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# Dinosaur fossils are often dated by using an element other than carbon, such as potassium-40, that has a longer half-life (in this case, approximately I.25 billion years). Suppose the minimum detectable amount is $0.1\%$ and a dinosaur is dated with $^{40}K$ to be 68 million years old. Is such a dating possible? In other words, what is the maximum age of a fossil that we could date using $^{40}K?$

## 12,457 million, or 12.457 billion years.

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In the previous couple of problems, we talked about carbon 14 dating and in this problem, we're talking about potassium 40 dating and potassium 40 has a much longer half life, 1.25 billion years. We're going to use our exponential decay model and we're going to find the value of K. And then that will complete our model. And we can figure out whether or not we could use this particular kind of dating to date a dinosaur fossil. Okay, so if the half life is 1.25 billion years, then we can put 1/2 of em Not. And for the final amount I m not, is the initial amount and the K value we don't know, but the time is 1.25 billion. So we're substituting all of those numbers into our function, and now we're gonna solve that for K. So we start by dividing both sides by I am not and we get 1/2 which alright is 0.5 equals e to the 1.25 billion times K. Now we'll take the natural log of both sides and we have the natural log of 0.5 equals 1.25 billion times K and will divide both sides by 1.25 billion. And we get Kay is the natural log of 0.5 divided by 1.25 billion. So our model then is going to be m equals m not times e to the natural log of 0.5 over 1.25 billion times T. Okay, so let's use that model and figure out how long we would have if the minimum detectable level is 0.1% so 0.1% as a decimal is 0.1 So that means our final amount is going to be 0.1 times I'm not. We'll substitute that into our equation and will solve her time. So let's divide both sides by I am not. And we get 0.1 equals e to the natural log of point 5/1 0.0.25 1,000,000,000 times T. Now we're going to take the natural log of both sides. Natural log of 0.1 equals and natural log of 0.5 over 1.25 billion times. T now to get t by itself. We want to multiply both sides of the equation by the reciprocal of this fraction, and we end up with T equals 1.25 billion times of natural log of 0.1 over the natural log a 0.5 and we put that into a calculator and we end up with approximately 1.25 times 10 to the 10th power, and that translates into 12.5 billion. So what that saying is that we could date something that was as old as 12.5 billion years old using this technique of using potassium 40. So could you date the 68 million year old dinosaur fossil? Yes, you could.

Oregon State University

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