
Exactly @Gerry well put, so what happens to the formulae when $Q=2A$ and then $Q=3A$? –  Autolatry Jun 25 '12 at 13:03
You were careless when you let $Q=3A$. What happened to the $t$ in $Q=Ae^{kt}$? –  Gerry Myerson Jun 26 '12 at 9:45
@GerryMyerson Right, included the $t$ this time but I am still stuck on what I am doing wrong. –  juleszero Jun 26 '12 at 9:59
There's still a mistake in that very first line after "let $Q=3A$." You've got $Q=Ae^{kt}$; $k=(1/25)\log2$; $Q=3A$. Put them together carefully and you'll get to the right answer. –  Gerry Myerson Jun 26 '12 at 10:04

Start with $Q = Ae^{kt}$. If the doubling time is 25 years, this translates to $$2A = Ae^{25k}.$$ You should be able to solve for $k$ and make a go of it now.
I've found that k is $\frac{\ln 2}{25}$. I'm not sure what to do now that I have k. –  juleszero Jun 25 '12 at 13:15
juleszero, did you read my comment? Doesn't it suggest what to do now that you have $k$? –  Gerry Myerson Jun 25 '12 at 13:28