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For the second final exam problem, we're looking at something that set the outside boundaries
of the material that we've looked at so far.
We're going to investigate the decay chain of Uranium-238.
This chain contains a number of different steps from one isotope down to the next.
Say for example we have Uranium-238 to a string of other things
that we're not going to think about right now until eventually we got to Bismuth-210
plus some radioactive radiation over here that we're not going to be concerned about.
Bismuth-210 has a half-life of 5 days and eventually decays to Plutonium-210 plus again some radiation
and this has a half-life of 138 days.
After sometime, it decays to Lead-206 and you can see that in general, the average lifetime
of the given isotope is equal to its half-life divided by the natural log of 2.
If you want to figure this out, you're welcome to or you can just trust me that it's right.
It'll be useful for us to think about how the number of atoms at any given sample changes every time.
If you think that the initial amount of atoms is equal 100% of that amount
and then t equals the half-life, we have half of those atoms left.
Over here, well we see the atoms lifetime worked out.
Of course, there are still some atoms left, but many of them are already decayed.
Looking at the supplied code, we can see that they are actually ignoring the part of the chain
that starts with Uranium-238.
In the beginning, I set it with the Bismuth isotope.
All initial values showed up all of the atoms are Bismuth at first
and later the Plutonium and then the Lead.
What we would like you to do is to the backward Euler--
remember backward Euler is not the same as forward Euler--to show how this
decay process happens for each population of atoms.
Now more hint, think about how the ideal logistic growth applies to the situation
and how it's going to affect the different rates of change of these populations.