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?

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Answer 1 · 2017-05-24T23:10:23

There are about #61814# bacteria after 5 hours.

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 #x=0#, #y=500#.

#500 = Ce^(k*0)#
#500 = C#

So now we know our equation is #y=500e^(kx)#.

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

#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 #x=5# and solve for #y#.

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

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