 Hi, this is Gichu. Welcome to ASMR math. Now, what I'd like to do in this video is do a small maximizing revenue problem. Okay, and this is a concept that is basically talked about in my part world anyway in grade 11 where they're talking about quadratic functions and you learn what quadratic functions are and how to graph them and what they represent, right? They're basically parabolas and in series 3 of the language of mathematics series 3a and 3b, we did a ton of videos regarding this stuff So if you're unfamiliar with this concept, specifically quadratic functions and stuff and polynomials and factoring, there's a lot of videos there you can build up, but you know, we're gonna do a simple problem So you should be okay as long as you know your basic algebra regarding this topic, okay? Now what we're going to look at specifically is something that takes place, it's basically economics, right? mathematics of economics, right? And basically it's something that takes place in our society, in our present economic system, which is basically companies look at a certain product and they try to figure out how to maximize their revenue, right? What's the highest price that they can sell something at where where it can be absorbed, right? Where the consumers will not freak out because the price is too high, right? And they won't be selling it at a really low price where they're not making enough revenue to make profits so they can invest in R&D and allow the company to grow and give bonuses or whatever else commitments they have to people, right? So basically what we're going to do is take a look at a system that exists, a model that exists within our economic system, which as far as I know every company uses maybe not when they're first starting out, right? But very quickly, most companies try to maximize their revenue by setting a price on a certain product that the industry can absorb, right? And this is the way we're going to approach it. Basically we're going to look at how we figure out what revenue is, right? And figuring out what revenue is is pretty simple, right? If you, you know, if I have, here, I'm going to grab this example that I use with my students all the time, right? Just to just to make it understood how simple calculating revenue is, right? Let's say we have five pens, right? And I'm going to sell each pen to you for two dollars, right? Five pens. How much is that going to cost you? Ten dollars, right? Five times two. The price of the item and the number of items decides the money you're getting in, what you're, what you're generating, right? And you can think about this as selling anything, right? It could be DVDs, it could be CDs, it could be comic books, it could be cars, it could be food, right? It doesn't have to be per item. It could be per weight, right? That's where food comes in a lot, right? A lot of stuff that you're buying is per weight, per gram, per kilogram, per per pound or whatever it might be, right? So basically you can think about it as your revenue, right? I'm going to call this R. Is equal to the number of items you've sold and how much you sold them for, right? So number of items, number items times the price, right? That's revenue. Okay. Now, in mathematics we're not going to go number of items and dollar sign to represent how much you're selling something for, right? We want to come up with numbers. We want to come up with variables. We want to be able to quantify this concept, right? So we come up with symbols to represent each one of these, right? And in series 3 and 3b we talked about functions, right? Polynomial functions and stuff like this. Basically the way we represent revenue, right? We're going to represent revenue as R of X, okay? And X is being what the revenues dependent on, right? And there's going to be an X in the number of items sold and there's going to be an X in the cost, the price that we're selling each item for, right? So X is something that spans all of our basically distinct concepts here, right? That's what everything is going to be dependent on, okay? So R of X, which is basically function notation, right? Revenue is dependent on X is going to be based on number of items, right? So we're going to call this I of X number of items and that's going to be based on X or variable times, right? The price. Let's call that P of X. That's how much we're going to sell each item for, right? Now to come up for equations like this, the way it works like, you know, like we said, if I have five pens, right? If I have five pens and I'm going to sell each one for $2, it's going to be $10, right? So my revenue, let's use this black one. So my revenue is going to be the number of items is going to be five pens, right? Times the price that I'm selling for, which is $2, right? So my revenue is going to be equal to $10, right? Oops, we're way below here. So let's make this a little bit smaller. So this is going to be $5 times $2, which is, or five items times $2 is going to be $10. That's how much we generated, or you put a symbol up here, $10. Now the way maximizing revenue problems work, how companies decide how much you're going to sell an item for, how do I know that I'm going to sell each one of these for $2, right? How do I know that people are willing to buy this for me for $2 each, right? The way they do this in general is they run marketing campaigns, right? First of all, they figure out how much a cost costs them to make this item. They figure out their overhead. They figure out shipping. There's a lot of other factors involved, right? So there's a cost to this item that includes a whole bunch of different other factors, right? That depends on the raw materials, shipping expenses, salaries, rent, whatnot, power, electricity, right? So let's assume there's a cost associated for this, right? Once they figure out the cost associated for this, they try to figure out how much profits they can, you know, the industry is able to absorb, right? And then they price their item. For me, I'm pricing this item at $2, right? I'm not telling you how much it costs me to make this, right? I'm just saying, I'm just saying I want