 When talking about a physical quantity, we've already discussed the importance of choosing the right units and dimensions to describe that physical quantity, and also being aware of how accurate and precise your knowledge of that quantity is. And obviously making measurements is the right way to go about making sure you're right about all those details. But the next section is about the enormous and surprising power of guessing. It's amazing how well what we already know can be used to extrapolate to answer questions about things that we haven't thought about. But before we do that, we have to talk about scaling. Something scale just means its size. Often when we try and think about something unfamiliar, we can relate it to something that we are familiar with, but just it might be much bigger or smaller. For example, suppose I go to the shops and I take a certain amount of money and I buy a certain amount of chocolate. But I have dreams and one day I dream of owning an entire truck full of chocolate. Now I've never experienced amounts of chocolate in that kind of quantity before at that scale. So I have to be a little bit careful about extrapolating my knowledge of how big a chocolate bar is and how much that costs all the way up to that scale. Now I've bought a lot of single chocolate bars and so I know roughly how big a single chocolate bar is. A single chocolate bar here is something like 10 to 15 centimetres long. I don't know roughly how big a truck is. The business end of a truck is maybe 4 metres long. Sure there are smaller and larger trucks, but that's how big a truck I want to fill with chocolate for my dream to be satisfied. So if I want to figure out how many chocolate bars fit into that truck, I say alright, well it's 4 metres long. I'm trying to fit things that are 10 centimetres in there so I should get this many along the bottom. And if I want to convert from centimetres to metres, which I'm going to need to do, that's going to be equal to... And remember all I'm doing here is multiplying by one, so that's got to be right. I've got 100 centimetres in a metre and so the centimetres cancel and the metres cancel and end up with a ratio... 400 divided by 10 so I end up with 40. So I should be able to get 40 of those chocolate bars along the bottom of the truck. Does that mean I need 40 times as much money? Unfortunately no, because the truck is also wider than a chocolate bar. And if the truck is wider than a chocolate bar, I have to think about how the amount of chocolate in that truck scales as I make the thing not just longer but also wider and taller. So there's a line. Now supposing I make an exact copy of that and so I have a line that's twice as big, obviously if I was to have to pay per unit length I would have to pay twice as much for that. But supposing I started with a square and now I make that length twice as big, if I have two of those squares, now obviously that's not a square. In order to make a square that's twice as long on each side, I actually have to make four of those squares. And so I've doubled this length, the length of a side, and I've doubled the length of the other side. So in effect I've doubled twice, and 2 times 2 is 4. And so I've doubled the length and I've quadrupled the area. You can see that algebraically very easily, the area for a rectangle is just the height times the width. So if I double the height and I double the width, I'm going to get 4 times the height times the width, which is 4 times the original area. And if you know the formula for the area of different shapes, you can show that the same thing works for other shapes too. You might be able to do a circle, or you might be able to do a triangle and so forth. But in fact, this is true for absolutely any shape. If you have any shape at all, you can always break it up into little squares, and the total area of the shape is just the total area of adding up all these little squares. So if I were to double the size of the shape in all directions, then I would double the size of all those squares in both directions, and so I would quadruple the area. And indeed, if I take any length in here at all, what I can say is that the area scales as L squared. So this symbol here just means proportional to. So the area is some number times L squared. And so if I were to double that length L, then I would quadruple the area. Or if I was to increase that L by 1.5, then I would increase the area by 2.25, which is 1.5 squared. Now you don't buy chocolate by the area, by the square meter or square centimeter or something like that. You buy chocolate by the kilogram or by the gram. And the mass of something is proportional to its volume. So it doesn't just matter how wide and how long the chocolate bar is, it also matters how thick it is. Now if I take a cube and double its length, width and height, you can see that I have 8 times as many cubes. And that's not surprising algebraically. You can see that if we have a volume and it's a height times a width times a depth, and if we double all of those, then we're going to end up with 8 times the height times the width times the depth, which is 8 times the volume. And again that works for any shape at all. So if I have some solid, I can always break my solid up into little cubes. And if I change some characteristic length scale of my solid, then when I change that length scale, I'm going to change the size of all the cubes. All the cubes are going to go as L cubed, the cube of that length L. And so my total volume, when I add it all up, is going to go as the cube of length L. Okay, so let's go back to the original question. How much would it cost me to buy my truck full of chocolate? Well, if I had a chocolate bar that was 4 meters long. So it looked exactly like a chocolate bar. It was just scaled up to be 4 meters long. It already worked out. That's about 40 times as big in one length. And so if I scaled up my chocolate bar by a factor of 40 in length, then the total volume, and therefore the total mass of chocolate, will go up as 40 cubed, which is 64,000 times as much chocolate, and presumably would cost about 64,000 times as much, although I would be hoping for a bulk discount by that point. Now in fact, it's even worse than that because my chocolate truck isn't quite the same shape as a block of chocolate. If I had a block of chocolate that was 40 meters long, it might actually look about as wide as a truck. So in other words, if I look from the truck above, that shape looks roughly like a chocolate bar. But if we were to look from the side, the chocolate bar would look a lot thinner than the truck. In other words, the shape of a truck looks a lot more like several chocolate bars on top of each other. And if I just look at my drawing here, I might have a chocolate bar, a chocolate bar, a chocolate bar. I'd have three or four of these. So in fact, my first guess is I'm going to need about, say, three times 64,000 times as much chocolate. So that's about 192,000 times as much. Now I haven't been very accurate, so I'm not going to use these significant figures. So I'll just say it's 2 by 10 to the 5 times as much chocolate or 200,000 times as much chocolate. When you're making guesstimates, you've got to be very wary of carrying too many significant figures because we were making stuff up left, right, and center there. So really the only number that we can be reasonably comfortable with is this one. Even that too is suspect because we're making big guesses. But it does tell me that I'm going to need something like a few hundred thousand times the price of a single block of chocolate in order to buy my chocolate truck, which I must admit is a little bit disappointing, but at least I have some concrete aims for when I finally become a multimillionaire.