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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.