The half life of carbon 14 is 5600 years if a piece of charcoal

Last updated: 7/8/2022

The half-life of carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 67% of the carbon-14 expected in living matter, when did the tree die? The tree died about years ago.

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Video Transcript

For this problem, we are given the half life of carbon 14. Now the half life of anything is how long it takes for that substance to become half of what it was. And it seems like for carbon 14, it takes 5600 years now. Since we were given that information. And we were asked to use that information to determine how old was a tree when it died. We are working with an exponential decay problem. And so what I have here up on the screen is the general form for an equation that deals with exponential decay. So I have the two A's over here. A sub zero or a knot. Is the initial amount. A by itself is a future amount at a certain time. T and R. Is the rate at which the substance is decaying in this case. So really the rate is what is unique to a particular substance and that's what we need first. We need to find the rate here. Okay, so this is the crucial thing we need to find first and we'll be finding it. Using the information from the first sentence about the half life. So let's get started doing that. When again we have a half life, it means that the time it takes for the substance to become half of what it was. So if it took Um if we initially had a sub zero of carbon 14, so we don't actually know how much initially we have, but let's say it's just a subzero then half of that would be half a sub zero, simple as that. We can just say it's half of what it was. And then we'll do E to the R. R. Again is what we want to find. And then T will be that half life information. So 5600 years. Okay, So what we're going to do is we're going to solve for our first. Now, how do we go about doing that? First thing I want to do is realize that I have a sub zero on both sides of the equation and in both sides we have this multiplication going on so I can essentially divide both sides by a sub zero, thus canceling it out on both sides. So I'm left with one half is equal to E. To the this power which I'm just going to reorder it. It's a little more standard E to the 5600 r. Now to get to this are, what we're going to do is we're going to introduce a natural log to both sides of the equation. So, natural log of 1/2 is equal to natural log of E to this power. Now, what's really nice about having a natural log now is that if we have a power within the natural log, such as we do right here. One of the natural one of the log rules really tells us that we can take this power, bring it outside and make it more of a multiplier. So what we're going to end up with is 5600 R. Times the natural log of E. And the left hand side of the equation stays the same now the natural log of E and this will be the reason why we chose the natural log. The natural log of e ends up being just one because natural log has a base E. So a few times anything by one, it doesn't really change it. So since we can ignore that part And all we're left with is 5600 times r is equal to this natural log of one half. So to get the are by itself, Let's divide both sides by this 5600. And that will be our our now this would be considered maybe more exact are more exact answer. I'm going to just quickly throw this into my calculator and then see what it is approximately in decimal form. So in decimal form our would be approximately negative 0.000123776. And it goes on forever more. But that's how many places that my calculator shows me. So this is a good sign that our rate is negative because negative tells us we're dealing with a d K. The mountains growing smaller over time. So that's good. Okay, so now that we have our we can actually use that information to answer the question of this problem, which it's telling us about a piece of charcoal Um which only shows 66 of what was expected in the living matter. So the question is when did the tree die? Okay so let us actually work on the question itself, We have 66% of what was expected. And the question is when did the tree die? Okay so again seeing similar situation where we didn't know how much of the substance, how much of the I guess plant matter. Did we start with how much of the wood we started with? So start with a sub zero. But we know that we have 66% of it. So 66%. We'll just write that as a decimal 66% of what we started what was expected Can be written this way so .66 times 80. We'll have that E from before just kind of throwing everything back into this form. We now have our our our rate. So that was I'm just going to use the exact one because I am a little scared about round off errors. So I like using exact answers if I can. And then what we're looking for is we're looking for T. So right now we are solv solving for T. Which is different from what we did before because last time we're solving for R. And we knew T. This time it's the opposite now we don't know what T. Is and we're trying to figure that out. Okay so very similar fashion though of solving I can divide both sides by 8:00 thus getting rid of them. I'll have this 0.66 is equal to E. To this power. And then we can go ahead and take the natural log of both sides, just like before we can take the power, even though it's getting a little uglier looking, but we can take this power, bring it outside, make it a multiplier. And then we'll have times natural log E. And then the left hand side didn't change. And again, just like before natural log of E. Is just one. So I don't have to worry about it. Okay, so now we have this we have natural log of .66 is equal to this fraction times t. So how do we get T by itself? Well, if you have a fraction in front of the variable, you're trying to isolate or solve for what you can do is you can multiply both sides of the equation by its reciprocal or that fraction basically upside down. So we can times both sides by 5600 over natural log of 0.5, you know already on this side as well. So we can see really why that will be helpful to us. So if you notice that we have all of this going on here on the right hand side of the equation um essentially you can say either lots of things canceled Or we end up with just one T. Which is just t okay, so that's how you can think about that. So we have we do have this, that is again an exact answer, which for the sake of this problem doesn't really make as much sense. So it would be a good idea to try to approximate this. Um So plug this into your calculator, which I'm doing right now into my calculator and then I'm getting a answer of tea being about 3300 about I'm going to round it to the nearest whole number, about 3357. And since the time that we were using earlier was in years, this will be in years as well. So we can say that this tree died about 3,357 years ago, um which is quite amazing if you think about it.