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# Density word problem: blimp

Sal solves a word problem about the number of passengers a blimp can carry, using his knowledge of volume and density. Created by Sal Khan.

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.