A model for the number of bacteria in a culture after t hours is given by P(t)=P_0e^kt. After 3 hours it is observed that 400 bacteria are present. After 10 hours 2000 bacteria are present. What was the initial number of bacteria? please show work.. the answer is in the back of the book which is 201.. but dont know how to get there! thank you!!

Question

anonymous · Answer

@RadEn  any idea? you helped me last time

anonymous · Answer

@TuringTest

anonymous · Answer

@Preetha  can yoj help me?

anonymous · Answer

Start by observing that 
$$P(3) = 400$$
$$P(10) = 2000$$

We can derive a system of equations from these facts to solve for the growth constant k. So,
$$P(0)e ^{3k} = 400$$
$$P(0)e ^{10k} = 2000$$

Solve for P(0) and you'll have
$$P(0) = 400e ^{-3k}$$
$$P(0) = 2000e ^{-10k}$$

Subtracting the second equation from the first and simplifying, we have
$$e ^{-3k}- 5e^{-10k} = 0$$

Using some log rules, we find that 
$$k = 1/7\ln(5)$$
Now, all we have to do is plug in k to one of our given equations to find that
$$P(0) = 400e^{-3(1/7\ln(5))} = 200.6787642... \approx 201$$