Question

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?

Answer 1

`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 * 2 ->` after `10` minutes

• `(x_0 * 2) * 2 = x_0 * 2^color(red)(2) ->` after `color(red)(2) * 10` mintues

• `(x_0 * 2^2) * 2 = x_0 * 2^color(red)(3) ->` after `color(red)(3) * 10` minutes

• `(x_0 * 2^3) * 2= x_0 * 2^color(red)(4) ->` after `color(red)(4) * 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

`color(purple)(|bar(ul(color(white)(a/a)color(black)(x = x_0 * 2^n)color(white)(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 = (150 color(red)(cancel(color(black)("min"))))/(10color(red)(cancel(color(black)("min")))) = 15`

Since you start with a single bacterium in a Petri dish, you have `x_0 = 1` and

`color(green)(|bar(ul(color(white)(a/a)color(black)(x = "1 bacterium" * 2^15 = "32,768 bacteria")color(white)(a/a)|)))`