Volume with fractional cubes

0:00- [Instructor] So I have this rectangular prism here, it's kind of the shape of a brick or a fish tank, and it's made up of these unit cubes. And each of these unit cubes, we're saying is 1/4 of a foot by 1/4 of a foot by 1/4 of a foot. So you could almost imagine that this is, so let me write it this way, this is a fourth of a foot, by 1/4 of a foot, by 1/4 of a foot.

0:26Those are its length, height, and width or depth, whatever you want to call it. So given that, what is the volume of this entire rectangular prism going to be? So I'm assuming you've given a go at it. So there's a couple of ways to think about it. You could first think about the volume of each unit cube, and then think about how many unit cubes there are.

0:48So let's do that. The unit cube, its volume is going to be 1/4 of a foot times 1/4 of a foot times 1/4 of a foot. Or another way to think about it is it's going to be 1/4 times 1/4 times 1/4 cubic feet, which is often written as feet to the third power cubic feet.

1:16So 1/4 times 1/4 is 1/16 times 1/4 is 1/64. So this is going to be 1/64 cubic feet, or 1/64 of a cubic foot. That's the volume of each of these. That's the volume of each of these unit cubes. Now, how many of them are there? Well, you could view them as kind of these two layers.

1:40The first layer has one, two, three, four, five, six, seven, eight. That's this first layer right over here. That's this first layer right over here. And then we have the second layer down here, which would be another eight. So it's going to be 8 plus 8 or 16.

1:59So the total volume here, the total volume is going to be 16 times 1/64 of a cubic foot, which is going to be equal to 16/64. 16/64 cubic feet, cubic feet, which is the same thing, 16/64 is the same thing as 1/4, divide the numerator and the denominator by 16.

2:28This is the same thing as 1/4 of a cubic foot, of a cubic foot, and that's our volume. Now, there's other ways that you could have done this. You could have just thought about the dimensions of the length, the width, and the height. The width right over here is going to be two times 1/4 feet, which is equal to one half of a foot.

2:52The height here is the same thing, it's two times, so it's gonna be two times 1/4 of a foot, which is equal to 2/4 or 1/2 of a foot. And then the length, the length here is 4 times 1/4 of a foot. 4 times 1/4 of a foot, well, that's equal to 4/4 of a foot, which is equal to one foot.

3:18So to figure out the volume, we could multiply. We could multiply the length times the width, the length times the width times the height. Times the height, and I mean these little dots here, these aren't decimals, I've written them a little higher. These are another way, it's a shorthand for multiplication.

3:38Instead of writing a, this kind of X looking thing, this cross looking thing. so the length is one, the width is half of a foot, so times one half. And then the height is another half. Let me do it this way. The height is another half. So what's 1 times 1/2 times 1/2?

4:00Well, that's going to be equal to 1/4. And this is a foot, this is a foot, this is a foot. So foot, times foot, times foot, that's gonna be feet to the third power or cubic feet, 1/4 of a cubic foot. Either way, we got the same result, which is good.