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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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