Question

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?

Answer 1

Use P(t)=P_0^e(kt) formula

Answer 2

we can set 8:00 am as t=0, so we can write:
\[P\left( 0 \right) = 2000\]

Answer 3

no, at noon, namely t=4 hours, we get:
\[\Large 3100 = 2000 \times {e^{k4}}\]

Answer 4

so we have:
\[\Large {e^{4k}} = \frac{{3100}}{{2000}} = 1.55\]

Answer 5

now, at midnight, namely at t= 16 hours, we have:
\[\Large P\left( {16} \right) = 2000 \times {e^{k16}} = 2000 \times {\left( {{e^{4k}}} \right)^4} = ...\]

Answer 6

hint:
\[\Large \begin{gathered}
P\left( {16} \right) = 2000 \times {e^{k16}} = 2000 \times {\left( {{e^{4k}}} \right)^4} = \hfill \\
\hfill \\
= 2000 \times {\left( {1.55} \right)^4} = ...? \hfill \\
\end{gathered} \]

Answer 7

oh--thank you!

Answer 8

:)