Area density

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 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 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 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 let me write it here area is going to be 12 kilometers the height of the trapezoid times the average of the two the the two parallel sides i guess you could say so the average of those so it's 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 so 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 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. 3 times 4 is 12 3 times 4 is 12 that gives us to 13 3 times 100 300 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 18 72. so this is 18 72 cats in total and we are done