# A bacteria culture starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours there are 9,000 bacteria. How do you find the number of bacteria after 5 hours?

May 24, 2017

There are about $61814$ bacteria after 5 hours.

#### Explanation:

The population of bacteria can be represented by the generic exponential formula:

$y = C {e}^{k x}$

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To find the specific exponential formula for this problem, let's plug in the two data values we have now.

We know that when $x = 0$, $y = 500$.

$500 = C {e}^{k \cdot 0}$
$500 = C$

So now we know our equation is $y = 500 {e}^{k x}$.

We know that when $x = 3$, $y = 9000$.

$9000 = 500 {e}^{3 k}$

$18 = {e}^{3 k}$

$\ln \left(18\right) = 3 k$

$\frac{1}{3} \ln 18 = k$

Therefore, our equation is:

$y = 500 {e}^{\frac{1}{3} x \ln 18}$

Which can be simplified to:

$y = 500 {\left(18\right)}^{\frac{x}{3}}$

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Now, just plug in $x = 5$ and solve for $y$.

$y = 500 {\left(18\right)}^{\frac{5}{3}} = 61814$

So there are about $61814$ bacteria after 5 hours.