# Drug dosage

So what can you do with exponential growth and shrinkage models? Turns out, an awful lot. The following pages contain issues from ecology, environment, economics, medicine ... Remember that ANYTIME a population or collection of things changes by a constant PERCENTAGE, that’s exponential. Have fun!

Over the counter pain medications – like ibuprofen (the main ingredient in a leading brand I don’t care to name) usually have dosing instructions on the bottle that say something like “take one or two tablets every 6 hours”. So you take the pills, and pain goes away, but pain starts coming back gradually before the 6 hours are up, increasing until you can take another dose. Turns out, you’re metabolizing the ibuprofen and the concentration is declining exponentially.

### Ibuprofen metabolizes at approximately 29% each hour. If you start out with a 500mg dose, what will the concentration in your body be after 6 hours?

(To make this problem interactive, turn on javascript!)

- I need a hint: We’re dealing with shrinkage here. What's the multiplier?
- ... another hint ... : The multiplier is 100% - 29% = 71% = 0.71.
- ... another hint ... : The exponent is the number of hours...

#### I think I have the answer: 500×0.71^{6} = 64mg

### What is the concentration in your body after 3˝ hours (when you started to feel pain again)?

(To make this problem interactive, turn on javascript!)

- I need a hint: Still using the same multiplier
- ... another hint ...: What should the exponent be?

#### I think I have the answer: 500×0.71^{3.5} = 151mg

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