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The answers to most of the following exercises are in terms of logarithms and exponentials. A calculator can be helpful, enabling you to express the answers in decimal form.The mean life of a radioactive nucleus Physicists using the radioactivity equation $y=y_{0} e^{-k t}$ call the number $1 / k$ the mean life of a radioactive nucleus. The mean life of a radon nucleus is about $1 / 0.18=5.6$ days. The mean life of a carbon- 14 nucleus is more than 8000 years. Show that $95 \%$ of the radioactive nuclei originally present in a sample will disintegrate within three mean lifetimes, i.e., by time $t=3 / k .$ Thus, the mean life of a nucleus gives a quick way to estimate how long the radioactivity of a sample will last.

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Calculus 1 / AB

Calculus 2 / BC

Chapter 7

Integrals and Transcendental Functions

Section 2

Exponential Change and Separable Differential Equations

Functions

Trig Integrals

Campbell University

Baylor University

University of Michigan - Ann Arbor

Boston College

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In mathematics, integratio…

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So we're told about this mean lifetime being TZ would've won over K for radioactive isotopes and what we want to show is that when t is equal to three over K, which is three mean lifetimes, that 95% of the original nucleus will have disintegrated. So let's go ahead and see what we can do for this. Well, let's say we weren't told that it was equal to tease even 23 K. And let's just solve for so at least 90% disintegrate disintegrated will mean that we should have 0.5% left. So we have 0.5 Why not is equal to why not e to the negative k t now we were divided side by Why not those, counselor? Then take the natural log. So we have natural log of 0.5 equal to negative K t. Now we would go ahead and divide each side by negative K would get t is equal to the natural log of that. I'm gonna rewrite um 0.5 So first, this is 1/20. So that means we can write that as negative natural log of 20 and then it would be over. Negative case of the negatives. Cancel. And we're gonna have that. Tea is equal to the natural log of 20. Over. Okay, so now let's see. What is this about? So this is approximately, well natural. August 20 is about 2.99 over K, which is less than three over K. So it does seem to be that if we have a time that's larger than it, then we should have that 95% actually disintegrated.

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