Hey, guys. Hopefully, you got a chance to check out this example problem. We've got an iron cube with a mass. We're told what the density of this iron cube is, and we're going to figure out the length of the sides of the cube. So, the first thing that I like to do is just draw a quick little sketch or diagram. We're told we have an iron cube, so I'm going to just draw a little cube here. A cube is basically like a special case of a rectangular prism in which all the lengths of the sides are the same. So I'm going to call this side s here. You don't have to say l, length, width, and height because they're all the same exact number.
What else do I know? I'm told the mass, the mass is just equal to 0.515. It's in kilograms, so it's the right unit. I'm also told the density of iron, 7.87×103 kilograms per cubic meters. And so I'm going to use this information to figure out what the side lengths of the cube are. That's just the letter s. So how do I use this to figure out what s is? That's my target variable.
Well, like we said, every time we have a density, or like densities, masses, and volumes involved, just start off with a density equation. So I've got my ρ is equal to mass divided by volume. So how does this get me any closer to figuring out what the side lengths are? Well, I already know what the density is, and I also already know what the mass is. So it's going to have to do something with this volume term. So the volume of a rectangular prism, remember, is just given by this equation, length times width times height. But in a special case of a cube, where all these letters are the same, length, width, and height, it's actually an even simpler equation. It's just the side length cubed. So this is my target variable. So this is actually what I have to figure out. So I can relate this back to the volume, and the volume I can get from using the density equation. So that's what I'm going to do.
So I'm going to use this density equation to figure out my volume and then basically just pass it back into this equation and then figure out the side length. So let's go ahead and do that. If I'm looking for the volume, I just need to move it over to one side. So what I can do is I can trade places with this ρ, with this density, and I can say that volume is equal to mass divided by the density. So my volume is just my mass, which is 0.515 divided by the density, 7.87×103. And notice how I've already checked the units and I don't have to do any unit conversions because they're all compatible with each other. So I just go ahead and plug this in and what I get is 6.54×10-5m³.
So now, basically, exactly like what we said, we can pass this number back into this equation and then figure out the side length. So that means my volume now, which is 6.54×10-5, is equal to the side length cubed. So, how do I get rid of this cubed, and how do I just get s? Well, I have to take the cube root of both sides. So I have to take the cube root here. So what happens is the cube root and the cube will cancel out, leaving me just the side length. And so if you plug in the cube root of this number, 6.54×10-5, you're just going to get 0.04 meters or, that would just be also 4 centimeters. So either one of these is correct. It just depends on which unit you would have to express it in. But that's really it. So let me know if you guys have any questions. That's it for this.
