WEBVTT mathematics/pre-calculus/selhorst-jones 00:00:00.000 --> 00:00:02.100 Hi--welcome back to Educator.com. 00:00:02.100 --> 00:00:07.100 Today, we are going to talk about variation--both direct variation and inverse variation. 00:00:07.100 --> 00:00:10.700 Variation is a way of talking about how different things relate to each other. 00:00:10.700 --> 00:00:14.900 Depending on the type of the variation, we will know what form the relationship takes. 00:00:14.900 --> 00:00:18.900 However, we should point out that variation isn't connected very closely to functions. 00:00:18.900 --> 00:00:23.800 While this sits in the section on functions, it is only sort of connected to functions. 00:00:23.800 --> 00:00:29.000 We can describe it in the language of functions; but honestly, it is much easier to talk about it with equations. 00:00:29.000 --> 00:00:35.600 It is much easier to talk about variation using equations; so that is what we are going to end up using, that y =...stuff involving x, 00:00:35.600 --> 00:00:38.300 what we are used to from doing lots of algebra for it. 00:00:38.300 --> 00:00:40.500 That is what we are going to see for our variations. 00:00:40.500 --> 00:00:46.300 In any case, variation comes up a lot in a wide variety of real-world situations; so it is important to understand. 00:00:46.300 --> 00:00:54.300 If you are interested in physics, chemistry, economics, astronomy, scientific fields in general...you are going to probably need to be familiar with variation. 00:00:54.300 --> 00:01:01.400 It gets tossed around in those fields; also, it tends to show up on standardized tests a lot--things like the SAT, the ACT, the GRE... 00:01:01.400 --> 00:01:08.000 All of these have a tendency to have one or two questions about variation, either direct or inverse variation. 00:01:08.000 --> 00:01:13.200 So, it is important to know this thing a little if you are going to be taking one of these standardized tests at some point. 00:01:13.200 --> 00:01:19.400 All right, let's take a look at it: direct variation: this is the simplest form of variation, but it is also the most common. 00:01:19.400 --> 00:01:25.000 Direct variation says that two things are directly related to each other: if one goes up, the other goes up. 00:01:25.000 --> 00:01:30.300 If the red one goes up, we know that the blue one will also go up. 00:01:30.300 --> 00:01:37.600 Similarly, if the other goes down, if the red one goes down, then we know that the blue one also has to go down. 00:01:37.600 --> 00:01:43.000 They will go at different rates; for example, in this picture we have here, the blue one always goes at a smaller rate, 00:01:43.000 --> 00:01:46.100 and the red one goes at a bigger rate--it goes faster than the blue one. 00:01:46.100 --> 00:01:54.000 But they always go in the same direction; so direct variation is the same direction--one goes up; the other goes up; one goes down; the other goes down. 00:01:54.000 --> 00:01:57.200 They are linked; they move in lockstep. 00:01:57.200 --> 00:02:01.700 We see this kind of relationship in a lot of situations; here is a common, everyday example. 00:02:01.700 --> 00:02:11.200 Say it costs you two dollars to buy a loaf of bread; if you buy 10, 10 would cost \$20; and if you bought only 2 loaves, it costs \$4. 00:02:11.200 --> 00:02:15.000 The more bread you buy, the more cost; the less bread, the less cost. 00:02:15.000 --> 00:02:19.400 The total cost of the bread and the number of loaves you buy are in direct variation. 00:02:19.400 --> 00:02:23.600 It seems kind of obvious; but direct variation is actually a pretty simple concept to get around. 00:02:23.600 --> 00:02:29.500 It is just two things that are linked: one goes up; the other one goes up; one goes down; the other goes down--direct variation. 00:02:29.500 --> 00:02:35.200 There are lots of different ways to say that two things (let's call them x and y) are in direct variation. 00:02:35.200 --> 00:02:40.300 We could say "x and y are in direct variation"; we could say "x and y vary directly," 00:02:40.300 --> 00:02:47.300 "y varies directly as x," which is to say "as x would do something," but we don't know what x is doing yet until later on; 00:02:47.300 --> 00:02:54.000 so it is "y varies directly as x"; "x and y are directly proportional," "y is directly proportional to x." 00:02:54.000 --> 00:02:58.100 There are lots of different ways to say it; but they all mean the exact same thing mathematically. 