How volume changes from changing dimensions

- [Tutor] I have a rectangular prism here. 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. And what we're gonna do in this video is think about how does the volume of this rectangular prism change as we change the height. So, let's make a little table here. So, let me make my table. So, this is going to be our height, and this is going to be our volume, V for volume. And 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 can say it's going to be the area of this square, so it's the width times the depth which is six times the height. So, that would be two times three times five. So, two times three times five which is equal to six times five 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. Alright, 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 gonna have two times three, two times three times our new height, times 10. So, now it's gonna be six times 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 gonna be two times three times 20, two times three times 20 which is equal to six times 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 can think of it the other way. If you were to halve, if you were to go from 20 to 10, so if you halve one of the dimensions, it halves the volume. You 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 two by three 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. This is gonna look like this. 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 gonna look something like this. So, our height is going to be a 10. I haven't drawn it perfectly to scale. Hopefully, you get the idea. So, this is our height at 10. What is the volume gonna be now? Pause this video and see if you can figure it out. Well, four times three is 12 times 10 is 120. So notice, when we doubled two of the dimensions, we actually quadrupled, we actually quadrupled our total volume. 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 of the dimensions, what does that do to the volume? Or if you halve all of the dimensions, what does that do to the volume?