# 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?

##### 1 Answer

There are about

#### Explanation:

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

#y = Ce^(kx)#

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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

#500 = Ce^(k*0)#

#500 = C#

So now we know our equation is

We know that when

#9000 = 500e^(3k)#

#18 = e^(3k)#

#ln(18) = 3k#

#1/3ln18 = k#

Therefore, our equation is:

#y = 500e^(1/3xln18)#

Which can be simplified to:

#y = 500(18)^(x/3)#

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Now, just plug in

#y = 500(18)^(5/3) = 61814#

So there are about