00:02:58.100 --> 00:03:05.100 In mathematics, it means y = k times x, where k is a constant. 00:03:05.100 --> 00:03:10.500 k is just a constant; it is called the proportionality constant or the constant of variation. 00:03:10.500 --> 00:03:13.200 It is the rate at which the two things are connected. 00:03:13.200 --> 00:03:16.200 Remember, with the arrows, the red arrow is bigger than the blue arrow; 00:03:16.200 --> 00:03:22.100 there is some rate that says the red arrow grows faster than the blue arrow, whether it is going up or going down. 00:03:22.100 --> 00:03:27.000 In the previous example about loaves of bread, k would be representing the price of one loaf. 00:03:27.000 --> 00:03:31.600 That is the rate of connection between loaves of bread and the cost of the bread. 00:03:31.600 --> 00:03:40.200 It is the price; price is what is connecting them--that would be our rate k for that loaves of bread example--it would be based on the price. 00:03:40.200 --> 00:03:48.000 All right, inverse variation: the idea of inverse variation is the opposite of direct variation--not a big surprise there. 00:03:48.000 --> 00:03:57.600 It says that two things are inversely related to each other; this means that if one goes up, the other one goes down. 00:03:57.600 --> 00:04:04.500 But if one goes down, the other one has to go up. 00:04:04.500 --> 00:04:08.800 They are usually going to go at different rates; once again, they have different growths in the red and blue arrow. 00:04:08.800 --> 00:04:15.100 But they are going in opposite directions; that is the really key idea to get across from inverse variation. 00:04:15.100 --> 00:04:25.200 It is that in inverse variation, they go in opposite directions: if one grows, the other one shrinks; if one shrinks, the other one grows. 00:04:25.200 --> 00:04:33.500 And also, I want to warn you here: inverse variation is not related to inverse functions--it is not connected to inverse functions. 00:04:33.500 --> 00:04:39.500 It is based on the idea of multiplicative inverses, those reciprocals, like 3 and 1/3. 00:04:39.500 --> 00:04:44.400 3 and 1/3 are multiplicative inverses, because they cancel each other out; so that is what we are talking about. 00:04:44.400 --> 00:04:48.300 We are not talking about functions canceling each other out--that kind of inverse function; 00:04:48.300 --> 00:04:54.100 we are talking about the multiplicative inverse; that is where inverse variation is getting its term from. 00:04:54.100 --> 00:04:59.200 All right, this idea comes up a lot less in everyday situations; we aren't going to see it with bread. 00:04:59.200 --> 00:05:03.000 But it is pretty common in the sciences--for example, consider gravity. 00:05:03.000 --> 00:05:07.100 Gravity is an inverse variation effect with distance. 00:05:07.100 --> 00:05:12.200 The force of gravity that Earth exerts is inversely related to the distance from Earth. 00:05:12.200 --> 00:05:16.600 The more distance you have from the earth, the less gravity you experience. 00:05:16.600 --> 00:05:20.800 The closer you are to Earth--the less distance you have--the more gravity you experience. 00:05:20.800 --> 00:05:29.500 Gravity and distance are in inverse variation; one of them goes up; the other one goes down; the first one goes down--the other one must go up. 00:05:29.500 --> 00:05:38.600 And technically, just so we have this on the books officially, it is not actually distance that is the inverse; it is the distance squared; but it is the same idea. 00:05:38.600 --> 00:05:44.000 One is going up; the other one is going down; the first one goes down; the other one goes up. 00:05:44.000 --> 00:05:49.600 Technically, it is d², but you can learn about that in a physics class; don't worry about that right now. 00:05:49.600 --> 00:05:53.900 Inverse variation--there are lots of different ways to call it out, just like direct variation has many names. 00:05:53.900 --> 00:06:00.800 We can say "x and y vary inversely," "y varies inversely as x," "x and y are inversely proportional," 00:06:00.800 --> 00:06:06.600 "y is inversely proportional to x," or "x and y are in inverse variation." 00:06:06.600 --> 00:06:14.700 And we also will sometimes call the relationship reciprocal proportion (remember, because we are talking about flips, the 3 to 1/3 kind of reciprocal). 00:06:14.700 --> 00:06:20.600 Or confusingly, sometimes, once in a while, you might hear somebody call it indirect variation. 