1. ## Population

how would someone work this out???

In a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 4 hours. After 12 hours, it is estimated that there are 1 million bacteria in the culture.
a. How many bacteria were present initially
b. What is the doubling time for the bacteria population?

2. Originally Posted by runner07
how would someone work this out???

In a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 4 hours. After 12 hours, it is estimated that there are 1 million bacteria in the culture.
a. How many bacteria were present initially
b. What is the doubling time for the bacteria population?

We use the exponential growth formula here.

$\displaystyle P(t) = P_0e^{rt}$

where $\displaystyle P(t)$ is the population at time $\displaystyle t$, $\displaystyle P_0$ is the initial population, $\displaystyle r$ is the rate of growth, and $\displaystyle t$ is time.

(a)
we are told that after 4 hours, the population triples. so if we start with $\displaystyle P_0$ at $\displaystyle t=0$, we will have $\displaystyle 3P_0$ when $\displaystyle t=4$. So let's plug those values into our equation.

$\displaystyle 3P_0 = P_0e^{4r}$

$\displaystyle \Rightarrow 3 = e^{4r}$

$\displaystyle \Rightarrow r = \frac {\ln 3}{4}$

So we have, $\displaystyle P(t) = P_0e^{\frac {\ln 3}{4}t}$

we are also told that when $\displaystyle t=12$, the population is 1000000

so we have:

$\displaystyle 1000000 = P_0e^{\frac {\ln 3}{4}(12)}$

now solve for $\displaystyle P_0$

once you find $\displaystyle P_0$, you will have the equation that models this growth, part (b) won't be difficult after that. So find $\displaystyle P_0$ and try part (b) and we'll see how far you get

EDIT: This is my 33th post!!!