## anonymous one year ago 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?

1. anonymous

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

2. Michele_Laino

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

3. Michele_Laino

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

4. Michele_Laino

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

5. Michele_Laino

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

6. Michele_Laino

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

7. anonymous

oh--thank you!

8. Michele_Laino

:)