Math 2300 Section 005 – Calculus II – Fall 2010
Webwork 11.5 Hints – Wednesday, November 3, 2010
HalfLife Example

I'll modify an example I've used before. Assume that at time I consume 300mg of caffeine (I believe this is the largest amount legally allowed in a single dose). The amount of caffeine in my system is then a function of time that satisfies some proportionality constant . That is, the amount of caffeine in my system at a time is a solution to the differential equation
We don't know what is, but maybe we know that the halflife of caffeine in my system is 1.5 hours. So, what we want to do is as follows: The differential equation above represents caffeine decay in my system in general. In the specific situation (corresponding to an initial condition) when I digest 300mg at time , there is a specific solution. We need to find that specific solution. Let's do that, and then solve for .
Now, we know that at we have mg. So, we can solve for in our specific solution to the given differential equation. We have and so . We know that at ( is in hours here) we will have half of our original amount. So, we solve for as follows.