# During a certain epidemic, the number of people that are infected at any time increases at a rate...

## Question:

During a certain epidemic, the number of people that are infected at any time increases at a rate proportional to the number of people that are infected at that time. 1,000 people are infected when the epidemic is first discovered, and 1,200 are infected 7 days later. Write an exponential growth model for the epidemic. Let t represent time in days.

## Exponential Growth

When constructing a model for a quantity in biology, we often turn to an exponential model. This is because quantities like population increase more when the value of this quantity is larger. We can construct an exponential model using the following formula.

{eq}p(t) = p_0 e^{kt} {/eq}

We can use the given information in order to construct an exponential growth model for this epidemic. Since we have the initial amount of people who are infected, we have the value of one of the constants in this model. We can use the second set of information to find the value of the growth constant.

{eq}1200 = 1000e^{k(7)}\\ 1.2 = e^{7k}\\ \ln 1.2 = \ln e^{7k}\\ 7k = \ln 1.2\\ k = \frac{1}{7} \ln 1.2 \approx 0.0260459 {/eq}

Therefore, we can model this epidemic using the exponential function: {eq}p(t) = 1000e^{0.026t} {/eq}.