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Scale and Indirect Measurement

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Scale and Indirect Measurement

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Let's do some scale and indirect measurement problems.
So they say use the scale diagram to the right to
determine-- I think this is the scale diagram they're
talking-- it's really above the question.
And they want to determine-- let's see first, the length of
the helicopter.
When they say the scale diagram, you notice this
helicopter looks pretty small.
I don't know how big your screen is, but looks maybe,
one or two inches, depending on what resolution you're
watching this video on.
But they're saying this helicopter is
much bigger than that.
They're saying just this little distance right here--
just that little distance, that little white box there--
that's actually a foot on the real helicopter.
So what they're able to do is to draw a small version of the
helicopter and essentially give us a ruler and tell us
that that little distance is equivalent to a foot.
And this whole distance right here, that whole distance
right there, is equivalent to 1, 2, 3, 4, 5, 6, 7 feet.
Did I get that right?
1, 2, 3, 4, 5, 6, 7.
So this thing is equal to 7 feet.
So given that, let's try to answer their question.
So the first one, the length of the helicopter.
So that's from cabin to tail, is that distance right there.
Let's see how many of these things it is.
So if I take that right there, let me see if I can copy and
paste that.
So if I-- let's see, if I copy it and then I paste it,
there's one right there.
One.
Edit, paste.
Two right there.
Edit, paste.
And then we get not quite three.
Almost three, but not quite.
We get about that far.
So it looks like about-- so this right here is-- this
right here is 7 feet.
That right there is another 7 feet.
And I know that, because that is 7 feet
according to the scale.
And then this looks like we got about that far.
So it looks like this is maybe another 5 feet,
if I were to estimate.
Actually, if we went to go all the way to the end of the
tail-- if we went to the end of here-- it actually looks
like another 7 feet.
We actually complete three 7 feets.
So I would say that part a, I would go it's 27 feet.
Sorry, not 27.
21 feet.
7 feet times 3 is equal to 21.
I'd go part one.
Because if you go all the way to this tippy, tippy of the
rear rotor right there, then you're going to get 21.
And you're going to get three of these lengths.
Let's do part b.
The height of the helicopter.
Floor to rotors.
So from here-- from floor to rotors.
Well, I'll go from down here to the rotors.
They might be talking about this floor, but who knows?
So what is this distance right here in feet?
What is that distance?
We could just eyeball a little bit.
They didn't give us a vertical scale.
We could just-- this scale looks like about that high.
So that would be about 7 feet.
Let me copy and paste that.
So let me copy it.
Let me paste it.
So that would be 7 feet right there.
Then if I were do another one, we're getting about another
1/2 of 7 feet.
So that'd be another 3 and 1/2 feet.
So I would say that this is 7 right there.
And then we get about 3 and 1/2.
So 7 plus 3 and 1/2 is 10.5 feet.
The length of one main rotor-- and I think the main rotor--
they're talking about this length right there.
That length right there.
That is the length of one main rotor.
Or actually, this length right here.
That is the length of one main rotor.
And let's compare it to this scale over here.
So let's compare it.
That's 7 feet.
So let's copy it.
And then let's paste it.
I lost it.
So let me copy it.
Let me copy and then paste.
So that's one length.
So we want one rear rotor.
So that's 7 feet right there.
7 feet right there.
And we can do another one.
The important thing is to get the idea of it.
You don't have to get the exact measurement.
We're not about to go out there and build a helicopter.
So what is this length?
We have 7 feet.
And then we have 1, 2, 3, about another 4 feet.
So I would go with part c, I'd go with 11 feet for one rotor
if we're getting to see this actually head on.
Maybe it's a little bit longer if it's at an angle.
But that gives us our general idea.
Actually it looks a little bit longer than that.
But the idea is as long as you understand how I'm using the
scale is the main thing.
The width of the cabin.
