# A common inhabitant of human intestines is the bacterium E. coli. A ccell of this bacterium in a...

## Question:

A common inhabitant of human intestines is the bacterium E. coli. A ccell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes, and its initial population is 80 cells.

Find the number of cells after 4 hours.

## Doubling Time

Populations often grow at an exponential rate. If we know the doubling time of a population, we can model it using an exponential function with a base of 2.

{eq}P(t) = p_0 2^{\frac{t}{d}} {/eq}

With both an initial population and a doubling time, we can construct a function that models the population of this bacteria culture at any time. The initial population of 80 is the coefficient in front of the exponential base of 2, and we divide the variable by the doubling time, 20 minutes, inside of the exponent.

{eq}P(t) = 80 \cdot 2^{\frac{t}{20}} {/eq}

We can calculate the population after 4 hours using this function. However, by using the doubling time of 20 minutes, we have defined this function to be in minutes. Thus, we can't evaluate this function at 4, but instead, we need to convert this to minutes. There are 240 minutes in 4 hours, so let's evaluate this at 240.

{eq}P(240) = 80 \cdot 2^{\frac{240}{20}} = 327680 {/eq}

The population grows to 327,680 cells after 4 hours. 