00:01
We're asked to find how old a carbon -14 sample with 2 % of the initial amount remaining is, given that the half -life of carbon 14 is 5 ,600 years.
00:11
In a lot of these kinds of problems, we write a differential equation and then find a general solution.
00:18
Then we work out some constants, and then once we have a function, we can answer the specific question.
00:23
For this one, we're going to let y represent the amount of carbon 14, and t be the time.
00:33
That's elapsed in years since we're given the half -life in years.
00:37
So let's start with our differential equation.
00:42
For a radioactive decay problem, we know that d -y -d -t, the rate at which the amount of carbon -14 is changing, is proportional with some constant of proportionality k, to how much carbon -14 is present.
01:03
Now that we have this first -order linear differential equation, we know that the solution is y of t equals a e to the k t where a is the initial amount since y of zero would be a e to the zero or just a now we can work out some constants to get our specific function first we'll let a equal one since we weren't given an initial amount we can instead let the units of y be a portion of what the whole or the initial amount was.
01:53
You can also just leave a as an unknown constant and it'll cancel later i promise but for simplicity's sake we can let a equal 1 and then our half -life statement can be translated into y of 5600 years equals a half and then we can use that to solve for k.
02:18
Since one -half, half should then equal e to the k times 5600.
02:26
Again, leaving a as one...