# Let P(t) be the population of a colony of fruit flies, where the time t is measured in days. The...

## Question:

Let {eq}P(t) {/eq} be the population of a colony of fruit flies, where the time {eq}t {/eq} is measured in days. The initial size of the population doubles in 4 days. Assume exponential growth. What is the size of the population after 12 days?

## Doubling Time and Exponential Growth

If a quantity doubles in a set amount of time, regardless of how much of the quantity originally exists, then we can model this growth with an exponential function. This function has the following form.

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

Let's assume that the initial size of this population is some constant {eq}p_0 {/eq}. Since we have the doubling time of this population, we can find the population at any time by constructing the function below.

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

Therefore, we can find the size of the population after 12 days as follows.

{eq}\begin{align*} P(12) &= p_0 \cdot 2^{\frac{12}{4}}\\ &= p_0 \cdot 2^3\\ &= 8p_0 \end{align*} {/eq}

Therefore, the population has increased by a factor of 8 after 12 days. 