00:06:20.600 --> 00:06:29.600 This is kind of weird, because direct/indirect...I think it is kind of weird; I think "inverse" and "reciprocal" really get the idea across much better than "indirect." 00:06:29.600 --> 00:06:33.700 But once in a while, we will hear people use that; so it is important that we are aware of it. 00:06:33.700 --> 00:06:38.600 In any case, even if it has all of these different names, they all mean the exact same thing mathematically. 00:06:38.600 --> 00:06:46.300 y = k/x; and once again, k is a constant--it serves the exact same purpose as it did for direct variation. 00:06:46.300 --> 00:06:49.800 It gives the rate at which the two things are connected. 00:06:49.800 --> 00:06:55.900 How much is that blue arrow going to grow or go down, depending on how much the red arrow is changing? 00:06:55.900 --> 00:07:04.100 The red arrow and blue arrow, on the last one, where we talked about inverse variation--they didn't go at the same speed; they went at different speeds. 00:07:04.100 --> 00:07:08.500 The red one grew a lot, and the blue one went down a little bit. 00:07:08.500 --> 00:07:14.000 The red one went down a lot, and the blue one went up a little bit; so we have that same rate thing going on with this k. 00:07:14.000 --> 00:07:20.000 And if we wanted to, we could also see this as y = k times 1/x. 00:07:20.000 --> 00:07:24.500 That would be perfectly reasonable, as well; but I prefer k/x, just because it seems a little more compact. 00:07:24.500 --> 00:07:27.000 But they are really just equivalent statements. 00:07:27.000 --> 00:07:34.500 OK, joint variation: if we want, we could have multiple direct variations going on at the same time. 00:07:34.500 --> 00:07:41.400 If we want to do this, we use the term joint variation; so we could say "z varies jointly as x and y," 00:07:41.400 --> 00:07:46.200 which says that z has a direct variation with x and a direct variation with y. 00:07:46.200 --> 00:07:52.000 Or, we could also say that z is jointly proportional to x and y; these are our two ways of saying it. 00:07:52.000 --> 00:07:56.100 But they are both going to mean the same thing: z = k times x times y. 00:07:56.100 --> 00:08:03.600 And once again, k is a proportionality constant; it is the thing that is linking the rates of change between various things. 00:08:03.600 --> 00:08:12.800 Now, really quickly: you might wonder, "Wait a second; if z is connected to x by direct variation, then that would be z = kx." 00:08:12.800 --> 00:08:26.300 But then, we also have direct variation with y, so that is z = ky; so let's use a different letter; let's call them k₁ and k₂. 00:08:26.300 --> 00:08:31.300 So, k₁ and k₂ are both constants; we are going to need different constants, because x and y are different things. 00:08:31.300 --> 00:08:42.500 If we put these together, we would really get z = k₁ times k₂ times xy. 00:08:42.500 --> 00:08:48.900 We can put those constants together, k₁ and k₂; so why is it that is shows up as just k, as opposed to 2 constants? 00:08:48.900 --> 00:08:56.700 Well, remember: if k₁ and k₂ are both constants, then that means that, 00:08:56.700 --> 00:09:04.300 when we multiply them together, k₁ times k₂ is constant, as well. 00:09:04.300 --> 00:09:08.900 So, if k₁ times k₂ is just a constant, then that means... 00:09:08.900 --> 00:09:13.500 let's just give it a new name and say k₁ times k₂ equals just k. 00:09:13.500 --> 00:09:18.100 And that is why we only have to see k there--because it is two constants combined into one constant. 00:09:18.100 --> 00:09:21.400 So, we aren't going to have to worry about keeping track of two separate constants. 00:09:21.400 --> 00:09:25.800 We can just merge it into a single proportionality constant; and that is why we only have the one of them. 00:09:25.800 --> 00:09:30.400 So, we have direct variation; we have joint variation (multiple direct variations going on at once)-- 00:09:30.400 --> 00:09:36.100 it is all of the things we are varying with one proportionality constant, k; great. 00:09:36.100 --> 00:09:44.000 We can also combine multiple types of variation; we can stack direct and inverse variation if we have these sorts of relationships going on simultaneously. 00:09:44.000 --> 00:09:50.000 For example, here is a gas law from chemistry and from physics; you can probably learn it in both courses, depending on how the course is taught. 