# A particular strain of bacteria doubles in population every 10 minutes. Assuming you start with a single bacterium in a petri dish, how many bacteria will there be in 2.5 hours?

Aug 3, 2016

$32 , 768$

#### Explanation:

The trick here is to realize that you can express the increase in population as a power of $2$.

You know that every $10$ minutes, the number of bacteria will double. If you take ${x}_{0}$ to be the initial number of bacteria, you can say that

• ${x}_{0} \cdot 2 \to$ after $10$ minutes

• $\left({x}_{0} \cdot 2\right) \cdot 2 = {x}_{0} \cdot {2}^{\textcolor{red}{2}} \to$ after $\textcolor{red}{2} \cdot 10$ mintues

• $\left({x}_{0} \cdot {2}^{2}\right) \cdot 2 = {x}_{0} \cdot {2}^{\textcolor{red}{3}} \to$ after $\textcolor{red}{3} \cdot 10$ minutes

• $\left({x}_{0} \cdot {2}^{3}\right) \cdot 2 = {x}_{0} \cdot {2}^{\textcolor{red}{4}} \to$ after $\textcolor{red}{4} \cdot 10$ minutes
$\vdots$

and so on. As you can see, you can say that the number of bacteria present after $t$ minutes, $x$, will be

$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{x = {x}_{0} \cdot {2}^{n}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Here

$n$ - the number of $10$-minute intervals that pass in $t$ minutes

In your case, you know that $t$ is equal to

2.5 color(red)(cancel(color(black)("h"))) * "60 min"/(1color(red)(cancel(color(black)("h")))) = "150 minutes"

So, how many $10$-minute intervals do you have here?

$n = \left(150 \textcolor{red}{\cancel{\textcolor{b l a c k}{\text{min"))))/(10color(red)(cancel(color(black)("min}}}}\right) = 15$

Since you start with a single bacterium in a Petri dish, you have ${x}_{0} = 1$ and

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{x = \text{1 bacterium" * 2^15 = "32,768 bacteria}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$