Interpreting expressions with multiple variables: Cylinder

0:00- [Instructor] We're told that given the height h and volume V of a certain cylinder, Jill uses the formula r is equal to the square root of V over pi h to compute its radius to be 20 meters. If a second cylinder has the same volume of the first, but is 100 times taller, what is its radius?

0:20Pause this video, and see if you can figure this out on your own. All right, now let's do this together. So first, I always like to approach things intuitively. So let's say the first cylinder looks something like this, like this. And then the second cylinder, here, it's 100 times taller.

0:36I would have trouble drawing something that's 100 times taller. But if it has the same volume, it's going to have to be a lot thinner. So if you want to maintain the volume, as you make the cylinder taller, and I'm not going anywhere close to 100 times as tall here, you're going to have to decrease the radius.

0:54So we would expect a radius to be a good bit less than 20 meters. So that's just the first intuition, just to make sure that we somehow don't get some number that's larger than 20 meters. But how do we figure out what that could be? Well, now we can go back to the formula, and we know that Jill calculated that 20 meters is the radius.

1:12So 20 is equal to the square root of V over pi h. And if this formula looks unfamiliar to you, just remember the volume of a cylinder is the area of either the top or the bottom, so pi r squared times the height. And if you were to just solve this for r, you would have this exact formula that Jill uses.

1:36So this isn't some new formula. This is probably something that you have seen already. So we know that 20 meters is equal to this, and now we're talking about a situation where we're at a height that is 100 times taller. So this other cylinder is going to have a radius of square root of, V is going to be the same, so let's just write that V there.

1:57Pi doesn't change. It's always going to be pi. And now instead of h, we have something that is 100 times taller. So we could write that as 100h. And then what's another way to write this? Well, what I'm going to do is try to bring out the 100, so I still get the square root of V over pi h.

2:13So I could rewrite this as the square root of one over 100 times V over pi h, which I could write as the square root of one over 100, and I'm just using properties of radicals here, times the square root of V over pi h. Now, we know what the square root of V over pi h is.

2:40We know that that is 20, and our units are in meters. So this is 20. And then what's the square root of one over 100? Well, this is the same thing as one over the square root of 100. And of course, now it's going to be times 20, times 20. Well, the square root of 100, I should say the principal root of 100 is 10.

2:59So the radius of our new cylinder, of the second cylinder is going to be 1/10 of 20, which is equal to two meters. And we're done. The second cylinder is going to have a radius of two meters, which meets our intuition. If we increase our height by a factor of 100, then our radius decreases by a factor of 10.

3:20The reason why is because you square the radius right over here. So if height increases by a factor of 100, if radius just decreases by a factor of 10, it'll make this whole expression still have the same volume.