Area density

- [Narrator] In this video, we're going to talk about density in the context of area. And the simplest way of thinking about it is density is going to be some quantity per unit area. So for example, let's say that I have a football field right over here and I have another identical football field right over here. Now they have the same area, but if I have let's say five people on this football field actually six people on this football field. And I only have three people on this football field the density of people per average unit area or the density of people I should say per football field is going to be higher in this left example. So it's always going to be quantity per area. Now, with that out of the way, let's do a worked example that helps us understand this idea a little bit better. So here we're told the town of Tigersville has a population density of 13 cats per square kilometer. So they're giving us the density let me write that 13 cats. So the quantity is quantity and cats per square kilometer. That's the density right over there. The town is shaped like a perfect isosceles trapezoid. So it looks something like this say perfect isosceles trapezoid. It's gonna look something like that, with two parallel boundaries, 12 kilometers apart. So this distance right over here is 12 kilometers one measuring eight kilometers. So this side over here is eight kilometers the other is 16 that's a longer one over there. How many cats are in Tigersville? So they give us the density here and they give us, I think enough information to figure out the area. And they want us to figure out how many cats we have. So what is the quantity? So pause this video and see if you can figure that out. Well, just as we said the density is equal to quantity divided by area. If we multiply both sides of this equation by area you get area times density is going to be equal to quantity. And we know the density. It's 13 cats per square kilometer, and we can figure out the area and then just multiply the two. So what's the area of this right over here? Well, the area of a trapezoid is going to be, write here area is going to be 12 kilometers. The height of the trapezoid times the average of the two parallel sides I guess you could say. So the average of those it's going to be eight kilometers plus 16 kilometers over two. So this is going to be equal to 12 kilometers times eight plus 16 is 24 divided by two is 12. So times 12 kilometers. So this gives us 144 square kilometers. Now we know we have 13 cats per square kilometer. So let me do this year in another color. So if I multiply 13 cats per kilometer squared and I multiply that times this business right over here, times 144 square kilometers. And you might also notice that the units cancel out the same way that variables might. So that cancels out with that. You're going to get 13 times 144. And the units that you're left with is just cats. So 144 times 13, three times four is 12 that gives us to 13, three times a hundred, 300 plus another hundred is 400. Now I'm just gonna multiply 144 essentially by 10 which is going to be 1440. And so if I add up all of that together I'm gonna jump down to here. I get 1872. So this is 1872 cats in total and we are done.