Density word problem: blimp

CCSS Math: HSG.MG.A.2

Video transcript

A helium-filled blimp stays aloft since helium is less dense than regular air. So helium will rise above regular air. That was a strangely-phrased first sentence, but they're just saying helium is less dense, so helium will rise relative to regular air. The density of helium is about 1.1 kilograms per meter cubed less than regular air. Which means a cubic meter of helium can lift about 1.1 kilograms. Because the density of helium-- so for a cubic meter of helium, so if this is a cubic meter of helium, and here is a cubic meter of regular air, they're saying that the mass of this cubic meter of helium is going to be 1.1 kilograms less than the mass of this cubic meter of regular air. And that's what allows it, because of the buoyancy, that's what allows a cubic meter of helium to be able to lift about 1.1 kilograms. A blimp with a mass of 2.11 times 10 to the fifth kilograms can carry as much as 2 times 10 to the fifth cubic meters of helium. If the average mass of a person with luggage is 10 to the second kilograms, what is the number of passengers the blimp can carry? So I encourage you to pause this video, and I'll give you a little structure for this. Think about how much mass 2 times 10 to the fifth cubic meters of helium can carry. And then just realize it has to carry the blimp, and then, what's left over, is what we can use to carry people. So I'm assuming you've given a go at it. So the first step is to say, look, we have 2 times 10 to the fifth cubic meters of helium. And helium, a cubic meter of helium, can lift 1.1 kilograms. So we could say that we can lift 1.1 kilograms per cubic meter. So what is this going to give us? Well our units cancel nicely. This gives us the mass of kilograms that we can lift. Meters cubed canceling with meters cubed. You'd have a meter cubed in the numerator, meter cubed in the denominator. Our dimensional analysis allows our units to cancel just like numbers would in fractions. And so we're left with 2 times 10 to the fifth times 1.1. Well 2 times 1.1 is 2.2. So this is going to give us 2.2 times 10 to the fifth kilograms is how much we can lift. So this is how much we can lift. Once again, and this is probably the most important step, this is our mass of helium in terms of cubic meters, and this is how many kilograms we can lift per cubic meter. So you multiply those two, we get in kilograms how much mass we can lift. So now let's subtract out the things that we need to lift. We need to subtract out the mass of the actual blimp, the actual structure that is holding the helium. So let's subtract that. So let's subtract 2.11 times 10 to the fifth kilograms. And so after lifting the blimp, what do we have left over? Well let's see. 2.2-- maybe I should, well let me make it-- 2.2 minus 2.11. We might be able to do that in our heads. That's the same thing as 2.20 minus 2.11. 20 minus 11 is 9. So we could think of it that way. It's 0.09. Or if you want, you could say well, 0 minus 1, well, let's do a little bit of regrouping here. 10 minus 1 is 9, 1 minus 1 is 0, 2 minus 2 is 0. So this gives us 0.09 kilograms of lifting capacity after we've netted out the mass of the blimp. Actually, let me be careful. 0.09 times 10 to the fifth kilograms that we can lift after netting out the mass of the blimp. This is what we can use to lift the people and their luggage with. So they tell us that the people and luggage-- that each person plus that person's luggage-- is 10 to the second kilograms. So let's divide this by 10 to the second. And I guess we could say kilograms per person. And then what do we end up with? Well this is going to be equal to 10 to the fifth divided by 10 squared is going to be 10 to the third. So this is going to simplify to 10 to the third. So we get 0.09 times 10 to the third. And then kilograms divided by kilograms per person. If we just think about the dimensional analysis, this is the same thing as kilograms times the reciprocal of this, of people, or person per kilogram. Kilogram gets canceled out, and you're just left with people, or persons. So this is how many people, and when we say people, we're saying the people plus their luggage. So what is 0.09 times 10 to the third? Well that's 0.09 times 1,000. So let's multiply it. If we start with 0.09 and we multiply it by 10 once, we get to 0.9. Multiply it by 10 again, we get to 9. Multiply it by 10 again, we get to 90. So this right over here is the same thing as 90 people. So we could lift, the maximum load for this blimp is 90 people and their luggage.