to sell this to you for $2. Now, what if I have more commitments? I want to either give out dividends. I want to give bonuses to everybody. I want to spend some money on R&D. And $2 is not enough for me, and I want to know how much I can increase the price of this item for to get in more revenue, right? Or I just basically want to look at how much can I price this item for, these pens for, to maximize my revenue, right? So the way this works is this. Let's take this down. And we're going to write down our equation up top, right? Our equation is going to be this. Let's do blue. So our equation right now is this. R of X, our revenue is dependent on the number of items sold times the price that we're selling, right? P of X. So we can't do anything with this right now. What we need to do is come up with an equation. We need to come up with an equation for the number of items. We want to have to come up with an equation for the price so we can vary it, we want to add variability to this. Because as soon as I start changing the price on this, I'm not going to sell the same number of items, right? If I was selling each one of these for $2 and I could sell five for $2 each getting $10 in, and let's say it took me a week to do this, if I lower the price on this to a dollar, then the odds are people are going to buy this faster, sooner, right? So it might only take me a couple of days to sell five of these. If I increase the price to $3, then what's going to happen is it might take me longer to sell five pounds, right? It might take me two weeks. So there's a sweet, sweet mark there where I can either increase the price or lower the price where it will maximize my revenue by increasing. If I lower the price, I increase the number of items sold. If I increase the price, the number of items sold, the odds are decreases, right? That makes sense. Something's more expensive less people buy, something's cheaper, more people buy, as long as the quality is there, right? So we're going to vary both the price of the item and the number of items based on an equation, okay? And what we have to do is have a model to begin with, right? And you could be selling pens, you could be selling cars, food, you know, the item doesn't have to be, what you're selling doesn't have to be item based. It could be weight based, right? It could be time based. It could be your time, right? How much are you selling your time? What's the price that you charge per hour of work, right? That could be your revenue coming in. That's your money that you're making. Well, salary is usually yearly based, but that's how much money you're making working, right? So this revenue model is not dependent on any specific type of item that you're selling. It applies to all types of businesses, right? Now, instead of picking something arbitrary, let me tell you, let's do it on something that I've done in the past, things that I've sold in the past, because I wish I knew this model when I was selling this stuff, okay? Because that's something familiar and that's something I've always wondered about what the maximum price could have been that the market would have absorbed, right? I should have run a sort of an experiment which I did. Now, what we're going to do is we're going to take a look at something I did in high school, okay? In grade eight, right? For me, I've always been into fireworks and firecrackers and stuff like this, things that go boom, right? Fun things that go boom, right? So every Halloween, I used to go, you know, trek out to Chinatown and this is way back in the 80s, right? So fireworks and firecrackers and stuff like this, you sort of have to find your way to the right places to be able to buy them as a kid, especially firecrackers, because firecrackers are illegal in Canada. So you have to make deals on the black market and fireworks, there's an age limit. So you sort of have to find your way to a place where you can buy some fireworks and stuff, right? If you're a kid, if you've been to have parents with you. And one thing I did when I was in grade eight, I find myself, I would go to Chinatown basically, because everyone knew in Chinatown you could get fireworks, because in China I think fireworks are illegal, which is legit, right? If you're a kid, you want to play with fireworks, right? So I made my way to Chinatown and there was little bits of fireworks that they sold in packages, and I think they were called poppers or something, where you could buy, like it was a small box, right? So it was a, you know, small box about, it wasn't even this big, like size-wise, it would have been like this big, right? Really, really small that they sold, I can't remember how much they sold them for. I think they sold them for either 25 cents, 50 cents, or way less than a dollar, right? So let's assume these little packages were being sold for 50 cents, right? So I would buy these things for 50 cents, right? And I'm going to work in dollars, so cents are going to be decimals, right? And these things contain 100 items. So there was 100 of these little popper things in a box of these poppers, I guess, I don't know what you call them. They were packed up, there were like little bits of, you know, I opened them up and figured out what's in them, but there were little bits of paper, something like, I don't know what it was, something like, I don't know if it was gunpowder, with sulfur in there, whatever it was, that it was wrapped up in paper, and then when you threw them against the wall to my pop, right? So 100 of these little guys were in a little box like this. So I would buy these and I would go to school and throw this stuff around, you know, you're in grade A, the kid does what a kid does, right? Had a fun time with them. And then at some point, someone, you know, most of the kids in grade A, they didn't have access to go to China though, right? Especially by themselves. So some kids wanted to buy this stuff for me, right? So I was the only one that had them, so I sold these things, right? Initially what I did, I sold these things for five cents each, right? So I had 100, right? I sold these things for five cents