Interpreting expressions with multiple variables: Cylinder | Modeling | Algebra 2 | Khan Academy

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 ( \frac{v}{\pi h} ) to compute its radius to be 20 meters. If a second cylinder has the same volume as the first but is 100 times taller, what is its radius? Pause 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 a hundred times taller. I 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, 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. So we would expect the 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. So 20 is equal to the square root of ( \frac{v}{\pi h} ). If this formula looks unfamiliar to you, just remember the volume of a cylinder is the area of one of the either the top or the bottom, so ( \pi r^2 \times h ), and if you were to just solve this for ( r ), you would have this exact formula that Jill uses.

So this isn't coming, 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 ( \sqrt{v} ) that is the same. So let's just write that ( v ) there.

( \pi ) doesn't change; it's always going to be ( \pi ). And now instead of ( h ), we have something that is a hundred times taller, so we could write that as ( 100h ). Then what's another way to write this? Well, what I'm going to do is try to bring out the hundreds. So I still get the square root of ( \frac{v}{\pi h} ), so I could rewrite this as the square root of ( \frac{1}{100} \times \frac{v}{\pi h} ), which I could write as ( \sqrt{\frac{1}{100}} \times \sqrt{\frac{v}{\pi h}} ).

Now we know what the square root of ( \frac{v}{\pi h} ) is; we know that that is 20, and our units are meters. So this is 20, and then what's the square root of ( \frac{1}{100} )? Well, this is the same thing as ( \frac{1}{\sqrt{100}} ), and of course now it's going to be times 20. Well, the square root of 100, I should say the principal root of 100, is 10.

So the radius of our new cylinder, of the second cylinder, is going to be ( \frac{1}{10} \times 20 ), which is equal to 2 meters.

And we're done! The second cylinder is going to have a radius of 2 meters, which meets our intuition. If we increase our height by a factor of 100, then our radius decreases by a factor of 10. The 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.