00:09:50.000 --> 00:09:57.100 Given a fixed amount of gas (which is to say, just air, like the kind of air in a room, not "gas" like gasoline/petroleum)-- 00:09:57.100 --> 00:10:02.300 given a fixed amount of air or gas in a container, the pressure, P, of the gas 00:10:02.300 --> 00:10:09.500 varies directly as the temperature, T, and inversely as the volume, V. 00:10:09.500 --> 00:10:15.600 So, the inverse thing would show up down here by the "divide by V"; and our direct would be just T. 00:10:15.600 --> 00:10:25.600 So, it is k times T for the direct, and then k/V for the indirect; and once again, we put them together; 00:10:25.600 --> 00:10:31.400 and they just combine, and they become a single constant of variation. 00:10:31.400 --> 00:10:36.700 So, we just get a single constant, k; so we just put the various kinds of variation that we have...we just stack them all together. 00:10:36.700 --> 00:10:46.200 We do them all at once, and we put only one single constant, k; and that will be enough, for the exact same reasons we talked about on the previous slide. 00:10:46.200 --> 00:10:49.600 All right, that is everything; we have everything we need to get to the examples. 00:10:49.600 --> 00:10:57.400 a and b are in direct variation; if a and b are in direct variation, we know that a = kb. 00:10:57.400 --> 00:11:02.000 Or, we could also write it as b = ka; but notice that they will end up having the same effect. 00:11:02.000 --> 00:11:07.800 When a equals 13, b equals 5; so let's plug this in, and let's find out what k is. 00:11:07.800 --> 00:11:13.600 If we want to know what b is when a equals 52, we are going to need to know what k is, to be able to figure that out. 00:11:13.600 --> 00:11:22.400 We start by figuring out what k is: 13 = k(5); so that means that 13/5 = k. 00:11:22.400 --> 00:11:32.000 Great; if that is the case, we can go back, and we have the a = 13/5k; now we plug in a = 52, and we figure it out for a different b. 00:11:32.000 --> 00:11:45.400 If a is 52, then we have 13/5 times k; we multiply by 5 divided by 13 to cancel out that fraction on the right side; we get (5/13)52 =... 00:11:45.400 --> 00:11:52.700 oh, oops, sorry; not k; I made a mistake; a = 13/5 times b; my apologies. 00:11:52.700 --> 00:12:01.700 So, we plug in...now we are trying to figure out what b is; 5/13 times 52...13 actually goes directly into 52; 00:12:01.700 --> 00:12:12.100 52 can be broken up into 13 times 4, so we can come along and cancel this 13, and we just 5 times 4, which is 20. 00:12:12.100 --> 00:12:17.600 So, 20 is what b is when a is 52; great. 00:12:17.600 --> 00:12:25.500 In the next one, x and y are in inverse variation; if they are in inverse variation, we have y = k/x. 00:12:25.500 --> 00:12:29.800 We could also do it as x = k/y; it will have the same effect--we end up having different k's, 00:12:29.800 --> 00:12:32.900 but the important part is that we have this setup of the inverse like this. 00:12:32.900 --> 00:12:42.500 When x = 72, y = 3; once again, we want to figure out what the k is, because the next step is to figure out what x is when y is 24. 00:12:42.500 --> 00:12:44.600 So, we will need that information about what k is. 00:12:44.600 --> 00:12:51.800 So, x we put in as 72; y we put in as 3; so 3 = k/72. 00:12:51.800 --> 00:12:58.400 Multiply both sides by 72; it cancels out the denominator on the right side; 72 times 3 equals k. 00:12:58.400 --> 00:13:06.100 We use a calculator, and we get 216 = k; you probably don't even need a calculator for that one--you might be able to do that one in your head. 00:13:06.100 --> 00:13:16.900 y =...replace the k now; y = 216/x, so now we put in y = 24; 24 = 216/x. 00:13:16.900 --> 00:13:24.200 So, we want to figure out what x is; we multiply the x over on the left side, and we divide by the 24; we get 216/24. 00:13:24.200 --> 00:13:32.200 You probably want to plug that one into a calculator; and we will get x = 9. 00:13:32.200 --> 00:13:34.500 And there we are--our answers; great. 00:13:34.500 --> 00:13:39.200 The next example: p varies jointly as m and n; what does that mean? 00:13:39.200 --> 00:13:49.500 "Jointly as m and n" means that these two are going to be on the right; so p = k (we have to have that proportionality constant), times m, times n. 00:13:49.500 --> 00:13:56.100 When m equals 4 and n equals 8, p equals 2; then we want to find out what n is when p equals 60 and m equals 24. 