Relating height and volume of cylinders

[00:00:00]We're told that Chris is pouring water into two cylinders, A and B. The graph shows how the height of the water changes as the volume increases in the two cylinders, A and B. Which cylinder has a larger radius? Pause this video and have a go at that.

[00:00:13]Okay, so let's just think about this.

[00:00:14]This is relating height to volume of cylinders. Let's think about how those relate. If we just think about if we have a cylinder like this.

[00:00:23]If we have a cylinder like this, just drawing it, hand drawing it.

[00:00:29]You have the radius of either the base or the top. That's the radius. We can figure out the area of the base. That's going to be pi r squared. Pi r squared is going to be the area of the base. And then if you multiply that times the height of the cylinder, so if you say height times pi r squared, that's going to be equal to the volume of the cylinder.

[00:00:51]Now, we for given volume, we have different amounts of or we have different heights here. And actually I like I let me pick a point that looks like it's awfully close. So let's pick those two points there. We don't have to get exact because these are so far apart. Let's It seems It seems like at a volume of 2 ml we're at a height of roughly 1 cm for B, and we're at a height of 4 cm for A. So let's just think about what what that's actually what that's actually saying. So, let me do this.

[00:01:23]Let me do this for A and for B. So for A at a volume of 2 ml, so the volume is 2 ml, that's going to be equal to the height of 4. We're dealing with A right over here. It's going to be equal to the height of 4 times pi r squared. And for B, we're going to have, once again, a volume of 2, and that's going to be equal to the height, in this case it's 1 times pi r squared. We could solve for r in either case. If we divide both sides by well, in this case 4 pi, we will get that r squared is equal to 2 over 4 pi, which is the same thing as 1 over 2 pi.

[00:02:07]While over here, if we divide both sides by 1 pi, we will get that r squared is equal to 2 over pi.

[00:02:16]2 over pi.

[00:02:17]Now, you might say, "Hey, I'm not having solved for r yet." But, it we're assuming r is only positive values.

[00:02:23]We're talking about a dimension here.

[00:02:25]So, if we just compare which one has a smaller r squared or a larger r squared, we want to figure out the larger radius.

[00:02:31]The the larger r squared is also going to be the larger r. So, what's larger? 2 over pi or 1 over or 1/2 over pi? Well, we could write it this way. This is equal to 1/2 * 1 over pi. This is equal to 2 * 1 over pi.

[00:02:51]So, if you compare, both are being multiplied by 1 over pi, but 2 is clearly larger. So, B has the larger radius. And you can actually even think about that intuitively, not mathematically. If you had two If you had two cylinders, one that's a little bit tall and skinny, like this, and then let's call that one A, and then you had another one that is a little bit squatter and wide with a big radius, like this, you would need to have for the same volume of water, you would have to fill it up much higher on this one than you would have to do on this one to for the same volume of water, because your base here is much, much bigger. So, you need the less height to have that same volume. So, that also makes sense that B's height is going to increase at a much slower rate.