Question

Soon after taking an aspirin, a patient has absorbed 300 mg of the drug. After 2 hours, only 75 mg remain. Find an exponential model for the amount of aspirin in the bloodstream after t hours.

Use your model to find the amount of aspirin in the bloodstream after 4 hours.

Answer 1

[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

`(75)/(300) = e^(2r) \rightarrow \ln\left( (75)/(300) \right) = 2r \rightarrow r = \frac12 \ln\left( (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: