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Finding the Side Length of a Cube

Patrick Ford
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Hey, guys, hopefully get a chance to check out this example. Problem. We've got an iron cube with the mass. We're told, with the density of this Iron Cube is and we're gonna figure out the length of the sides of the Cube. So the first thing I'd like to do is just draw a quick little sketch are diagram. We're told we have an iron cubes. I'm gonna draw a little cube here. And a cube is basically, like a special case of rectangular Prism in which all the lengths of the sides are the same. So I'm gonna call The side s here. You don't have to say l length, width and height because they're all the same exact number. So 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. Also told the density of iron 7.87 times 10 to the third. This is also in the correct units, kilograms per meters cube. And so I'm gonna 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 like densities, masses and volumes involved just started with the density equation. So I brought My row is equal to mass divided by volume. So how does this get me any closer to figuring out what the side links are? Well, I already know what the what the density is. And I also would already know what the masses. So it's gonna have to do something with this volume term. So the volume of a rectangular prison, remember, is just given by this equation length, times with 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 cube. 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 could get from using the density equation. So that's what I'm gonna dio. So I'm gonna 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 row 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 times, 10 to the third and notice how I've already checked the units. And I don't have to you do any union conversions because they're all in the same sort of like they all compatible with each other. So I just go ahead and plug this in and what I get is 6.54 times 10 to the minus 5 m cube. 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 times 10 to the minus fifth, is equal to the side length cube. 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, and I'm gonna 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 roots of this number 6.54 times 10 to the minus five, you're just gonna get 0.4 m or that would just be also four centimeters. So either one of these is correct. Just depends on which unit you would have to express it in. Um, but that's really it. So let me know if you guys have any questions, that's about this one.