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)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now, just plug in
#y = 500(18)^(5/3) = 61814#
So there are about