\( \def\dfrac#1#2{\displaystyle\frac{#1}{#2}} \def\solve#1{\begin{array}{rcl}#1\end{array} } \)

Home / 09 Exponential Functions / 11 Solving Applied Exponential Question With Graphing Device

Example: When studying outbreaks, it is not always obvious when the first day of infection occurs in a population. A researcher in the field observed that when they first arrived, there were 40 people infected in the population. Two days later, that number jumped to 120. Using the model \(I=Pe^{{rt}}\), determine the relative rate of infection as well as an approximation for the number of days since the first infection.

Solution

Treat the data as points to simplify the problem: we have two points, \((0,40)\) and \((2,120)\) and we want to fit these points to the model \(I=Pe^{rt}\). Since this is an exponential model, we actually already know \(P=40\) because the coefficient of the model is always the same as the vertical intercept. To determine the relative rate, we set up the equation: \[ 120 = 40e^{2r} \] While we can use logarithms to solve this, for now, we will rely on a calculator instead: Thus, we can see that the rate is approximately \(r=\approx 54.9\%\). Using the model, we can then back track to see how many days ago the infection started: Here we can see that the infection started about 7 days before the initial observation of 40 infected people.