00:01
Hi there, so for this problem we are told that to suppose that a sample of radiative element decades from 10 grams to 2 grams in 2 hours.
00:22
Now we are also told now that radiative decay is always exponential decay.
00:30
Now for part a of this problem, we need to give a formula for the amount of the sample that remains a after the hours.
00:41
So in this case, the values that we are given are the initial values.
00:47
We know that the amount at 10 and sorry, the initial amount at zero is equal to 10 grams and the amount after two hours is equal to two grams.
01:08
Now, and we are, the problem states that radiate decay is always exponentially decay.
01:15
So the amount of the element is equal to the initial amount times the exponential of minus a constant k that we need to determine times the time.
01:28
Now with the values that we are given, we need to determine the initial amount and the constant k.
01:36
So first we use the initial value at zero.
01:40
So we know that at zero, a time, equals to zero, the amount is 10 grams.
01:48
So we substitute that, so it's going to be minus k times zero.
01:52
So we know that this is this penitial of zero that we know is just simply equal to one.
01:59
So from this we obtain that initial amount is equal to 10 grams.
02:05
Now, to find the value of the constant k, we use the second condition.
02:13
And the second condition states that after two, hours there is two grams we substitute a zero by 10 and we will have the esponimittal of minus k times two so from this if we start solving we will find that this is 1 over 5 what we have done is to pass the 10 to the left side to divide the 2 so we will have 2 over 10 which is the same as 1 over 5 now to get rid of this 2 right here we just simply take the square root to both sides.
02:54
So we will have 1 over 5 elevated to 1 over 2 and this is equal to the spenumatial of minus k.
03:01
That we can already live it like that because we know that we can write this as a 0 times the s penitial of minus k elevated to the time.
03:14
Now we can write this as the following.
03:19
We know that the square root of 1 is just 1 and the square root of 5.
03:24
Is the square root of five...