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Volume of rectangles inside rectangles

Video transcript
The Ubas are moving from Houston to Egypt. They pack their belongings in rectangular crates and hire a boxcar to ship the crates across land and sea. The crates are made specifically to fit inside the boxcar with their bases facing down. Each crate has a base 5 meters long by 1.5 meters wide. So let me draw that. So crate is 5 meters long, and 1.5 meters wide, and has a height of 2 meters. So its height might look something like this. So it has a height of 2 meters. So that's each of the crates. And they're designed to fit inside a boxcar. So this is a crate right over here. I'll do my best to draw a crate. And they give us the dimensions of the boxcar. A boxcar is 15 meters long. So let me draw a boxcar here. So it's 15 meters long. Maybe I'll try to make sure I can fit it on the page. So that this whole distance would be 5 meters, and then another 5 meters here, and then another 5 meters here. So that would be 15 meters long. So you could fit three of the crates along an edge like that. And then, they tell us that it is 3 meters wide. So this is 1.5 meters wide. So you could put two of these to get you to 3 meters. Let me draw this so we can see what's going on behind the scenes. So you could go 3 meters wide for a boxcar, and then it is 4 meters high. So each of these are 2 meters high. So you could stack one more. And so you have 2 meters plus 2 meters. This entire distance right over here is going to be 4 meters. And I could draw the rest of the boxcar like this. So there's a couple of ways to think about how many crates you could fit in a boxcar. One way would be just the way that we're doing it right now. We could visualize. How many can you fit in this direction along the length? How many can you fit along the width? And how many can you fit along the height? And essentially, if we multiply those three numbers, we would have counted the number of crates that could fit inside. So you could fit 1, 2, 3 along the length. So that'd be 3. You could fit 2 along the width. 1.5 and 1.5 gets you to 3 meters, so times 2. And then you could fit 2 along the height, so times 2, gets us to 3 times 2 is 6 times 2 is 12. You can fit 12 crates in the boxcar. Now, another way you could have done it is you could say, OK, they're telling us that these are designed to fit. So we really just have to compare the volumes. How many times more is the volume of the box car than the crate? I like doing it this way more, just to make sure that the dimensions actually work out, so that you could actually squeeze these in. Because if the dimensions aren't right, even if the boxcar is 12 times the volume of one of the crates, if the crates don't have the right dimensions, you might not be able to squeeze exactly 12 crates in there. But they're telling us that it is the exact dimensions. So we could figure out the dimensions of the boxcar, then the dimensions of the crate. And then we could figure out how much larger the boxcar is, how many times larger. Let's do the boxcar in this blue color. The boxcar is 15 meters long, 3 meters wide, and 4 meters high. So boxcar volume is equal to 15 in cubic meters. So there's 15 meters times 3 meters times 4 meters. So this is going to be in cubic meters. So this is going to be-- let's see, 15 times 3 is 45. 45 times 4 is 180 cubic meters. That's the boxcar volume. And then what's the volume of the crate? Well, the crate volume-- if we do our math right, it should come out to 1/12 of this, because that's what we just figured out, is 5 times 1.5 times 2. So 5 times 1.5 times 2. Well, 1.5 times 2 is 3, times 5 is 15, so 15 cubic meters. So how many times larger is the boxcar than the crate? Well, what's 180 divided by 15? Well, it's exactly 12. 10 times 15 is 150. And then 2 times 15 is 30. 150 plus 30 is 180. So notice, 180 divided by 15 is 12. So either way, however you think about it-- I find this one to be a little bit easier to kind of just visualize the boxes-- you can fit 12 crates in the boxcar.