Area density

In this video, we're going to talk about density in the context of area. 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 that out of the way, let's do a worked example that helps us understand this idea a little bit better. 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 in 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. It's a perfect isosceles trapezoid; it's going to 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 the 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, that 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. 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—let me write it here—area is going to be 12 kilometers, the height of the trapezoid, times the average of the two, the two parallel sides, I guess you could say.

So, the average of those is going to be: eight kilometers plus 16 kilometers over two. So, this is going to be equal to 12 kilometers times 8 plus 16 is 24. Divided by 2 is 12. So, times 12 kilometers. This gives us 144 square kilometers.

Now, we know we have 13 cats per square kilometer. So, let me do this here in another color. 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 are just cats. So, 144 times 13. Three times 4 is 12. Three times 4 is 12; that gives us 2, 13. Three times 100 is 300, plus another 100 is 400. Now, I'm just going to multiply 144 essentially by 10, which is just going to be 1440.

And so, if I add up all of that together—I'm going to jump down to here—I get 1872. So, this is 1872 cats in total, and we are done.