each, right? Now, if you do the calculations for this, I bought 100 of them for 50 cents, right? That means I was paying to figure out how much it was per popper. All you do is take 50 cents and divide it by 100, right? And all that does, really, is just move the decimal place over two. So I was selling each one of these for 0.005 cents, right? Per item. Per. 0.00 per. And I ended up selling them for five cents per popper. That's a 10 times markup, right? So I was buying basically the packages for 50 cents and I ended up selling them for five bucks, right? Because 100 of these, if you're going to sell each one for five cents, all you do, you multiply the price by the number of item, right? Number of items times the price per item, right? So, and if you multiply five cents by 100, all you do is just move the decimal place over here, right? By two. And I was selling a box, right? A box for five dollars. Box, right? And this was per box, this goes with this and that goes with that, right? And what ended up happening when I was selling these little guys? I had to line up. People wanted to buy all of them and I didn't have myself to have fun with, right? So I kicked up the price. I kicked it up to 10 cents, okay? And what happened when I kicked up the price to 10 cents? There was a bunch of people that bought them, right? And there were some people that couldn't afford to buy them anymore because people were sometimes buying a bunch and throwing a bunch in, you know, it wasn't good for the teachers and stuff like this and they quickly banned these poppers. You would get detention if you were caught with them, right? So my life as someone who bought products and sold products that are profit lasted for a few weeks, building up to Halloween and I stocked up on these things. Once I realized this was happening, I went back to the Chinese store in Chinatown and I bought a whole bunch of these, right? So I stocked up on these so I could supply, supply for a while, right? And I basically went through all of them over an extended period of time. Detention was sometimes the price was worth it to use these things, but a lot of kids didn't. So this thing lasted a short period for a few weeks, building up to Halloween. But what I basically ended up doing, I kicked up the price to 10 cents, right? 10 cents. And 10 cents is 0.1, right? And what that meant, right, is 10 cents per item, for 100 items, it ended up being $10. And that's where I kept that out because I was keeping enough supply for myself and there's a lot of other kids doing well, some other kids doing this as well. So, you know, the finger was not always pointed at me, you know, Chichu was the one doing it, that could have been someone else, right? So there was multiple benefits to this endeavor, okay? Now, if I knew mathematics, or I didn't know math then, but I didn't know this type of math in grade 8, if I knew this type of math in grade 8, maybe I would have run a little analysis to see what the market could absorb, right? Maybe I could have kicked up the price to 12 cents, or 15 cents. I never tried 15 cents, actually, I don't know why. Because I was happy, I was basically, I wasn't doing this to make money, I was doing this because the situation arrived and it was paying for my own supply, right? So that was a pretty good thing to do. So what I ended up doing, stabilized at selling them for 10 cents each, and there were 100 in each box, okay? So my revenue would have been 10. 100 items times 10 cents gives you $10, okay? And that's the model we're going to start with. Now, what we have to do is we're going to assume, I was in grade 8 and I knew about this thing, maximizing revenue, and I was going to experiment with price differences, right? And this is what something that a lot of companies do, right? What you're paying for something in general is not the price they came out with right off the bat, and they decided the market could absorb that, and they started selling that. Usually, most companies run little experiments in different regions, right? They run a little experiment selling something for a certain price, run a little experiment selling something for a certain price, selling something for a certain price. They collect data and they do analysis on it, right? So what we're going to do is do it mathematically, quantify this, right? And what we're going to do is use something called a let statement in mathematics. And let statement, again, I think we talked about it in in series 3 and 3B, and it's a very powerful tool because what it does, it sets up your model. And what we're going to do, we're going to call x, right? Let, oops, let x, let x equal number of price changes, number price changes, the increment, right? How many times we're going to change the price? Okay. And what we're going to do, we're going to try to increase the price by two cents every time, right? I'm not going to do five cents because five cents, I think it's going to be too much, right? I'm not going to go selling it from 10 cents to 15 cents because that way I probably wouldn't have sold as much, right? Triple the price of initially introducing something might not go over well with my customers, right? But what I'm going to do is I'm going to start kicking off the price increments of two cents each to see how many I'm going to sell, right? So if I kick up the price from 10 cents to 12 cents, that's one price increment change, one price increment increase. And what's going to happen is the odds are I'm going to sell lecitos, right? So I just don't want to do, you know, run an experiment for a week because my time is limited. I wasn't allowed to do this for a very long, right? Rules were brought in that prevented me from doing this anymore, right? So I'm not going to run an experiment for a week selling at a 12 cents and another week selling at 14 cents, another week selling at a 16 cents. What I'm going to do is I'm going to create a model where I'm going to make the increments for the price of the item two cents a pop, right? So X is going to be the number of price changes I'm going to implement to my model, okay? Now, I was able