00:01
So in this problem, we're working with plutonium 239, yep, pu -239.
00:14
And we're given that the half -life is 24 ,360 years.
00:30
We're first asked to find the equation.
00:32
It says the rule, but it means the equation that models this radioactive.
00:39
Decay, the domain and the range of that function, and then we're asked how long it takes from 10 grams to go down to one gram, to decay down to one gram.
00:54
Okay, so these decay equations, these radioactive decays, follow an exponential decay formula, which is in the amount that you have is equal to n -0, e.
01:16
To the k t so let's define what some of these variables are now in is the amount at time t and not it's going to be the initial amount in other words what we start with t is going to be the number of years and k is a decay factor for the formula so let's plug in what we know.
02:04
Well, first of all, half -life.
02:07
Let's define what half -life means.
02:11
Half -life is the time for half of the original amount, in other words what we started with, to achieve half of it remaining.
02:44
So what does that mean? that means the time for in to be, half of n0 or n over n0 is equal to one half okay so that means i can set this up one half is equal to e to the k times 24 ,360 because that was my t that was the amount of time said for the half life now if we haven't ever taken natural logs or any kind of logarithms we can use this relationship ln of e is 1.
03:34
So take ln, the natural log, of both sides.
03:44
So we have ln of one half is equal to the ln of e to the k times 24 360.
03:54
Well, by logarithm rules, this exponent can move out front.
04:03
So this is equal to k 24 360 ln of e.
04:10
But we just said here, lna is 1.
04:14
So this is 1.
04:18
So now i can isolate k, because all i have it is multiplied times 24 -360, correct? because 1 times anything is itself.
04:28
So this means that k then is the ln of 1 -half over 24 ,360, which when you do that in your calculator, you get negative 0 .000 -400 -400 -400 ,000, 28, 4 -0 .28 .4 .4 .2 .8 .4.
04:46
454.
04:49
All right.
04:51
So that means i have this rule or this relationship...