Volume with mini cubes
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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-- a 1/4 of a foot by 1/4 of a foot by 1/4 of a foot. Those 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 units cubes there are. So 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. So 1/4 times 1/4 is 1/16, times 1/4 is 1/64. So this is going to be 1 over 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. The first layer has 1, 2, 3, 4, 5, 6, 7, 8. That's this first layer right over here. And then we have the second layer down here, which would be another 8. So it's going to be 8 plus 8, or 16. So the total volume here is going to be 16 times 1/64 of a cubic foot, which is going to be equal to 16/64 cubic feet, which is the same thing. 16/64 is the same thing as 1/4. Divide the numerator and the denominator by 16. This is the same thing as 1/4 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 2 times 1/4 feet, which is equal to 1/2 of a foot. The height here is the same thing. So it's going to be 2 times 1/4 of a foot, which is equal to 2/4, or 1/2 of a foot. And then the length here is 4 times 1/4 of a foot. Well, that's equal to 4/4 of a foot, which is equal to 1 foot. So to figure out the volume, we could multiply the length times the width times the height, and 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, instead of writing this kind of x-looking thing, this cross-looking thing. So the length is 1. The width is 1/2 of a foot, so times 1/2. And then the height is another 1/2. Let me do it this way. The height is another 1/2, so what's 1 times 1/2 times 1/2. Well, 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 going to 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.