to sell my revenue was initially 100 items at 10 cents each, right? Times 10 cents. So I was making $10. Now I'm going to change this, change this model. I'm going to introduce price hike to each popper. I'm going to see how many items I'm going to sell. Now, if I do a little time travel back to when I was in grade eight, I'm not sure, you know, what happened, how many, how many less people bought from me and how many less poppers I sold when I went from a price hike of five cents to 10 cents. But I'm going to assume for every two cents increase in price, I'm going to sell 10 less poppers, right? So if I have 100, if I was selling 100 items for 10 cents, now I'm going to sell 100 items minus 10 X, where X is a number of times I'm going to change the price. So if I change the price by two cents, if I change the price once is a two cent increment, then I'm going to sell 90 items. If I change the price twice, which is a four cents increase in price, I'm going to sell 80 items, right? Two times 10 is 20. 100 minus 20 is 80, right? Now, as far as the price goes, initially, I was charging 10 cents and we're working in decimals, right? So 10 cents is 0.1. And I'm going to kick up the price. I'm going to raise the price two cents every time. So that's going to be plus 0.02 X, right? So if I'm going to lose sales, it's minus. If I'm going to increase prices plus, right? And this is the model we have for us to figure out what the maximum price is that we can sell these little firecrackers, these things that we threw against walls or on the ground. So we're going to take this model. Okay. I'm going to erase these. I'm going to put this thing up here, right? Transfer this equation. Let's do this in black, see if the black comes out nicer. So R of X, right, is equal to 100 minus 10 X, 0.1 plus 0.02 X. Okay. Now we can get rid of this guy. Now, the way we can figure out the maximum revenue that we can sell something like this is, and the way it's going to work is going to be an upside down parabola. What we're going to do is multiply this out, foil this out. And we talked about foiling, you know, series 3A and 3B, right? So we're going to just basically expand this. This guy multiplies this and this, and this guy multiplies this and this, and we combine like terms. 100 times 0.1 is 10. 100 times 0.02 X is going to be plus 2X, right? Because all we're going to do is move the decimal place over 2. 10X, negative 10X times 0.1 is just going to move this guy over 1. So it's going to be minus X, right? Negative 10X times 0.02 X is going to be negative 0.2 X times X is X squared. Okay. Now, what we're going to do is we're going to combine like terms. This guy asked that guy, and we're going to rearrange this with the highest power up first, right? So this is going to be negative 0.2 X squared. 2X minus X is plus X and plus 10. And what we're going to have to end up doing is graphing this. And when we end up graphing this, what we're going to get, I'm going to draw a generic one and then we'll graph it and we'll see what it looks like. My one is, well, let's just give you a heads up of what it's going to look like. It's basically going to look something like this. And our maximum revenue is going to be the vertex of the parabola. Simple as that. Okay. And our X is, with our let statement, is a number of times we're going to kick up the price, right? So if our vertex ends up being, let's say 3, right? The X part is 3. What's going to happen is it means we kicked up the price 3 increments of 2, which is 6 cents. So we're selling the item for 16 cents each, right? And if you put 3 here, 3 times 10 is 30. 100 minus 30 is going to be 7. That means we're only selling 70 per box instead of 100 of them at 10 cents a pop, right? So what we need to do is do the process where this is called completing the square. Okay. And I've done a little bit of it. I did a video where we graphed the parabola, sort of speedy Gonzales style for ASMR math with completing the square that I covered. I haven't done completing the square in depth for the language of mathematics yet. I don't believe I have. I did those series 3 and 3 a long time ago. But I will at some point, I haven't covered complete, we did graph parabolas, but I didn't cover the completing square in depth. But I will at some point show you what completing the square looks like in depth with all the little variations and especially the harder problems with when we have fractions involved in this, right? But basically I'm just going to move this guy up because it's going to take a little bit of space to do this. And I want to do the whole process until we get, we put it in the form that we need to. And to be able to graph this, we need to put it in this form, right? Y is equal to AX minus P squared plus Q, right? And if you haven't seen this, this is basically the form that we can read a parabola, where the P and the Q become your vertex. And this guy tells you if it's vertical expand, he's just basically reading a roadmap really, right? It's like equation of a line. If you've done equation of a line, MX plus B, where the B is the Y intercept, M is the slope, A, P and Q have meaning in this as well, right? So let me transfer this guy up here. Here is negative 0.2 X squared plus X plus 10. Now for me, it's easier to work in fractions. I do have my calculator here, right? But I rather work in fractions. So basically graphing this thing, completing the square, that's what we need to do. And here I'm going to show you little marks for these. The way it works, we're completing a square. We have to make sure that the X squared term, there's no coefficient in front of it, right? So what I end up doing usually, I put brackets around this. And what you need to do, because you need to keep the X squared and the next term together inside the bracket, right? We're trying to force this to become a perfect square. That's why it's called completing the square, right? So what we then need to do is put brackets, contain the X squared and the X term. And then what we need to do is we've got to make sure that the coefficient in front of the X squared is 1. That means