Exponential vs. linear growth
Current time:0:00Total duration:3:26
- [Instructor] The table represents the cost of buying a small piece of land in a remote village since the year 1990. Which kind of function best models this relationship? And I'm using, this is an example from the Khan Academy exercises, and we're really trying to pick between whether a linear model or a linear function models this relationship or an exponential model or exponential function will model this relationship. So like always, pause this video, and see if you can figure it out on your own. All right, so now let's think about this together. So as the time goes by around this, the time variable right over here, we see that we keep incrementing it by two. Go from zero to two, two to four, four to six, so on and so forth. It keeps going up by two. So if this is a linear relationship, given that our change in time is constant, our change in cost should increase by a constant amount. Doesn't have to be this constant, but it has to be a constant amount. If we were dealing with an exponential relationship, we would multiply by the same amount for a constant change in time. Let's see what's going on here. Let's just first look at the difference between these numbers. To go from 30 to 36.9, you would have to add 6.9. And to go from 36.9 to 44.1, what do you have to add? You have to add 7.2. And now to go from 44.1 to 51.1, you would have to add seven. Now to go to 51.1 to 57.9, you are adding 6.8, 6.8. And then finally, going from 57.9 to 65.1, let's see, this is almost eight, 7.1, this is what, 7.2 we're adding, plus 7.2. So you might say, "Hey, wait, "we're not adding the exact same amount every time." But remember, this is intended to be data that you might actually get from a real-world situation. And the data that you get from a real-world situation will never be exactly a linear model or exactly an exponential model. But every time we add two years, it does look like we're getting pretty close to adding 7,000 dollars in cost. 6.9 is pretty close to seven. That's pretty close to seven. That is seven. This is pretty close to seven. That's pretty close to seven. So this is looking like a linear model to me. You could test whether it's an exponential model. You see, well, what factor am I multiplying each time? But that doesn't seem to be as, this doesn't seem to be growing exponentially. It doesn't seem like we're multiplying by the same factor every time. It seems like we're multiplying by a slightly lower factor, as we get to higher cost. So the linear model seems to be a pretty good thing. If I see every time I increase by two years, I'm increasing cost by 6.9 or 7.2 or seven. It's pretty close to seven. So it's not exactly the cost, but the model predicts it pretty well. If you were to plot these on a, on a coordinate plane and try to connect the dots, it would look pretty close to a line, or you could draw a line that gets pretty close to those dots.