Relating height and volume of cylinders
Around 89-95s the narrator says the volume is '2 millimeters' when the graph and earlier narration use 2 milliliters. The interventions use milliliters.
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A graph shows how the water height changes with volume in two cylinders, a and b. Using V = h x pi r^2 at a volume of 2 milliliters (height 4 cm for a, 1 cm for b), Sal finds r^2 = 1/(2 pi) for a and 2/pi for b, so b has the larger radius. He also explains intuitively that a wider cylinder needs less height for the same volume.
You'll be able to
- use V = pi r^2 h to relate height and volume in a cylinder.
- read two points from a height-volume graph and compare cylinders.
- explain why a wider cylinder's water level rises more slowly.
Key words
- cylinder
- A 3D shape with two equal circular ends joined by a curved side.
- radius
- The distance from the center of a circle to its edge.
- area of the base
- For a cylinder, the area of the circle at the bottom, pi r^2.
- volume
- How much space a shape holds; for a cylinder, base area times height.
Transcript
- [Instructor] 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.
Okay, so let's just think about 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, if we have a cylinder like this, just drawing it, hand drawing it, 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 gonna be pi r squared. Pi r squared is gonna 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 gonna be equal to the volume of the cylinder.
Now, for given volume, we have different amounts or we have different heights here. And actually I like, 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. It seems like at a volume of two milliliters, we're at a height of roughly 1 centimeter for b and we're at a height of 4 centimeters for a.
So let's just think about what that actually is saying. So let me do this. Let me do this for a and for b. So for a, at a volume of 2 millimeters, so the volume is 2 millimeters, that's going to be equal to the height of 4. We're dealing with a right over here.
It's gonna 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 gonna 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.
While over here, if we divide both sides by 1 pi, we will get that r squared is equal to 2 over pi. 2 over pi. Now you might say, "Hey, I haven't solved for r yet," but we're assuming r is only positive values. We're talking about a dimension here. 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.
The larger r squared is also gonna be the larger r. So what's larger? 2 over pi or 1/2 over pi? Well, we could write it this way. This is equal to 1/2 times 1 over pi. This is equal to 2 times 1 over pi. 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 could actually even think about that intuitively, not mathematically. 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 for the same volume of water because your base here is much, much bigger.
So you need 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.
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