if there's anything other than 1 here, we need to take that out of the bracket. And when we take that out of the bracket, we have to compensate for this, right? Whatever number is in front of the X, here it happens to be 1, right? But we need to compensate for this because if we take right now, we're going to have this. This is going to be negative. So I'm going to do each new step in red so you see what's going on. This is going to be negative 0.2 on the outside and we have our bracket here right now, right? And the X squared is X squared now, right? Because we took this out of the bracket. But we need to compensate for this. There's a 1 here and because this is inside the bracket, right? The X is inside the bracket. That 1 changes because if we didn't change the 1, then this number multiplying come in would change that to negative 0.2. It wouldn't be 1 anymore. We need it to be the original 1. So the way you do this is you take 1 here and you divide it by negative 0.2. Now to do this, I'll just do it with fractions instead of using the calculator, okay? It's going to be 1 divided by negative 0.2, right? Actually, let me do this here so you see it. But usually with calculations and stuff like this, I always do them on the side, right? And I never erase them because that way if there's anything that goes wrong, I can see what's going on, right? So I'm going to take 1 and I'm going to divide it by negative 0.2 and this means 1 divided by negative 0.2 which means 1 divided by negative 0.2 as a fraction is 2 over 10, right? And 2 over 10 becomes 1 over 5 if we reduce the fraction, right? And division means you change it to multiplication and flip this. So this becomes 1 times negative 5 over 1, which is just negative 5. So that's what goes here, okay? This number, negative 5, I hope you see it, is negative 5 here. Because if you go 5 times 0.2, you get 1 back, right? If you want, if you want, we can prove it or we can just do it so you see it. What is all? What is all? Cover everything here so you know what complete square is like, right? So if we go 5 times negative 0.2 or negative 5, remember the sign in front of the number goes with the number and we need it to be negative because we want the original to be positive, right? So this becomes negative 5 times negative 2 over 10, 5 goes into 10 twice, negative times the negative is positive, 2 over 2 is 1, right? We've got the original 1 back. So we got this right now and what we have, we got plus 10 here, right? Now what we're going to do is the process of forming, because we want to take the inside of the brackets here and convert it to a perfect square, hence completing the square. So the next step required, what we need to do is we need to take negative 5 divided by 2 and square it. So we take this, right? The sign in front of the number always goes with the number. We take negative 5 divided by 2 and if we can, we simplify this further like it's negative 2.5 but I'm going to stick with fractions, right? Negative 5 over 2 and then we square it. I circle it and then we square it, negative 5 over 2, negative 5 times negative 5 is 25, 2 times 2 is 4 and I circle that. Now I tell everyone, when you're doing this, circle this, circle this because we're going to use both of them, okay? Now what we end up doing is we take this number and we add and subtract it inside the bracket, okay? Now because we're running out of space, I'm going to erase these, I'm going to put these numbers here so we see them so we know what they are, right? But you wouldn't erase this, right? If you're doing this piece of paper, you continue to work down here, right? We're limited with our paper, we're limited with our space here. So this becomes negative 5 over 2, we're going to be using that number and we've got 25 over 4 and that's the squared number of that, right? So what we end up doing is the process is we're going to add, let me take this, erase this, which is just negative 5, right? We don't have to circle in that, negative 5. So what we end up doing right now is taking the squared term 25 over 4 and we're going to add and subtract it inside the bracket and we're going to add and subtract it because they have to cancel each other out. The only additional number, variable, anything new that we're going to add to an equation, it has to sum up to zero because we can't arbitrarily change the equation, right? This is our equation. So the net gain has to be zero or net loss has to be zero, right? They can't be any change rate. So what we end up doing is we're going to go negative 0.2 x squared minus 5x and we're going to add and subtract 25 over 4 inside the brackets. So this guy comes in, we're going to add it and we're going to subtract this. We're going to go plus 25 over 4 minus 25 over 4 and then we're going to close our bracket because we'll close it and then plus 10 and we're going to add it first and we're going to subtract it because the addition part stays within the equation. This one, the negative one, we're going to take out of the bracket. Now if you notice, what's going to happen is we're trying to force this thing to be a perfect square. So this part, I'm going to do this in red. So this part, okay, is now perfect square. Perfect square means when you factor it, you get the same thing times itself, right? So a simple version of a perfect square would be this, right? Factoring the trinomial, two numbers that multiply to give you 4 and add to give you 4, 2 and 2, right? And we talked a lot about factoring trinomials. So this would be x plus 2 times x plus 2. If we multiply this out, expand this out, we're going to get that. A shorter way of us writing this is going to be x plus 2 squared, okay? That's what we got here. That's a perfect square and let me show you what this looks like. Two numbers that multiply together to give you 25 over 4 and add to give you negative 5 or negative 5 over 2. The reason we circled both these numbers in this process when we took this divided by 2 and squared it, we get that? The reason I circled both of these and I tell everyone, circle both of these, because that way your eye automatically sees