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Problem 12 Medium Difficulty

Dinosaur fossils are too old to be reliably dated using
earbon-14. (See Exercise 11.) Suppose we had a 68-million-year-old dinosaur fossil. What fraction of the living dinosaur's $ ^{14}C $ would be remaining today? Suppose the minimum detectable amount is $ 0.1\%. $ What is the maximum age of a fossil that we could dale using $ ^{14}C? $

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DV

Diego V.

April 14, 2019

cant undertand half of what is being said and shown here

Video Transcript

All right, so in this problem, we're talking about carbon dating again. And so we have our exponential decay model. M equals M, not each of the K T. And from problem number 11 we have are carbon dating model in specific with its K value of natural log 110.5 over 5730. And that value of K came from the half life of carbon 14 being 5730 years. So what we want to do is show that this is unreliable as a method for finding the age of the dinosaur. So imagine that the dinosaur fossil is thought to be 68 million years old. So the time is 68 million. What will happen if we substitute that number into this model? So we have m equals m not times E to the natural log of 0.5, over 5730 times 68 million. What you're going to put in the calculator is not the M not but the rest of it. And what happens is you end up with zero now it's not technically zero, but as far as the calculator can handle, it thinks it zero. So that shows us that we can't use carbon dating for something that old. So then the question is, what's the limit on carbon dating? What's the minimal or the minimum detectable if the minimum detectable level is 0.1% than what's the maximum age? So let's let em be 0.1% of M not, And that would be M equals 0.1 times m Not if we substitute that into our model. We can solve for the age. So we get bring it over here. 0.1 am not equals m not times e to the natural log of 0.5 over 57 30 times t Let's divide both sides by m not and we have 0.1 equals e to the natural log of your 0.5 over 5730 times T. Now we're going to take the natural audible sites. So the natural law gives your 0.1 equals and natural log of 0.5 over 5730 times t. Now our goal is to solve for T, so we need to multiply both sides by the reciprocal of this fraction, and we will get T equals 5730 times the natural log of 0.1 over the natural log of 0.5. And when you put that into a calculator, you get approximately 57,104. So that means that the maximum age of a fossil that you could date using carbon 14 is 57,104 years.