During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 8:00AM there were 2,000 bacteria present in the culture. At noon, the number of bacteria grew 3,100. How many bacteria will there be at midnight?

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During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 8:00AM there were 2,000 bacteria present in the culture. At noon, the number of bacteria grew 3,100. How many bacteria will there be at midnight?

Mathematics
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Use P(t)=P_0^e(kt) formula
we can set 8:00 am as t=0, so we can write: \[P\left( 0 \right) = 2000\]
no, at noon, namely t=4 hours, we get: \[\Large 3100 = 2000 \times {e^{k4}}\]

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so we have: \[\Large {e^{4k}} = \frac{{3100}}{{2000}} = 1.55\]
now, at midnight, namely at t= 16 hours, we have: \[\Large P\left( {16} \right) = 2000 \times {e^{k16}} = 2000 \times {\left( {{e^{4k}}} \right)^4} = ...\]
hint: \[\Large \begin{gathered} P\left( {16} \right) = 2000 \times {e^{k16}} = 2000 \times {\left( {{e^{4k}}} \right)^4} = \hfill \\ \hfill \\ = 2000 \times {\left( {1.55} \right)^4} = ...? \hfill \\ \end{gathered} \]
oh--thank you!
:)

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