00:13:56.100 --> 00:14:00.600 So, once again, we need to figure out what that k is, what that proportionality constant is, 00:14:00.600 --> 00:14:04.400 if we are going to be able to figure out what n is in the second half of the question. 00:14:04.400 --> 00:14:17.600 So, we plug in all the values that we know from the beginning; we have p = 2; 2 = k times m at 4, and n at 8; 2 equals k times 32. 00:14:17.600 --> 00:14:26.600 Divide by 32; 2/32 = k; 1/16 = k; great. 00:14:26.600 --> 00:14:31.500 So, with that information, we can now take this; we will create a new thing that tells us the relationship in general. 00:14:31.500 --> 00:14:42.200 p = 1/16 times m times n--great; we know that p is 60; we know that m is 24 for the one where we want to figure out what the n is. 00:14:42.200 --> 00:14:51.900 So, we plug in p = 60; that equals 1/16; times...we don't know what n is, but we do know what m is: m is 24 times n. 00:14:51.900 --> 00:15:03.200 Now notice: there are some common factors between 16 and 24; 24 can be broken down into 8 times 3; 16 can be broken into 8 times 2. 00:15:03.200 --> 00:15:11.800 So, the 8's cancel out, and we are left with 60 = 3/2 times n. 00:15:11.800 --> 00:15:20.000 We can now multiply both sides by 2/3; 2/3 times 60 = n. 00:15:20.000 --> 00:15:27.900 So now, that cancels out the fraction on the right side; we can break down 60 into 3 times 20. 00:15:27.900 --> 00:15:33.700 The 3's cancel out, and we have 2 times 20, which is 40, equals n. 00:15:33.700 --> 00:15:39.300 There is our answer; when p is 60 and m is 24, n has to be 40. 00:15:39.300 --> 00:15:44.400 The next example: all right, this one is a little more complex, because it is a word problem; but we will be able to work through it, actually. 00:15:44.400 --> 00:15:53.000 Hooke's Law connects the force of a spring to its compression; it says that the distance, x, that a spring is compressed or stretched 00:15:53.000 --> 00:15:58.700 from its equilibrium natural length is directly proportional to the force, F, of the spring. 00:15:58.700 --> 00:16:04.200 First, let's understand what that means: the very first thing you should do, if you read a word problem, 00:16:04.200 --> 00:16:06.900 and you are not sure what a word means--go look it up. 00:16:06.900 --> 00:16:13.600 Either (if it is one of the words that you learned because it was in that lesson) go check what that definition meant in your math book, 00:16:13.600 --> 00:16:19.000 or from this lesson; or if it is just a word that you don't know, that didn't show up in the math before, just look it up in a dictionary. 00:16:19.000 --> 00:16:26.700 Equilibrium: say this is a spring--equilibrium is just what it is when it is at rest, when it is not being compressed or stretched. 00:16:26.700 --> 00:16:32.500 It says that the distance, x, a spring is compressed--if we push it in by some amount x, it compresses it. 00:16:32.500 --> 00:16:36.600 And then, that causes the spring to push back with some force, F. 00:16:36.600 --> 00:16:46.400 And this law, Hooke's Law, says that the force, F, is equal to k times that distance. 00:16:46.400 --> 00:16:54.900 It is directly proportional; so it has that proportionality constant, times the thing that it is directly proportional to (direct variation). 00:16:54.900 --> 00:17:02.200 F = kx; now, once again, once we have that part figured out, it is just a normal problem from there on. 00:17:02.200 --> 00:17:08.200 Say a spring is stretched by .1 meters and has a force of 95 Newtons when stretched that far. 00:17:08.200 --> 00:17:12.600 And Newtons is just the unit of force in the metric system--just force. 00:17:12.600 --> 00:17:15.100 And then, what force would it have if stretched by .25 meters? 00:17:15.100 --> 00:17:22.500 Well, .1 meters is just x = 0.1, and force equals 95 for the first part of the problem. 00:17:22.500 --> 00:17:32.200 For the second part of the problem, we have x = 0.25; and we want to find out what the force is: F = ? 00:17:32.200 --> 00:17:37.700 So, we see this diagram; we push the spring in by some amount; and that will cause a force to appear, 00:17:37.700 --> 00:17:42.400 in reaction to that, where the spring is pushing back; and it will depend on the amount we have compressed it. 00:17:42.400 --> 00:17:46.500 If you take a spring in your hands, and you push a little bit on the spring, it pushes a little bit back. 00:17:46.500 --> 00:17:52.400 If you push the spring really, really heavily, it pushes really, really hard back; and that is why we have F = kx. 00:17:52.400 --> 00:17:57.500 How hard it pushes back is directly connected to how much it has been compressed (or stretched). 