Constructing linear models for real-world relationships
- [Voiceover] Karl filled up the tank of his truck with 400 liters of fuel and set out to deliver a shipment of bananas to Alaska. The truck consumed 0.8 liters of fuel or eight-tenths of a liter of fuel for each kilometer driven. Graph the amount of fuel remaining in the truck's tank in liters as a function of distance driven in kilometers. And right over here we have, we have a graph where we have a coordinate plane where our horizontal coordinate is distance in kilometers, and our vertical, our vertical axis is fuel in liters. So we can define the line by moving these two points around, because two points define a line. And so, let's just think about two points that we could figure out. Can we figure out the fuel at two different distances, and then that will help us define the line. Well, the first thing that we might want to think about is, well, what about before we've traveled at all? That might be the easy thing to figure out. What was the amount of fuel in the tank when we haven't traveled at all? And they tell us that in this passage. And I encourage you to pause the video and think about that. Well, they tell us Karl filled up the tank of his truck with 400 liters of fuel and then set out to deliver a shipment of bananas. So before he had driven at all, right after he'd filled his tank, he had 400 liters of fuel. So we could say when distance was zero kilometers, he had 400 liters of fuel. So we have one point on that line. Now we gotta think about where we might want to put, where we want to put this other point. And the way I think about it is, well, let's just, we know he's consuming, he's consuming eight-tenths of a liter of fuel for each kilometer driven. But they don't have, you know, we're not going by one kilometer, two kilometers. They're going by, this is like 50 kilometers, 100 kilometers. So let's think about how much fuel he would have consumed after driving 100 kilometers, and if he consumed that much, we would subtract that from the amount of fuel he started with, and then that would tell us, that would tell us where this point would be. It's going to be some place over here, and it's going to be, it's going to be below 400, 'cause we're consuming fuel. Fuel should be going down as distance increases. This should be a downward-sloping line. So I have my, I have my scratchpad here. Let me, let me get it out. And I have the same question there. It just gives us all the same information. But what I want to figure out is, so we already know, we already know that. So we have distance, distance. Let me, I'll just write. Actually, let me just write the whole thing. Give myself a little bit more space. Distance in kilometers. Distance in kilometers. And then you have fuel, you have fuel in liters. You have fuel in liters. And we already figured out that before he got on, right after he filled up his tank but before he set out on his trip, at distance zero kilometers he had 400 liters of fuel. And we've already actually plotted that. But then we said, well what happens at a hundred kilometers? At a hundred kilometers, how much fuel will he have? Well, they tell us that he consumes 0.8 liters of fuel for each kilometer. So, 0.8 liters per kilometer, and then we just multiply that times the number of kilometers. So, times 100 kilometers. The units work out, kilometers divided by kilometers. We're just going to be left with liters, and then we multiply the numbers. Eight-tenths of a hundred, well that's going to be equal to 80, and the units are liters, 80 liters. So at a distance of a hundred kilometers, he's going to have consumed, right, let me be careful here. He's going to have consumed 80 liters. He's going to have consumed 80 liters. So the fuel, the fuel is actually going to be what he started with, what he started with minus how much he consumed. So it's going to be minus 0.8, and if we want to write the units there. I might as well, so, you know, this is liters right over here. This is 400 liters. And I can write this kilometers, kilometers. It's going to be 400 liters minus 0.8 liters per kilometer times, times, let me make that clear, times 100, times 100 kilometers. And same thing, kilometers divided by kilometers, and we are left with 400 liters minus 0.8 liters times 100. Well, that's just going to be equal to 400 liters, which is how much he started with, minus eight-tenths of a hundred is 80, and the units left is 80 liters. So 400 liters minus 80 liters, that's going to be 320 liters. So when he has traveled a hundred kilometers, he will have 320 liters left in his tank. So let's plot that. So when he has traveled a hundred kilometers, actually, I just randomly had put the point there, he is going to have 320 liters left in his tank. And just like that, we have plotted the line that showed how much fuel he has in his tank as a function of, as a function of distance traveled. And you can even see from this that he's going to run out of fuel at the 500-kilometer mark.