Problem

A curve passes through the point $(0,5)$ and has …

02:43

Problem 11 Easy Difficulty

Scientists can determine the age of ancient objects by the
method of radiocarbon dating. The bombardment of the
upper atmosphere by cosmic rays converts nitrogen to a
radioactive isotope of carbon, $^{14} \mathrm{C},$ with a half-life of about
5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates $^{14} \mathrm{C}$ through food
chains. When a plant or animal dies, it stops replacing its
carbon and the amount of 14 $\mathrm{C}$ begins to decrease through
radioactive decay. Therefore the level of radioactivity must
also decay exponentially.
A parchment fragment was discovered that had about
74$\%$ as much $^{14}$ C radioactivity as does plant material on
the earth today. Estimate the age of the parchment.

Answer

2500 years

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Calculus 1 / AB

Essential Calculus Early Transcendentals

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Exponential Growth and Decay

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Video Transcript

Hey, guys, welcome back This problem it is given that the parchment has 1/2 life of 5000 730 years. So let's start figuring out what are constant Caisse. We can use her exponential growth equations, the change in y over the change in T. Siegel to K y. And when you any rate both sides, that becomes why is able to our initial value im e raised at the KT. So let's say Let let why so you could have one initial value. In that case, when the half life occurs, we will say that why of 5730 years, 1/2 life there's half of it left will be 1/2. So plugging that into our equation right here have 1/2 equal to one times E raised to the Katie T is 5730 and R K is unknown. Let's say the natural log of both sides left with natural log of 1/2 is equal to 5000 730 K K's an eagle to the natural log of 0.5, divided by 5000 700 and 30. And when you do that calculation, you get a value of approximately negative 0.121 So now we know that equation looks something like this. You have? Why? To our initial value. Weiss abo times erase the negative. 0.12 It's one to one multiplied by were asked how long was apartment there? If 74% of it remains so basically saying if why Siegel? To 74% of our initial value. What is to you? Well, we know that y zero Why are sorry twice about is one Because we set up here in this box. Answer. So then we're gonna let why? Equal to 0.74 So it was 0.74 Steel toe one e to the negative. 0.121 Bonus offorty. We need the natural log on Both sides were left with Ln point 74 is equal to negative 0.0 zero 0121 T teas Eagle to Ellen 0.74 over negative 0.0 1 to 1. And when you do the math there comes out to write about 2500. So that says that the age of department is right about t. Is it 25 100 years as our answer? Thanks for watching

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