00:17:57.500 --> 00:18:00.300 So, what force would it have if it was stretched by .25 meters? 00:18:00.300 --> 00:18:03.700 To figure out that, we are going to first have to figure out what k is. 00:18:03.700 --> 00:18:12.700 So, we do that by plugging in our numbers; our force is 95 when it has been stretched by 0.1. 00:18:12.700 --> 00:18:22.800 We divide both sides by 0.1: 90/0.1 = k; so that just moves the decimal place over 1, and we get 950 = k. 00:18:22.800 --> 00:18:29.800 So, that tells us, for this specific spring, the spring that this problem happens to be about--it has a constant of 950. 00:18:29.800 --> 00:18:37.200 It has a coefficient, a proportionality coefficient/constant, of 950 (technically, 950 Newton-meters--it is just that it has units). 00:18:37.200 --> 00:18:41.700 But we don't have to worry about that; 950 = k is perfectly enough. 00:18:41.700 --> 00:18:49.000 So, 950 = k is what it is for this spring; however, this is true, in general, for any spring: F = kx is a very good rule 00:18:49.000 --> 00:18:52.800 for describing any spring that can be compressed or stretched. 00:18:52.800 --> 00:18:55.500 But this specific spring that we are working with has 950. 00:18:55.500 --> 00:19:01.200 A smaller, easier-to-push-around spring would have a smaller k, and a harder, thicker, heavier, 00:19:01.200 --> 00:19:04.300 harder-to-push-around spring would have a larger k. 00:19:04.300 --> 00:19:12.900 So, 950 = k for this one; so now we have F for this specific spring: F = 950x. 00:19:12.900 --> 00:19:23.600 So, what is the x that we have for the second half of the question? x = 0.25, so F = 950(0.25). 00:19:23.600 --> 00:19:33.300 We plug that into a calculator; and 1/4 of 950 is 237.5; what are the units that we are using? 00:19:33.300 --> 00:19:44.400 We were told that the unit of force in the metric system is the Newton; so it is 237.5 Newtons. 00:19:44.400 --> 00:19:47.400 And there is our answer--great! 00:19:47.400 --> 00:19:55.200 All right, the last example: The maximum load a horizontal beam can support, if held up at both ends, 00:19:55.200 --> 00:20:02.400 is jointly proportional to the width of the beam and the square of its depth, while inversely proportional to its length. 00:20:02.400 --> 00:20:05.800 Wow, let's see if we can figure out what that means before we try to do this. 00:20:05.800 --> 00:20:14.200 First of all, we have some horizontal beam; we have some beam, and it is being supported at both ends. 00:20:14.200 --> 00:20:25.400 It says "the maximum load a horizontal beam can support..." so there is some big, heavy weight in the middle of it. 00:20:25.400 --> 00:20:34.000 And it is being supported on both ends; it is being held up just at the ends, and it is able to support an amount. 00:20:34.000 --> 00:20:44.000 The maximum load is jointly proportional to the width of the beam and the square of its depth, while inversely proportional to its length. 00:20:44.000 --> 00:20:48.800 What is length, width, and depth? Well, let's take another look at the beam. 00:20:48.800 --> 00:21:00.300 We could have a beam like this; its depth is how far down it goes. 00:21:00.300 --> 00:21:10.200 Here is d, the depth; we could also talk about what its width is; its width is going to be 00:21:10.200 --> 00:21:15.800 (sorry, that was just me trying to make an arrow pointing at the w, but it ended up looking like a w, as well) 00:21:15.800 --> 00:21:26.900 width...it is jointly proportional to the width; and then finally, the length of it is there. 00:21:26.900 --> 00:21:33.300 Now, this makes sense; it would make sense that it is going to be inversely proportional. 00:21:33.300 --> 00:21:40.700 The farther out the beam gets...we have a beam like this that is the exact same thickness and depth and everything. 00:21:40.700 --> 00:21:44.200 And we support it at the two ends; it makes more sense that it is going to be easier to snap it in the middle. 00:21:44.200 --> 00:21:48.700 The farther and farther we stretch it out on the sides, the easier it is going to be to snap it in the middle. 