[1] An exponential model forces us to use the following formula:
\(f(t) = Ae^{rt}\)
where A is a scaling variable, r is the growth rate, and t is time.
[2] Let's say that "soon" means t is basically 0. We know that the patient has 300 mg of the drug. In our formula, we must have
\(300 = Ae^{r\cdot 0} = A e^0 = A\)
Hey! We just found that A = 300. Cool.
[3] After 2 hours, we are told that only 75 mg is left. In our formula we must have
\(75 = A e^{r \cdot 2} = (300) e^{2r}\)
notice that since we know that A = 300 we have plugged that in. We can solve for the unknown r
\(\frac{75}{300} = e^{2r} \rightarrow \ln\left( \frac{75}{300} \right) = 2r \rightarrow r = \frac12 \ln\left( \frac{75}{300} \right) \approx -0.693\)
So, now we know that r = -0.693.
[4] Our finished model looks like this
\(f(t) = 300 e^{-0.693 t}\)
Congrats! You've just built a formula!
[5] To find the amount of aspirin after 4 hours, we use our newly created formula:
