00:01
So say we had an element with a half -life of 5 .24 years.
00:13
And say we have an initial amount of this element, so the initial amount, we'll say that that is 100 milligrams.
00:29
And say we want to know the amount left after 20, years so let that be question one here what's the amount left after 20 years and then another question we might want to answer is how long will it take for one milligram to remain when will there be one milligram remaining so this is an exponential decay model which has the form y equals y sub zero e to the decay t where y is the amount left, y sub zero is the initial amount, and then k is the decay constant, and t is the time in years.
01:31
So we know that half -life is 5 .24, we can use that to find the decay constant k.
01:37
So that's our first step.
01:38
We need to find the value of the k.
01:40
So we're going to substitute some values in here.
01:43
We know that one half of the end.
01:47
Initial value will be left whenever the time is 5 .24 years.
01:57
So we just substituted 5 .24 in for time, and we know that the final amounts, which is y here, will be one -half of the initial amount whenever that half -life passes.
02:08
So we just need to solve this for k.
02:10
Now first of all, we can divide both sides by y -sub -0 and that cancels.
02:14
We're just left with one -half equals e to the 5 .24.
02:20
So to solve for k, we can take a natural logarithm of each side, gives a natural logarithm of 1 half equals the natural logarithm of e to the 5 .24k...