00:21:48.700 --> 00:21:53.400 On the other hand, if it is all the same thing, and it is a very short thing, and it is very stout, 00:21:53.400 --> 00:21:59.000 it is going to be able to support a lot more load, if it is supported very, very closely, like this. 00:21:59.000 --> 00:22:08.000 So, that makes sense; it also makes sense that, if it is wider (there is more stuff there), then it is going to be able to support more. 00:22:08.000 --> 00:22:10.000 And its depth...it is going to be able to support more. 00:22:10.000 --> 00:22:14.700 Now, it turns out that it is not just the depth, but the square of the depth. 00:22:14.700 --> 00:22:20.900 So, we have to integrate that into our formula; so let's figure out how we can turn this into just a formula in math. 00:22:20.900 --> 00:22:28.400 The maximum load is equal to...the first thing we have to do is put in that proportionality constant, k. 00:22:28.400 --> 00:22:35.800 k times...what is it? It is jointly proportional to the width of the beam and the square of its depth, 00:22:35.800 --> 00:22:43.500 so times the depth squared, because it is the square of the depth; divided by (because it is inversely proportional to) the length. 00:22:43.500 --> 00:22:46.900 This gives us a formula for figuring out maximum load. 00:22:46.900 --> 00:22:51.100 We would have to know what k is to be able to use this formula; but we have a formula for doing that. 00:22:51.100 --> 00:22:56.300 All right, now let's continue with the problem and see if we can use that to figure out the rest of these questions. 00:22:56.300 --> 00:23:04.700 OK, so now that we have our maximum load formula here, if a given beam can support a maximum load of 750 kilograms, 00:23:04.700 --> 00:23:14.800 how much could it support if its length is tripled, or its width is doubled, or its length is doubled, its width is halved, and its depth is tripled? 00:23:14.800 --> 00:23:17.500 Wait a second; we don't have any information! 00:23:17.500 --> 00:23:21.900 The way we did all of these previous problems was that they gave us enough to figure out k, 00:23:21.900 --> 00:23:24.000 and then we used k to figure out the rest of these problems. 00:23:24.000 --> 00:23:26.800 We don't have any specific numbers to work with; so what are we going to do? 00:23:26.800 --> 00:23:31.900 Well, the first thing we need to do is not get scared; we have this problem, so there is a good chance that we can solve it. 00:23:31.900 --> 00:23:34.800 So, let's just try the things that we normally try with word problems. 00:23:34.800 --> 00:23:37.300 Let's try to just name things that we aren't sure of, at least. 00:23:37.300 --> 00:23:45.100 We don't have a specific number for the length that it is; a given beam must have a length; it must have a width; and it must have a depth, at first. 00:23:45.100 --> 00:23:47.600 We don't know what they are, but we can still give them names. 00:23:47.600 --> 00:23:51.600 This is a great thing to do--to just give names to the things that you don't know. 00:23:51.600 --> 00:24:03.800 So, let's say, right from the beginning, that we will name its initial width w<font size="-6">i</font>, its initial depth d<font size="-6">i</font>, and its initial length l<font size="-6">i</font>. 00:24:03.800 --> 00:24:17.400 These all end up being the initial width, depth, and length, respectively. 00:24:17.400 --> 00:24:20.600 Now, what do we know when we use the initial width, depth, and length, respectively? 00:24:20.600 --> 00:24:33.000 We know that 750, the maximum load, is equal to k (we don't know what k is), times w<font size="-6">i</font>, times d<font size="-6">i</font>², over l<font size="-6">i</font>. 00:24:33.000 --> 00:24:34.800 Well, wait; I still can't figure out what k is. 00:24:34.800 --> 00:24:38.200 So, what are we going to do? Once again, don't get scared yet. 00:24:38.200 --> 00:24:43.200 Let's actually try some of these out: the first one is if its length is tripled. 00:24:43.200 --> 00:24:49.700 So, if its length has tripled, we are going to have a different thing than l<font size="-6">i</font>; but it is going to be connected to l<font size="-6">i</font>. 00:24:49.700 --> 00:24:55.900 If we triple the length of l<font size="-6">i</font>, it is going to be 3l<font size="-6">i</font>, so it is going to be 3 times l<font size="-6">i</font>. 00:24:55.900 --> 00:25:08.000 What is it going to be if we have k times d<font size="-6">i</font> times w<font size="-6">i</font>, over l<font size="-6">i</font>(3l<font size="-6">i</font>)? 00:25:08.000 --> 00:25:16.600 Well, we don't know what that is; but oh, that looks a lot like this; that is just 1/3--we can pull that 1/3 out. 00:25:16.600 --> 00:25:26.600 We have 1/3 times (k times w<font size="-6">i</font> times d<font size="-6">i</font>)², over l<font size="-6">i</font>. 