it. You don't sit there and try to figure out what are two numbers that multiply to give you 25 over 4 and add to give you negative 5. It's difficult. It's not as easy as 4 and 4, right? So you don't have to think about it, right? In a sense it becomes, well, you know, it becomes foolproof, right? It's a guaranteed way of you getting it, because you're not going to get, and these numbers are pretty easy, pretty sweet, right? You're not going to get sweet numbers like this all the time. Sometimes you're going to get crazy numbers, decimals, you're going to get fractions that are absurd, right? So when you get those types of numbers, it's difficult to come up with what the factors are. You don't have to come up with what the factors are. It's whatever this was divided by 2, okay? The sign in front of the number goes in number. So this guy is a perfect square, and his factor is x minus 5 over 2 squared, okay? And while we're factoring this in the same step, in the same process, we're going to take this guy out of the bracket, and when it comes out of the bracket, whatever is in front of the bracket multiplies it. It has to, right? Whatever is standing guard in front of the bracket multiplies whatever is coming out of the bracket, right? So this guy multiplies by negative 0.2. So I'm going to do both these steps in the next step, right? Let's do this multiplication first and figure out what it is. That's going to be, and I might as well do it in red, negative 25 over 4 times negative 0.2 as a fraction that's negative 2 over 10, right? So let's erase this. Negative 2 over 10, and what I'm going to do now is simplify this, right? Multiplying fractions, we put out a video. The most important thing you need to know about multiplying fractions, simplify before you multiply, right? Makes life a lot easier. So 2 goes into 4 twice, 5 goes into 10 twice, 5 goes into 25 five times, right? So this becomes 5, negative and negative is positive, so it becomes 5 over 2 times 2 is 4, 5 over 4, right? That's what goes there, okay? So this becomes 10 plus 5 over 4, okay? Now this looks a little messy because I'm limited with my space, but what I'm going to do now is erase all these, and in the next line I'm just going to put down what it is, okay? So we have x minus 5 over 2 squared and we've got plus 5 over 4 here, right? Erase this. So what we got now is negative 0.2 x minus 5 over 2 squared plus 10 plus 5 over 4. Now I have to add these guys, right? Adding fractions 10 over 1, right? So common denominator is going to be 4 and I multiply that by 4, so I have to multiply that by 4. So 5, so just they have the same common denominator, so you can just add the top. So it becomes 45 over 4, right? So this just becomes 45 over 4, 45 over 4, okay? This is the equation for our revenue, for our maximizing revenue. What's going to happen if we kick up the price? What's going to happen if we lower the price? And this is R of x. Now what I'm going to do is I'm going to erase all these and I'm going to write down this equation here and then we're going to graph it so you see what's going on. And again, if you're doing this, you wouldn't be erasing all this. You keep this on board, right? So I'm going to take all this out and we got, so when we expanded this, foiled this out, completed to square, this is what we end up getting, right? Negative 0.2 x minus 5 over 2 squared plus 45 over 4. And what was our x? Our x is a number of times we're going to change the price, right? So what you end up doing when you're graphing this guy, yeah, let's do this. Actually, let's lay out the grid in green, I guess. Let's do green for the grid. Now, for graphing parabolas, what this ends up being is when you're graphing parabolas, the first thing you're going to graph is the vertex, so the vertex of this thing is the opposite sign of this, right? The x becomes 5 over 2 and the y becomes 45 over 4. And this is your x and this is your y, right? And your y is r of x. Your y-axis is your y, your r of x, right? Your function, oops, r of x, not f of x, r of x. Now, in our let statement, we said let x equal number of price changes, number of times we changed the price. Now, how much will we change in the price by? We're changing the price by two cents every time, right? So the number of times we're going to change the price is 5 divided by 2. 5 divided by 2 is 2.5. 2.5, you see this? Whoops, you're not seeing this. Let's do it here. So number of times we're going to change the price is 5 over 2, which equals 2.5, right? That's our x. So to figure out how much we increase the price by, we're going to go 2.5 times 0.02. 2.5 times 0.02. So this is going to be, should we do some fractions? Let's do some fractions. 25 over 10, right? 2.5 is 25 over 10, times 0.02 is 2 over 100, right? 2 goes into 10 five times, 25 goes into 100 four times, so this is going to be 1, 1 up top, so it's going to be 1 over 20, right? So 1 over 20, if you want to convert it to a decimal, take it to 100, right? You take it to 100, and if I multiply this 5.5 to get to 100, that becomes 5. So this becomes 0.05. I'm going to kick up the price by 5 cents. So I'm going to sell each of these poppers, right? These fire works I guess, for 15 cents. And if I do that, I increase the price two and a half times, I can figure out how many of these I'm going to sell. I'm going to go 2.5 times 10, which is 25. 100 minus 25 is 75. So I'm going to sell 75 of them for 15 cents instead of selling 100 of them for 10 cents, right? That's going to maximize my revenue. So that's going to give me more than $10, right? Because that's the maximum revenue. So what we're going to do, we're going to graph this thing. So I hope this is clear. I'm going to rewrite these again. I just wanted to do the calculation so you see what this is. This is about. Let me erase the grid as well, now that we know what the numbers are. So that was our let's statement. I'm going to erase this as well. We should know what our let's statement is now, right? Here's a grid. So my vertex is two and a half and what's this 45 over four? So we're mapping X versus R of X, right? So the grid here is this is X or X axis is X, right? And that's the number of price changes we're making. And we're making two and a half price increases. So we're increasing the price by five cents, where the total price is now going to be 15 cents. And our Y axis is going to be R of X. R of X is our revenue. So if this is our vertex, X is the number of times we're going to increase the price. That's our revenue. 