# Bacteria grow by cell division. If each cell divides into 2 in every 2 minutes how many cells will exist after 32 minutes? Assume there was only one cell at the beginning?

Apr 5, 2016

${\left(2\right)}^{{2}^{31}} = {2}^{2147483648}$.

#### Explanation:

After 1' the count is 2. After 2', the count is ${2}^{{2}^{1}}$, and so on. . After, 32 minutes, the cell population will be
$\left(\ldots . \left({\left({\left({\left({2}^{2}\right)}^{2}\right)}^{2}\right)}^{2}\right) \ldots 31 \times\right)$
$= {\left(2\right)}^{{2}^{31}}$
The count after N minutes is ${\left(2\right)}^{{2}^{N - 1}}$.
For cell-division arithmetic, double-precision format is a must.

Apr 5, 2016

$65536$ cells

#### Explanation:

Use the exponential growth formula:

color(blue)(|bar(ul(color(white)(a/a)y=a(b)^(t/d)color(white)(a/a) |)))

where:
$y =$final amount
$a =$inital amount
$b =$exponential growth
$y =$time elapsed
$d =$doubling-time

Start by setting up the equation.

• $y =$unknown
• $a = 1$ (starts with $1$ cell)
• $b = 2$ (represents doubling of population)
• $t = 32$ (how long bacteria population doubles for - $32$ minutes)
• $d = 2$ (doubling-time of $2$ minutes)

Thus:

$y = a {\left(b\right)}^{\frac{t}{d}}$

$y = 1 {\left(2\right)}^{\frac{32}{2}}$

Solve for $y$.

$y = {\left(2\right)}^{16}$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} y = 65536 \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\therefore$, $65535$ cells exist after $32$ minutes.