 ###### Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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# Exponential Growth and Decay - Problem 1

Carl Horowitz ###### Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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Archeologists typically use what's called carbon-14 dating to approximate the age of certain things. And how carbon-14 dating works is there is a certain amount of carbon in most elements; in a tree, in a person, whatever it maybe and they know that carbon-14 is a special kind of carbon that decays at a rate that archeologists know. So what they can do is compare the amount that should be in whatever they're looking at into the amount that's left and using a formula which is the exponential decay formula, they can figure out how old something is.

So for this example what we're going to be looking at is a stick in King Tuts tomb. They found a stick that had 71% of it's original carbon-14, so they know how much carbons should be in this stick say it's oak or whatever it maybe and there's lots of Math, so they know that some has decayed over time. Using carbon-14 dating, so basically exponential decay, and this particular rate, we're supposed to figure out how old the tomb is.

So let's a look. So we know our decay formula to be N is equal to N zero, e to the rt and they told us that our rate is a very small negative number and we have 71% of the original amount and we're supposed to find the time, we're supposed to find t.

The trick is that we don't know how much we started with, so we can't plug in a number, so we're still left with N sub 0, we're left with e to the -.00012t, because we don't know how much we started with, we also don't know how much we ended with, but we do know we have 71% of our original amount.

So this is our entire amount, if I said we had half of that we would just multiply this by a half. We have 71%, so what I can do is just say .71 times our original amount. If you have \$5 and I have 2/3 of the amount of money you have, you take 2/3 times 5. It doesn't really matter that we don't know the exact amount, we're still trying to solve the same exact way.

So now we have a exponential equation, except we have n zero on the same side which is a variable we don't know what it is so all we have to do is divide by that n zero, divide both sides it cancels way all together and we're just left with that 71% equals e to the small negative number t.

So now we're solving for a variable and the exponent, whenever we see that, we need to just take the natural log. We take a natural log because it's the base e, we could take the log, but then we'd be left with a log b, so we take the natural log, this is going to make our base to disappear. Natural log point 71 is equal to negative, it's very small decimal t, natural log of e, natural log of e is just 1, so to solve this out we just divide by this decimal natural log of .71 divided by .00012 is equal to t.

Finish up plug in your calculator, natural log .71 divided by negative .123, 12 and this gives us around 2854 years.

So it's just a little bit of an introduction into carbon-14 dating. Basically it's exponential decay when you know the amount of a substance remaining, you can figure out how long it has been decaying.