Carbon dating using exponential growth
At any given time, the real-world population contains a whole number of bacteria, although the model takes on noninteger values.
When using exponential growth models, we must always be careful to interpret the function values in the context of the phenomenon we are modeling.
And that's useful, but what if I care about how much carbon I have after 1/2 a year, or after 1/2 a half life, or after three billion years, or after 10 minutes? A general function, as a function of time, that tells me the number, or the amount, of my decaying substance I have.
So that's what we're going to do in this video.
After all, the more bacteria there are to reproduce, the faster the population grows.
[link] and [link] represent the growth of a population of bacteria with an initial population of Note that we are using a continuous function to model what is inherently discrete behavior.
dt as an infinitesimally small time, but let's say it's a change in time. And let's say over one second, you observe that this sample had, I don't know, let's say you saw 1000 carbon particles.
We say that such systems exhibit exponential decay, rather than exponential growth.
The model is nearly the same, except there is a negative sign in the exponent.
Thus, for some positive constant The following figure shows a graph of a representative exponential decay function.
So let's just think a little bit about the rate of change, or the probability, or the number particles that are changing at any given time.
So if we say, the difference or change in our number of particles, or the amount of particles, in any very small period of time, what's this going to be dependent on?