So here-- that's the cabin width right there.
This distance.
That distance right there is the cabin width.
And if we compare it to this right here, we already said
that this right here is 7 feet roughly.
Edit, copy, edit, paste.
So that is 7 feet.
That looks like about the width of the cabin.
It's a good estimate.
So the width of the cabin, I would say, is 7 feet.
And then find the diameter of the rear rotor system.
The rear rotor system is this right here.
That is the rear rotor system.
So if we were to copy and paste our 7 foot lengths--
actually, let me copy and paste the actual ruler.
So this is our actual ruler.
So how long is the rear rotor system?
Let's look at our ruler.
Looks like 1, 2, 3, 4 feet is my best estimate.
So I'd say the diameter of the rear rotor system is 4 feet.
So you might have already known how to do this, but now
you have a good sense.
Look, this is telling us that this little distance on the
picture is actually equivalent to a foot.
And you just compare it to the actual picture, and you know
how big the actual helicopter is.
That's all that's going on there.
Let's do another scale and indirect measurement problem.
So it says on a Sunday morning, the shadow of the
Empire State Building is 600 feet long.
At the same time, the shadow of a yard stick, which is 3
feet long, is 1 foot 5 1/4 inches.
How high is the Empire State Building?
So we could say height of Empire to length of shadow.
So length of the shadow of Empire should be equal to
taking the length of a yard stick, so height of yard
stick-- I'm assuming that they're pointing up the yard
stick like a skyscraper-- height of yard stick divided
by length of shadow of yard stick.
Now this problem looks a little bit hairier because we
have feet here, and then we have 1 foot 5 1/4 inches.
And you do everything in inches.
And you do everything in feet.
And the interesting thing is here.
You can look at the units.
As long as we calculate this in feet and we calculate this
in feet down here, these units are going to cancel out.
If you have x feet divided by 600 feet, in this case, the
feet are going to cancel out.
You'll have a unit-less quantity.
So it's really just going to be a pure ratio.
Same thing here.
You can do it in feet.
Or you can do it in inches.
And you could actually do this in feet and do this side in
inches, because you get feet divide by feet.
The feet cancel out.
The inches cancel out.
So they stay unit-less.
So it actually becomes a lot easier when you can keep the
right side in inches and you can keep the
left side in feet.
So let's say the height of the Empire State Building is x.
So we could say x feet over the length of the shadow.
The shadow of the Empire State Building is 600 feet.
Over 600 feet is equal to the length of a yard stick
standing up.
Or the height of a yard stick.
Let's do it in inches.
3 feet long is 36 inches.
36 inches.
And I want to show you that the units work out.
36 inches divided by the length of its shadow.
1 foot 5 1/4 inches.
What's that?
One foot is 12 inches.
12 plus 5 1/4 is 17 1/4.
Or 17.25 inches.
And notice the units cancel out.
Feet divided by feet.
Inches divided by inches.
So they just become pure ratios.
x/600 is equal to 36/17.25.
The reason why I could use feet here and inches here is
because the units cancel out.
I want to make that very, very, very clear.
Now we can just cross multiply, which is essentially
just multiplying both sides of the equation by 600 and 17.25.
So 17.25 times x is going to be equal to 36 times 600.
And then we can divide both sides by 17.25.
So x is equal to 36 times 600 over 17.25.
And let's get the calculator out.
So x is equal to 36 times 600 divided by
17.25 is equal to 1,252.2.
Let's just round it.
So x is equal to 1,252.2 feet, which is almost 1/4 of a mile.
So the Empire State Building is pretty high.
But this is pretty neat, because this is something that
you could actually do.
You could actually measure the shadow of a skyscraper and
then you could actually measure the
shadow of a yard stick.
And then doing that, you can actually figure out the height
of a skyscraper.
Which is pretty neat, because if you didn't know how to do
this math, you'd have to get into a helicopter, and drop a
huge-- I mean it'd be a much harder thing to do.
So hopefully, you found that pretty interesting.

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Scale and Indirect Measurement

Scale and Indirect Measurement