Constructing exponential models: half life

we're told carbon-14 is an element which loses exactly half of its mass every 5730 years the mass of a sample of carbon-14 can be modeled by a function M which depends on its age T in years we measure that we measure that the initial mass of a sample of carbon 14 is 741 grams write a function that models the mass of the carbon-14 sample remaining T years since the initial measurement all right so like always pause the video and see if you can come up with this function M that is going to be a function of T the the years since the initial measurement I let's work through it together and what I like to do is I always like to start up with a little bit of a table to get a sense of things so let's think about T how much time how many years have passed since the initial measurement and what the the amount of mass we're going to have well we know that the initial we know that the initial mass of a sample of carbon-14 is seven hundred and forty-one grams so T equals zero our mass is 741 now what's another interesting T that we could think about well we know at every 5730 years we lose exactly half of our mass of carbon-14 every five thousand seven hundred and thirty years so let's think about what happens when T is five thousand seven hundred and thirty well we're going to lose half of our mass so we're going to multiply this times one half so this is going to be seven hundred and forty one times one half I'm not even going to calculate what that is right now and then let's say we have another 5730 years take place so that's going to be and I'm just going to write 2 times 5 thousand seven hundred thirty I could calculate it was going to be ten thousand eleven thousand four hundred and sixty or something like that all right but let's just go at two times five thousand is it ten yeah ten thousand plus one thousand four hundred so eleven thousand four hundred plus sixty yeah so eleven thousand four hundred sixty but let's just leave it like this well then it's going to be this times one half so it's going to be seven hundred forty one times one-half times one-half so we're going to multiply by one half again and so this is the same thing as 741 times 1/2 squared and then let's just think about if we if we wait another 5730 years so three times five thousand seven hundred and thirty well then it's going to be 1/2 times this so it's going to be seven hundred and forty one this times one half is going to be 1/2 to the third power so you might notice a little bit of a pattern here however many half-life's we have we're going to multiply we're going to raise 1/2 to that power and then multiply it times our initial our initial mass this is one half life is gone by two half life so we have an exponent of two three half-lives we multiply by three sorry we multiply by one half three times so it's going to be a general way to express em of T well M of T is going to be our initial value 741 times and you might already be identifying this as an exponential function we're going to multiply times this number which we could call our common ratio as many half-lives if passed by so how do we know how many half-lives if passed by well we could take T and we can divide it by the half-life and try to try the test this at when T equals 0 is going to be 1/2 to the 0 power which is just 1 and we're just gonna have 741 when T is equal to 5730 this exponent is going to be 1 which we want it to be we're going to multiply our initial value by 1/2 once when this exponent is 2 times 5,000 set when T is 2 times 5730 well then the exponent is going to be 2 and we're going to multiply by 1/2 twice it's going to be 1/2 to the second power and it's going to work for everything in between when we are a fraction of a half-life along we're going to get a we're going to get a non integer exponent and that two will work out and so this is our function we are we are done we have we have written our function M that models the mass of carbon-14 remaining T years since the initial measurement