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# Applications of Exponential Functions

## Word problems with variables as exponents

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Medication Dose

Credit: e-Magine Art
Source: http://www.flickr.com/photos/50382026@N06/4742089272

Have you ever wondered why your little brother or sister was prescribed a lower dose of medicine than you were, even though you both had the same illness? It all has to do with how quickly the medication is absorbed by the body. And the mathematics behind that absorption is an example of exponential decay.

#### It’s a Weight-Related Thing

When you take a medication, whether in pill or liquid form, it enters into your bloodstream. As it is absorbed by the body, it gets to work to help you feel better. With the passage of time, however, as your body processes it, the medicine begins to wear off. How quickly it wears off is often based on your weight.

If we were to plot the medication levels in your sibling′s bloodstream over time, the curve representing that level would start off very high. It would then gradually dip lower and lower until it approached zero.

Credit: Laura Guerin
Source: CK-12 Foundation

Now see what happens when we plot the levels of the same dose of medication in you, someone who weighs 40 pounds more than your sibling.

Credit: Laura Guerin
Source: CK-12 Foundation

Notice how much more quickly the level drops for you. Within minutes, the same dosage that worked for your younger brother or sister is no longer effective for you. To maintain the right level of medicine to fight off the same illness as your sibling, you need a higher dose to begin with.

#### Explore More

Thinking of running a marathon or swimming the English Channel? After you achieve your feat, you must reduce your heavy training in a systematic way to ensure a steady recovery, a practice known as "tapering." Read the following research article about the effects of exponential-decay tapering on athlete performance:

What effects did exponentially decreasing training volume have on the performance of the runners and the swimmers mentioned in the article?

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