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# Linear vs. exponential growth: from data

CCSS.Math:

## Video transcript

the number of branches of an oak tree and a birch tree since 1950 are represented by the following tables so for the oak tree we see when time equals zero has 34 branches after three years it has 46 branches so on and so forth and the birch tree they give a similar data at the beginning has eight branches in ten years has 33 branches and they give us all of that and what I want to think about in this video is how should we model these if we want to model these with functions and the choices we'll give ourselves there are other options but the choices we'll give ourselves in this video are linear and linear versus exponential functions which of these are going to be better for modeling for modeling these this this data so let's first look at the oak tree and the key to realization is whenever have a fixed increase in time so each of these steps this is plus three years so it's a fixed increase in time what happens to my number of branches this is going to be a fixed change in which or roughly a fixed change in which case a linear model might be good or is it going to be a change that's dependent on where we were so what am I talking about so 34 to 46 that is plus 12 46 to 59 is plus 13 59 270 is plus 11 70 to 80 2 is plus 12 and so at first you might say well hey this this isn't an exactly fixed change these numbers they seem to average right around 12 but when you're looking at real-world data you're never going to get something that is exact the models are just going to give us a good fit are going to give us a good approximation of the behavior of the number of branches over time for me this is pretty close to a constant 12 branches a year so I would construct a linear model here I would say here branches as a function of time let me be clear this isn't 12 branches per year this is 12 branches every three years 12 this is 13 branches over three years this was 11 branches every three years we're going to average twelve branches over three years so the number of branches we have we're going to start at 34 branches and then - with 12 branches every three years that's four branches every or esterday plus four branches every year and you could test this out B of zero is going to give us thirty four branches B of twelve let's just really test out the extreme part of the model B of twelve is going to be 34 plus 48 which is equal to which is equal to 82 so this model works quite well it's going to have a couple of places we're sad exactly exactly fitting the data but it fits it quite quite well and so this is a linear model so this one is linear now let's look at the birch tree so time equals zero so fix change in time let me the wrong layer alright so we have a fixed change in time every time we are moving into the future a decade let's see our change in branches we go from 8 to 33 so what is that that is plus 25 branches then we go from 33 to 128 well that's way more than 25 branches that's going to be what 5 less than 5 less than 100 so that's going to be plus 95 branches so this clearly is not a linear model and so let's think in terms of an exponential model how much you have to multiply to go from I do that right 128 - yeah if it was 133 then it would be 100 it's 5 less than that yep okay so now let's think about it in terms of an exponential model in terms of the next exponential model we care what would we have to multiply for each for each step so if we have a constant step in time what do we multiply for are how much we increase our branches so to go to 8 - 33 that'll be approximately it's going to be approximately 4 it's going to be a little bit more than 4 33 to 128 well that's going to be a little bit less than 4 but it's approximately 4 33 times four be 132 so we're close 128 to 512 that's exactly four right that's exactly 120 times four is four eighty plus thirty-two yep that is exactly 4 so times four and so it looks like we keep we keep multiplying by four every ten years that go by so one way one way to think about it is we could say here B of T the branches of T our initial condition our initial state is eight and now we could say our common factor is four take is four but if we want t to be in years well that every 10 years we multiply by a factor of four so T has to go to ten before we increase the exponent to one or has to go to 20 until this exponent becomes 2 so 8 times 4 to the T over 10 power seems like a pretty good pretty good model and you could even verify this for yourself if you like try out what B of 30 is going to be B of 30 would be 8 times 4 to the 30 divided by 10 to the third power and what is that going to be that's going to be 4 to the 3rd is 64 64 8 times 64 is it's 480 plus 32 it is 512 so then once again this exponential model this exponential model for this data it does a pretty good job