Question

A bacteria culture grows exponentially. After 2 hours there are 600 bacteria and after 8 hours there are 75000 bacteria.

(A) Find the culture's initial population
(B) When will the population reach 200,000?

Answer 1

(a) 120 bacteria
(b) 9.219 hours

For (a), how can you find the initial population when you have two different unknown variables?

Answer 2

you can start counting at any time you like k

Answer 3

you can start at hour 2 and call that hour zero
then to find the initial amount, once you have your model, put \(t=-2\)

Answer 4

Oh, I see! Would I still start from 2 hours and call it zero for (B)?

Answer 5

really makes no difference
"when" is time, so you can say whatever hours after hour 2, or whatever hours plus 2 after hour zero
did you get a model for this?

Answer 6

75000=600xe^8k?

Answer 7

oh you want to solve it that way

Answer 8

that will work, but it is only 6 hours from hour 2 to hour 8, so you should solve
\[75000=600e^{6k}\] for \(k\)

Answer 9

Oh, ok, thanks!

Answer 10

you should get
\[k=\frac{\ln(5)}{2}\] or
\[k=.80472\] making your model
\[P(t)=600e^{.80472t}\]

Answer 11

then if you want the initial amount, put \(t=-2\)

Answer 12

you do in fact get 120 exactly

Answer 13

there is an easier way to do this, but that is ok, one way is good enough