# Exponential growth - apparently not getting something

"A certain bacteria population is known to quadruple every $90$ minutes. Suppose that there are initially $120$ bacteria.

What is the size of the population after $t$ hours?"

I've been using this formula: $$N(t) = ae^{rt}$$ Where $a$ is the starting amount and $r$ is the rate of growth and $t$ is the time

So since the population quadruples every $90$ minutes, we have to convert it to how much it would grow in an hour, right?

$$4 (\text{initial growth rate}) * 60/90 = 8/3 (\text{growth rate per hour})$$

So after plugging everything in, I get $N(t) = 120e^{(8/3)t}$

This is incorrect. What am I doing wrong?

• You could use in $90$ minutes we grow by $2^2$, so in $60$ we grow by $2^{(2)(60)(90)}=2^{4/3}$. That would be fine, but works well only with "nice" numbers, like $2^2$. Safer is the more mechanical procedure described in my answer. – André Nicolas Sep 12 '13 at 3:24

Let us measure time $t$ in hours.
Then $4=e^{(1.5)r}$, and therefore $$r=\frac{\ln 4}{1.5}.$$
So our formula becomes $$N(t)=a\cdot e^{(t\ln 4)/1.5}.\tag{1}$$
Since $e^{\ln 4}=4$, we can alternately write $$N(t)=a\cdot 4^{t/1.5}.$$ We could rewrite $4^{t/(1.5)}$ as $4^{2t/3}$ or $16^{t/3}$ or $2^{4t/3}$.
The form (1) may be the most generally useful one, though there could be arguments for using powers of $2$, or powers of $10$.