00:01
Hi there.
00:01
So for this problem, we are given that the half -life of sisyon is given, and the half -life is equal to 30 years.
00:12
Remember that the half -life corresponds when the amount of that element is half of the initial value.
00:21
Okay? so suppose that initially, the amount is equal to 100 milligrams.
00:29
So for part a of this problem, the question is to find a mass that remains after t years.
00:38
So first of all, we assume that the amount of this element at any given time is equal to the initial amount times the exponential of a concept of proportionality that we need to determine times the time.
00:50
So now, to obtain the expression for this, we already know that the initial amount is 100.
01:00
So we'll have 100 times the exponential of the concept of proportionality times the time.
01:06
So now we're going to use the condition that at the half time, we will have half of this.
01:12
So then that will be that this is 1 divided by 2 is equal to the exponential of the concept of proportionality.
01:20
And in here we use the half life, which is 30.
01:28
Then from this, what we can do is to solve for the exponential of k.
01:32
So that will be 1 divided by 2, and that elevated to 1 divided by 30.
01:39
And then we can substitute that in here.
01:42
So with that, the amount for any given time is equal to 100 times 1 divided by 2 times the time divided by 30.
01:53
And then, well, that's the solution for...
01:57
Oh, sorry, i forgot that this is 100, not 10.
02:02
So it is 100 times 1 divided by 2, elevated to the times divided by 30, okay? and that's the, yes, that's a solution for part a of this problem.
02:21
Now, for part b, the question is how much of the sample remains after 10 ,000 years? so what we need to do is to evaluate this function that we've came from before at 100 ,000, sorry, 100 years.
02:36
So that will be 100 times 1 divided by 2, and this elevated to 100 divided by 30.
02:44
Then using our calculator, we obtain a value of.
02:58
So the value that we obtained from this is 9 .92 in units of milligrams...