00:25:26.600 --> 00:25:34.900 We already know what this is; that is just 750, so it is 1/3 times 750; 1/3 times 750 is 250. 00:25:34.900 --> 00:25:38.100 What is the unit we are working with? We are working with kilograms as our unit. 00:25:38.100 --> 00:25:43.400 So, the maximum load, if we were to triple the length of this beam, would be 250 kilograms. 00:25:43.400 --> 00:25:46.300 Great; so now we have an understanding of how this is working out. 00:25:46.300 --> 00:25:50.200 We just plug it in, and we can use what we have here; we can use the information that we have. 00:25:50.200 --> 00:25:55.300 We don't have to know all of the numbers; knowing just one of them is enough, because we have a general form that it is working in. 00:25:55.300 --> 00:26:08.500 When the width is doubled, it would be 2 times the initial width, w<font size="-6">i</font>; so that would be k times 2w<font size="-6">i</font> times d<font size="-6">i</font> squared, over l<font size="-6">i</font>. 00:26:08.500 --> 00:26:18.200 So, we pull that 2 out; we have 2 times (k times w<font size="-6">i</font> times d<font size="-6">i</font>) squared, over l<font size="-6">i</font>. 00:26:18.200 --> 00:26:27.800 We plug in what we know that all is; that is 750, so it is 2(750), or 1500 kilograms; that would be the maximum. 00:26:27.800 --> 00:26:36.600 Great; on the last one, this is a lot of things: length is doubled (that would mean 2l<font size="-6">i</font>); 00:26:36.600 --> 00:26:46.700 width is halved (which would mean 1/2w<font size="-6">i</font>); and depth tripled would mean 3d<font size="-6">i</font>. 00:26:46.700 --> 00:26:57.800 OK, so how does this come out? We have k times w; what w are we using? We are using 1/2w<font size="-6">i</font>. 00:26:57.800 --> 00:27:03.100 Times d...what is our new d? It is 3d<font size="-6">i</font>; and this is important. 00:27:03.100 --> 00:27:09.500 Remember, we are not plugging in for just 3d<font size="-6">i</font> squared; we are plugging in all of this--it is what all of the new depth is. 00:27:09.500 --> 00:27:17.600 And the new depth is 3d<font size="-6">i</font>, so it goes in in parentheses: (3d<font size="-6">i</font>)², so that 3 is going to get squared, as well. 00:27:17.600 --> 00:27:25.200 Divided by 2l<font size="-6">i</font>: so k times 1/2...let's pull the 1/2 down; so we will get w<font size="-6">i</font> up here, 00:27:25.200 --> 00:27:30.500 but it will be divided by 4l<font size="-6">i</font> times 9d<font size="-6">i</font>². 00:27:30.500 --> 00:27:39.400 So, we can pull out the coefficients, and we will get 9/4k times w<font size="-6">i</font> times d<font size="-6">i</font>², over l<font size="-6">i</font>. 00:27:39.400 --> 00:27:44.300 We know what that one is; that one is 750; so 9/4 times 750... 00:27:44.300 --> 00:27:55.500 plug that into a calculator, and we get 1,687.5 kilograms; and there we are. 00:27:55.500 --> 00:28:02.000 One thing I would like to point out: notice that depth, by far, matters the most, because it is depth squared. 00:28:02.000 --> 00:28:05.500 So, you can get a lot more load by just having a larger depth. 00:28:05.500 --> 00:28:10.100 You have to increase width a lot more than increasing depth, because depth gets squared. 00:28:10.100 --> 00:28:14.900 This is why, if you have ever worked in roofing, or anything where you see a beam supporting a long horizontal length, 00:28:14.900 --> 00:28:17.600 if you get the chance to look up inside of an attic or inside of a roof, 00:28:17.600 --> 00:28:21.700 at what is actually holding up the structures, you will notice that the beams aren't flat like this. 00:28:21.700 --> 00:28:28.500 They are supported like this, so that they can have the most depth, because it is the depth squared that makes the strength. 00:28:28.500 --> 00:28:34.500 So, they are always supported on a long, deep kind of axis, because that gives them the most strength. 00:28:34.500 --> 00:28:37.000 So, if you have done construction, you have actually seen this before. 00:28:37.000 --> 00:28:43.500 You have seen something where you realize that what you are seeing there is mathematics in effect in the real world--pretty cool. 00:28:43.500 --> 00:28:47.500 All right, I hope variation makes sense; and we will get started on the next section in the next lesson. 00:28:47.500 --> 00:28:49.000 All right, see you at Educator.com later--goodbye!