### Word Problems: Exponential Decay

Carbon dating to determine the age of fossil remains Carbon dating is based upon the decay of 14C, a radioactive isotope of carbon with a relatively long half- life ( years). Problem 1- Calculate the amount ofC remaining in a sample. I do not get the − value, but perhaps my answer will help anyway. If we assume Carbon decays continuously, then. C (t) = C 0 e − k t,. where C 0 is the. Carbon 14 is a common form of carbon which decays over time. when the exponential function has the given base of, as in this problem. Solution: 1.

Where the amount of the element that's decaying, that we have at any period in time, is equal to the amount that we started with, times e to the minus kt.

Where the k value is specific to any certain element with a certain half-life, and sometimes they don't even give you the half-life.

- IM Commentary
- Your Answer

So let's try this interesting situation. Let's say that I have an element. Let me just give you a formula.

Let's say that I have some magic element here, where its formula is, its k value I give to you, k is equal to minus, let me think of a-- [coughs] Excuse me, I just had a lot of walnuts and my throat is dry. Let's say that k is equal to, well k, we're putting a minus in front of it, so I'll say the k value is a positive 0.

So its exponential decay formula would be the amount that you start off with, times e to the minus 0. My question to you is, given this, what is the half-life of the compound that we're talking about? What is the half-life? So let's do that.

## More exponential decay examples

So we're starting off with N sub 0 This is just some value, our initial starting point. We could put there. Actually, let's do that, just to keep things less abstract. So let's say we start with I'm just picking out of air. I could have left it abstract with N. Let's say I'm starting with And I take the times e to the minus 0. So this should be equal to We just solved for t.

Divide both sides by You get e to the minus 0. You take the natural log of both sides of this. The natural log of this, the natural log of that. And then you get-- the natural log of e to anything, I've said it before, is just the anything.

So it is minus 0. So let's figure out what that is.

### Carbon 14 Dating - Math Central

Actually, someone just made a comment, and I might as well do that. I could just put this minus up here. I could make this a plus, and this a minus, if I just multiply the numerator and the denominator by negative 1. And if I want to, just to make the calculator math a little easier, if you put a minus in front of a natural log, or any logarithm, that's the same thing as the log of the inverse of 2 over 0.

It makes the calculator math a little bit easier. So if I do 2 natural log, divided by 0. So when t is equal to And I'm assuming that we're dealing with time in years.

### Illustrative Mathematics

That tends to be the convention, although sometimes it could be something else and you'd always have to convert to years. But assuming that this original formula, where they gave this k value 0.

I just solved that after At this point, the overall amount of 14C in the organism begins to decay exponentially. Therefore, by knowing the amount of 14C in fossil remains, you can determine how long ago an organism died by examining the departure of the observed 12C to 14C ratio from the expected ratio for a living organism.

Decay of radioactive isotopes Radioactive isotopes, such as 14C, decay exponentially. The half-life of an isotope is defined as the amount of time it takes for there to be half the initial amount of the radioactive isotope present. Modeling the decay of 14C. Returning to our example of carbon, knowing that the half-life of 14C is years, we can use this to find the constant, k. Thus, we can write: Simplifying this expression by canceling the N0 on both sides of the equation gives.

Solving for the unknown, k, we take the natural logarithm of both sides. Thus, our equation for modeling the decay of 14C is given by.

**Exponential Equations: Half-Life Applications**

Other radioactive isotopes are also used to date fossils. The half-life for 14C is approximately years, therefore the 14C isotope is only useful for dating fossils up to about 50, years old.