MathBench > Environmental Science

What's in your Watershed?

Building the growth equation

As we just saw, bacteria multiply by dividing, so to speak. A single bacterium grows and lengthens, and then divides into 2. Those two "daughter" cells eventually divide in 2. And so on.

Starting with one cell dividing in half, the resulting microbe populationswould look like this:

# in gen0 --> 1

# in gen1 --> 1*2

# in gen2 --> 1*2*2

# in gen3 --> 1*2*2*2

# in gen4 --> 1*2*2*2*2

Instead of writing "# in gen 4", it is more usual to write N(4), which you can read as "the number in generation 4". Likewise, for the general equation we write N(t), or "the number in generation t". So, given the formulas above, which of the following is the correct equation for exponential growth starting from a single cell?

N(t) = t * 2t N(t) = 1 * 2t N(t) = 1 * 2t N(t) = 1 * t2

With a few more changes, we can make this look more official.

  1. Most importantly, we may not be so lucky as to start with one single, solitary cell. The culture could start with 2, or 7, or 10,000. Instead of the initial population being 1, let's call it N0, pronounced "n sub zero", meaning "the number of cells in generation 0".

  2. Less importantly, I removed the multiplication sign - it's still there, just invisible.

Our final equation looks like this: