# Webwork 11.5 Hints

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Math 2300 Section 005 – Calculus II – Fall 2010

Webwork 11.5 Hints – Wednesday, November 3, 2010

Half-Life Example

1. 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 half-life 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.