Linear vs. exponential growth: from data (example 2) | High School Math | Khan Academy

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. 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 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 from minute two to minute three, it's not going to be the exact same amount, but it should be the same factor of where you started. So let's think about this. Here, our change in time is two minutes. What is the absolute change in temperature? So our absolute change in temperature is negative, what, 15.7?

And what if we viewed it as a multiplication? So what do we multiply 80 by to get 64.3? Well, I can get a calculator out for that. So 64.3 divided by 80 is equal to approximately 0.8. So we could multiply by 0.8. This is going to be approximate. So to get from 80 to 64.3, I could either subtract by 15.7 if I'm dealing with a linear model, or I can multiply by 0.8.

Now, if I increase my time again by 2, I'm going from minute 2 to minute 4. So delta T is equal to 2. The absolute change here is what? This is going to be, not 12—this is going to be... my brain isn't functioning optimally. If this was 64.7, then this would be 12, but it's 4 less than that, so it's negative 11.6.

But if you look 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 52.7 divided by 64.3, divided by 64.3 is equal to about 0.82, so times 0.82. 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—it's not even close to being the same.

If this was like 15.6, 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 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 passed by. Now, this was very tempting, 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 two minutes. So what we really should say is this: one way to think about it is it takes two minutes to have a 0.8 change, or to be multiplied by 0.8. So the real way to describe this would be T over 2. Every two minutes when T is zero, we'd be at eighty. 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, actually, let's just verify that. We feel pretty good about this. So if we had something like this, so T and C of T, when T is 0, C of T is 80. When T is... well, let me just do the same data that we have here.

When T is 2, we have 80 times 2 over 2, which 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 times 0.8 squared, which is pretty close to what we have right over here. I can just calculate it for you. If I have 0.8 squared times 80, I get 51.2—getting pretty close. This is a pretty good approximation, a pretty good model.

So 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 0.8 to the one-half? So 0.8 is the same thing as the square root of 0.8—it's roughly 0.89.

So this is approximately 80 times 0.89 to the T power. And if you look at all of these choices, this one is pretty close to this. This model best fits the data, especially of the choices. This is pretty close to the model that I just thought about.

Now, another way of doing it, it 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 to say, okay, 80 is our initial state. All of these, whether you're talking about exponential or linear models, start with 80 when T is equal to 0.

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 0.8. So we're going to have an exponential model. So you say, okay, it'd be one of these two choices. Now this one down here you could rule out because we're not changing by a factor of 0.8 or 0.81 every minute.

We're changing by a factor of 0.81 every two minutes, so you could have ruled that one out. Then you could have deduced this right over here. And you could say, look, if I'm changing by a factor of 0.9 every minute, then that would be 0.81 every 2 minutes, which is pretty close to what we're seeing here—changing by a factor of about 0.8 or 0.81 every two minutes.

So once again, that's why we like that first choice.