Hey guys. So you may run into some problems involving density. In this video, I'm just going to give you a quick refresher on how we solve density problems just in case it's been some time since you've seen this stuff. So guys, remember that density is defined as mass, which is the amount of stuff that something has, divided by the volume, which is the amount of space that it takes up. The symbol for density is this Greek letter rho, which kinda looks like a p but it's a little curved. And it's defined as mass over volume \( m \) over \( v \). Now in the SI system, the units are kilograms per cubic meter. And really guys, all these problems are just going to relate these three variables inside of this equation, density, mass, and volume, for just different geometric shapes like rectangular prisms, cubes, spheres, cylinders, all that stuff. And then eventually, you may also have to convert some units. So that's really all there is to it. Let's just jump straight into an example.

So we've got the average density of the Earth is 55100. We're going to assume that's the sphere, which might be news to some of you. Contrary to what you see on the Internet, the Earth is actually round. The radius is 3960 and we're going to calculate what the mass is. So first things first, I'm just going to draw a quick little sketch of the Earth. It's a sphere so I'm going to just draw it like this. So here's a sphere and the radius, which I'm going to call capital R, we already know what that is. It's 3960. So I know the density of the Earth is 55100 and I know the radius of the Earth is 3960, but it's in miles. And the density is in kilograms per cubic meter. I'm going to use this information to find out what the mass of the Earth is. This is my target variable. So here's what we're going to do. And these problems involving density, you're always just going to start off just with the by writing the density equation. So remember that the density equation, rho, is just equal to mass over volume. So if I'm trying to figure out what the mass is, basically all I need to do is just rearrange the equation. So I can move this volume term up here and just say that rho times the volume is equal to the mass. Now I already know what the density is and it's in the right units, kilograms per cubic meter. So I can use this to basically figure out what the mass is. All I have to do is just figure out what the volume of the Earth is. And so notice all we have is the radius. We don't actually have the volume. So let's just go over here and figure out what that volume is. So if I can figure out the volume of the Earth, which is just modeled after a sphere, so we're going to use this volume equation up here, which is \( \frac{4}{3} \pi \) times the radius of the Earth cubed. So we can say that this is the radius of the Earth. If I can figure out this volume here, I can plug it back into my equation for density and figure out the mass. Unfortunately, what happens is that if I do this and if I use the unit that I have for radius, this is given to me in miles, which unfortunately is not the SI unit. So first, I actually have to do a unit conversion. So I'm told here that one mile is equal to approximately 1609 meters. So what that means is I can set up a little unit conversion here. I can say that 3960 times a conversion factor will give me meters. So to do that, I want to set up my conversion factor so that the miles are on the bottom, so that it'll cancel out. And so 1 mile, I'm told here, is equal to 1609 meters. And so if I work this out, the miles cancel, and I'm just left with meters, which is the SI unit that I want. So if you work this out in your calculators, you're going to get 637 or \( 6.37 \times 10^6 \), and that's in meters. You could also just display it as a full number. That's totally fine. I just chose to do it in scientific notation because it's a little bit easier that way. So here are my units in meters. So now I've got the radius. I'm just going to plug it back into this equation and figure out the volume.

So the volume of the Earth is just equal to \( \frac{4}{3} \times \pi \times (6.37 \times 10^6)^3 \), and you have to cube that. Make sure that you plug this into your calculator. Do this first and then multiply by this raised to the 3rd power. And your calculator should be handling the rest. And when you do that, you're going to get the volume of the earth is equal to \( 1.08 \times 10^{21} \) in cubic meters. So now we just plug it back into this equation. So now we have the density, which is 55100, and we have the volume which is \( 1.08 \times 10^{21} \), and you're going to get the mass of the Earth is \( 5.94 \times 10^{24} \) kilograms. So that's what we get for the mass of the Earth. It's actually pretty close. It might be just slightly off from the real value just in case you go look it up, and that's because we rounded a bit. But that's basically the right answer for the mass of the Earth. That's it for this one, guys. Let me know if you have any questions.