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Linear vs. exponential growth: from data (example 2)

Video transcript

the temperature of a glass of warm water after it's put in a freezer is represented by the following table so we have time in minutes and then we have the corresponding temperature at different times in minutes which model for C of T the temperature of the glass of water T minutes after it's served best fits the data so pause the video and see which of these models best fit the data all right now let's work through this let's work through this together so in order for it we see these choices some of these are exponential models some of these are linear models in order for it to be a linear in order for a linear model to be a good description when you have a fixed change in time you should have a fixed change in temperature if you're dealing with an exponential model then as you have a fixed change in time you should be changing by the same factor so the amount you change from say minute one to minute two or for a minute one to two minute three it's not going to be the exact same amount but should be the same factor of where you started so let's think about this so here our change in our change in time is two minutes what is the absolute change in temperature so our absolute change in temperature is negative what fifteen point seven negative fifteen point seven and what if we if we viewed it as a multiplication so what do we multiply 80 by to get sixty four point three well I can get a calculator out for that so sixty four point three divided by eighty is equal to zero point eight I'll just say approximately point eight so we could multiply by point zero point eight this is going to be approximate so to get from eighty to sixty four point three I could either subtract by fifteen point seven if I'm dealing with a linear model or I can multiply by zero point eight now if I increase my time again by two away from minute to two minute for so delta T is equal to two the absolute change here is what this is going to be not 12 this is going to be my brain isn't functioning optimally this was sixty four point seven then this would be twelve but it's for less than that so it's eleven point six negative eleven point six but if you looked at it as multiplying it by a factor what would you have to multiply it by approximately let's get the calculator back out so if I said fifty two point seven divided by sixty four point three divided by sixty four point three is equal to about point eight two so times zero point eight two so just by looking at this I could keep going but it looks like for a given change in time my absolute change in the number is not going is not even close to being the same if this was like fifteen point six I'd be like okay there's a little bit of error here data that you're collecting the real world is never going to be perfect these are models that try to get us close to describing the data but over here we keep multiplying it by a factor of roughly 0.8 roughly 0.8 now you might be tempted to immediately say okay well that means that C of T is going to be equal to our initial temperature 80 times a common ratio of 0.8 to the number of minutes that pass by now this was very tempting at and it would be the case if this was one minute and if this was two minutes but our change in temperature each time is our change in temperature each time is two minutes so what we really should say is this is one way to think about is that it takes it takes two minutes to have a 0.8 change so or to be multiplied by 0.8 so the real way to describe this would be T over to every two minutes when T is zero we'd be at 80 after two minutes we would take 80 times 0.8 which is what we got over here after four minutes it would be 80 times 0.8 squared in fact lecture let's just verify that we feel pretty good about this so if we had something like this so T and C if T so when T is zero C of T is a T when T is well let me just do the same data that we have here when T is 2 we have a T times 2 over 2 is 1 so it's 80 times 0.8 which is pretty close to what we have over here when T is 4 it would be 80 point 0.8 squared which is pretty close to what we have right over here I can just calculate it for you if I have point 8 squared times 80 51.2 getting pretty close this is a pretty good approximation pretty good model so I'm liking I'm liking this model this isn't exactly one of the choices so how do we manipulate this a little bit well we can remind ourselves that this is the same thing as 80 times 0.8 to the one-half and then that to the T power and what's point eight to the one-half so 0.8 that's the same thing saying the square root of point eight it's roughly 0.89 so this is approximately 80 times zero point eight nine to the T power and if you look at all of these choices this one is pretty close to this this list model best fits the data especially of the choice is going this is pretty close to the model that I just thought about now another way of doing it there might have been a little bit simpler I like to do it this way because even if I didn't have choices we would have gotten to something reasonable another way to do it is say okay 80 is our initial State all of these whether you're talking about exponential or linear model start with a T when T is equal to zero but it's clearly not a linear model because we're not changing by even roughly the same amount every time but it looks like every two minutes we're changing by a factor of point eight so we're going to have an exponential model so you say okay it be one of these two choices now this one down here you could rule out because we're not changing by a factor of zero point eight or zero point eight one every minute we're changing by a factor of zero point eight one every two minutes so you could have ruled that one out and then you could have deduced to this right over here and you can set look if I'm changing by a factor of 0.9 and minute then that would be 0.81 every two minutes which is pretty close to what we're seeing here changing by a factor of about 0.8 or 0.8 one every two minutes so once again that's why we like that first choice