How volume changes from changing dimensions

So, I have a rectangular prism here and we're given two of the dimensions. The width is two, the depth is three, and this height here, we're just representing with an h. What we're going to do in this video is think about how the volume of this rectangular prism changes as we change the height.

So, let's make a little table here. So, let me make my table. This is going to be our height, and this is going to be our volume (V for volume).

So, let's say that the height is five. What is the volume going to be? Pause this video and see if you can figure it out.

Well, the volume is just going to be the base times height times depth, or you could say it's going to be the area of this square. So, it's the width times the depth, which is 6, times the height. So, that would be 2 * 3 * 5.

So, 2 * 3 * 5, which is equal to 6 * 5, which is equal to 30.

30 cubic units! We're assuming that these are given in some units, so this would be the units cubed.

All right, now let's think about it. If we were to double the height, what is going to happen to our volume? So, if we double the height, our height is 10. What is the volume? Pause this video and see if you can figure it out.

Well, in this situation, we're still going to have 2 * 3. 2 * 3 * our new height times 10. So now, it's going to be 6 * 10, which is equal to 60.

Notice, when we doubled the height, if we just double one dimension, we are going to double the volume.

Let's see if that holds up. Let's double it again. So, what happens when our height is 20 units? Well, here our volume is still going to be 2 * 3 * 20.

2 * 3 * 20, which is equal to 6 * 20, which is equal to 120.

So, once again, if you double one of the dimensions (in this case, the height), it doubles the volume. You could think of it the other way: if you were to have volume go from 120 to 60.

Now, let's think about something interesting. Let's think about what happens if we double two of the dimensions. So let's say, so we know. I'll just draw these really fast.

We know that if we have a situation where we have 2 by 3 and this height is five, we know the volume here is 30, 30 cubic units. But now, let's double two of the dimensions. Let's make this into a 10 and let's make this into a four.

So, it's going to look like this, and then this is going to be a four. This is still going to be a three, and our height is going to be a 10.

So, it's going to look something like this. So, our height is going to be a 10. I haven't drawn it perfectly to scale, but hopefully, you get the idea.

So, this is our height at 10. What is the volume going to be now? Pause this video and see if you can figure it out.

Well, 4 * 3 is 12, and 12 * 10 is 120.

So, notice when we doubled two of the dimensions, we actually quadrupled our total volume. Think about it. Pause this video and think about why did that happen.

Well, if you double one dimension, you double the volume. But here, we're doubling one dimension and then another dimension, so you're multiplying by two twice.

So think about what would happen if we doubled all of the dimensions. How much would that increase the volume? Pause the video and see if you can do that on your own.

In general, if you double all the dimensions, what does that do to the volume? Or if you have all of the dimensions, what does that do to the volume?