45 divided by four? 45 divided by four? How many times was four going to 45? 11 times, because 11 times four is 44, right? So 11 times and we have one over four left. One over four is 0.25 is a quarter, right? Quarter 25 cents. So this becomes $11 and 25 cents. So instead of making $10, right? By selling 100 of the fireworks for 10 cents each, I could have been making $11 and 25 cents and only sold 70 of them for 15 cents each. So even though my revenue, you also have to consider this, right? Our revenue increased to $11 and 25 cents. My inventory also increased because I only sold 70 of them, not 100 of them, right? And based on how I was selling stuff, I could have sold those three as well. So this equation really doesn't take into consideration the extra inventory you're left with, right? Which is also to a certain degree profit. If it has value later on, if it depreciates right away, if it's food, for example, you can't hold on to that too long, right? Or rot, right? So that really you can't put towards being profitals only very short term, right? So let's graph this thing and see what this looks like. If we graph this thing, let's call our, you know, this is 2.5 and this is 11.25. So we're going to put this on the graph. So let's assume this is one, two, three, four, five. These are the price increases, right? So 2.5 is here, 2.5. Let's call this 5, 10, 15. Okay. So 11.5, 11.25 is around here. So this is going to be my vertex. Okay. And if you know how to graph parabolas, you're going to go, oh, okay. This is negative. So the parabola opens down, right? So this guy opens down. I usually put like little arrows here, because I really don't know where it's going to cross the y-axis yet, right? And I don't know where it's going to cross the x-axis yet, right? We can figure it out. We are going to figure it out, right? But right now we don't know yet, right? So first thing I always do is try to figure out where the x-axis is, or sorry, where it crosses the y-axis, right? And you cross the y-axis when x is zero, right? When x is zero, right? This is x is zero, right? To cross the y-axis, where you're on the y-axis, you set x is equal to zero original equation, or here. You can do it here, but that becomes more difficult. I could do it here. x is zero, this guy's gone, x is zero, that guy's gone. 100 times 10 cents is $10. That's where we cross the y-axis. So my graph is not very good, because I have a horizontal stretch happening here because the scales aren't the same, right? So I'm going to redraw this, and I'm going to go 11 and a half, I'm going to put, or 11 and a quarter I'm going to put here, and my graph is going to look like this, right? Here's $10, and that's the vertex, which is 2.5 and, oops, 2.5, 2.5, 2.5, and 11.25, okay? So we found our y-axis. Since this has to be symmetrical, parabolas are always symmetrical, I'm two and a half units away from the axis of symmetry, right? Because we talked about this, series 3A, 3B, and a little bit of ASMR math, where parabolas are symmetrical, right? So however far you are away from the axis of symmetry on a horizontal, you have to be the same distance away this way. So if this goes back two and a half, you go this way two and a half and you're at five, and that's your parabola there. Now, my drawing on the parabola isn't the best, right? I'm just trying to make this symmetrical a little bit. So this guy looks like this, and what we should do is take this guy, at least our axis here, right? Okay, and that's our x-axis. So graphically, what does this mean? If I increase the price two and a half times, I'm going to make $11.25. If I increase the price two and a half times to figure out how much I'm going to be selling the item for, I have to go back to my POVax here, which is going to be two and a half times .02 is .05, right? .05 plus .1 is going to be .15, right? So our price, POVax, originally we wrote it, hopefully you wouldn't have erased it if you're doing the work, so you're going to look back up and see what that is. Our POVax was 0.1 plus 0.02 x, right? And x is 2.5 for our maximum revenue. So you sub in 2.5 here. So this becomes times 2.5. 2.5 times this is 0.05. That plus that is 0.15. Each item is being sold for 15 cents, each one of those fireworks, right? And you can do the same type of calculation for your, or for my revenue or how many items I'm going to sell. The number of items sold, right? Again, you wouldn't erase that sentence that I just wrote there. You would keep it on your work. It would be part of your solution, most likely. The number of item sold is going to be I of x, which is going to be 100 minus 10 x and x for the vertex is 2.5, right? Again, the x is the same thing, right? We narrowed everything down to two variables of revenue and the number of price increment changes is going to be 2.5. 10 times 2.5 is 25. 