# A bacteria culture starts with 900 bacteria; the population doubles every 4 hours. Find an...

## Question:

A bacteria culture starts with 900 bacteria; the population doubles every 4 hours. Find an expression for the number of bacteria after t hours.

## Doubling Time and Exponential Growth

A function that describes exponential growth can be written in a few different ways. This is because we can use different bases for an exponential function. If we know how much time it takes for a quantity to double, we can use a base of 2 to write a function modeling this quantity as follows.

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

We can use an exponential model with a base of 2 to express this population. The initial quantity of bacteria at a time of zero is 900, so this is the value of the coefficient outside of the function. Since the population doubles every 4 hours, we will divide our variable by 4 inside the exponent. This yields the following function that calculates the population at any time, based on the formula for exponential growth using a base of 2.

{eq}P(t) = 900 \cdot 2^{\frac{t}{4}} {/eq}

We could also express this as a function with a base of e, but we would need to know the value of our growth constant. This requires a calculation, putting this model at a disadvantage over the one we chose. However, if we ran this calculation, we would find that the growth constant would equal {eq}k = \frac{\ln 2}{4} \approx 0.17329 {/eq}, making our model {eq}P(t) = 900e^{0.17329t} {/eq}.