100 minus 25 is 75, right? I sold 75. I would have sold 75 of those fireworks, right? The little things that go, okay. That's how you would do your, you know, a maximum revenue problem. As far as what the y-intercept means, the y-intercept means I haven't done any price changes. That means I make $10 and I sell all of it, 100. When do I not sell anything? Which is sort of, you know, according to the graph, it's sort of weird concept. But if I decrease the price, I can find my x-intercepts. Let's find the x-intercepts, right? Let's find these points. If you want to find the x-intercepts, you set your revenue equal to zero, right? You want this point and this point, right? When will I not sell any of these things, right? Where I price myself out of the market, right? That's what it means. When do I price myself out of the market? Here and here because I don't sell any. I don't make any revenue, right? R of x, y-axis is zero. So I'm going to set this equal to zero. So we're going to have that times that equals zero. So we're going to have 100 minus 10 x times 0.1 plus 0.02 x is equal to zero. So I set my revenue R of x equal to zero. So I'm trying to find my x-intercepts and we talked about this in a previous ASMR math video. We talked about the power of zero, right? The power of zero is how can you have two things multiplied together to give you zero or two or more things? The only way that's possible is if at least one of them is equal to zero, very powerful. We wouldn't be able to do mathematics without this property of zero, right? So all we do is set each one of these equal to zero, right? So I'm just going to do them separately. And again, you wouldn't be racing your work. You would do it, right? So I'm going to set each one of these equal to zero. 100 minus 10 x is equal to zero. And 0.1 plus 0.02 x is equal to zero, okay? Let's erase this. It doesn't get too much. So all we're going to do is just solve for x now, right? We bring 100 over. It becomes negative 10 x is equal to negative 100 divided by negative 10. So x is equal to x is equal to 10. So if I kick up the price, this is where 10 is, right? If I kick up the price 10 times, right? That means if I'm increasing the price by two cents every time, 10 times would be, I kick it up 20 cents, right? So now I would be selling the items for 30 cents each, right? If I sold the items for 30 cents each, my revenue would be zero because no one would be buying them. That's the way it would work, right? If I put 10 in here, maybe 10 times 0.02, 0.2, that's 20 cents, 0.2 plus 0.1 is 0.3, which is 30 cents. So if I sold the items for 30 cents, I kicked up the price 10 times. I put 10 here, 10 times 10 because I'm selling 10 less items for every two cents increase in price. 10 times 10, each increment increase, 10 times 10 is 100, 100 minus 100 is zero. I don't sell any. Oops, I priced myself out of the market, right? To figure out this number here, bring this guy over. So 0.02x is equal to negative 0.1, right? And then divide by 0.02, divide by 0.02, x is equal to, are we off the market? We're off the board. So I bring this guy over, so it becomes 0.02x is equal to negative 0.1, and I divide both sides by 0.02. So let's do this little calculation here, right? So it's going to be negative 0.1 divided by 0.02, right? So it's going to be negative 1 over 10 divided by 2 over 100, which equals negative 1 over 10 times 100 over 2. 10 goes into 100, 10 times 10 divided by 2 is negative 5. So this guy becomes negative 5. What does negative 5 mean? Negative 5 times 0.2 is going to be 0.1, right? Negative 0.1, negative 0.1, minus 0.1 is zero. That means I'm pricing the thing at zero cents, right? If I'm pricing the thing at zero cents, my revenue is going to be zero, irrelevant to how many I would sell, right? How many of these I'm going to sell, right? It's going to be 5, negative 5 times negative 10 is 50. I'm going to be able to give away 150 at zero the cost, right? It's not unlimited. Everyone won't take one, right? There's going to be a limit to this. So I'm going to sell 150 items, right? If I put 5 in here, let's do this. What does this mean? The negative increment changes here, right? My price, p of x is going to be p of negative 5, right? Let's put negative 5 here, because I'm substituting negative 5 for x, is going to be 0.1 plus 0.02 times negative 5. Negative 5 times 0.02 is negative 0.1. So it's going to be 0.0 minus 0.1 is going to be zero. So I'm selling them for nothing. I'm giving away, right? Zero times anything is going to be zero revenue. That's why it's an x intercept, it's an r of x, right? And how many items am I going to be able to give away for free? Well, that's i of x, right? So i of negative 5 is going to be 100 minus 10 times negative 5. Negative 5 times 10 is negative 15, negative and negative becomes positive. So that becomes 100 plus 50, which is going to be 150. So I could give away 150 of these, right? Hopefully you can see this. Let's put it here so you see it. So I'm going to give away 150 of these for zero, right? Great chaos in school. All these little poppers go enough everywhere, right? This is a maximum revenue problem, a simple one, interesting one, right? It deals with a lot of things that we can, a lot of events that can happen for you to maximize your revenue or lose customers, right? Make nothing, right? And you can change the price, right? Increase your increments, decrease your increments, right? I would still sell 100 if I increase my increment to this level, right? Because that's at the same level as this, right? That was mirrored here. So two and a half increment changes. I could do five increment changes. This would be five. That means five times 0.02, five 2 cent increment changes. That's going to be 10 cent increment increase. So I could sell each item for 20 cents and I would end up selling 50 items, right? My revenue would still be $10, but five times 10 is 50. 100 minus 50 is 50. I could sell 50 of these poppers and make $10, right? Instead of selling 100 of these things, I would sell them for 20 cents each. I'd sell 50, make the same amount of money as selling them for 10 cents, and I would sell 100, right? I hope that's clear how this works. It's interesting. It's modeling things like this. It gives you a lot of power. It allows you to analyze your business, right? It makes you appreciate the price that you're paying for certain items that you're buying in the real world. It gives you a better appreciation for it because most things are not priced based on, you know, our economic model is a little weird, right? Most things aren't based on how much this thing costs to make, right? This thing doesn't, the cost of this thing, it doesn't even take into consideration many other factors that should be included in the cost of this thing, right? They're going to come up into the future, right? As, you know, because it's based on resources and stuff like this and resources doing the certain price of things are going to go up, right? That's one reason people are, a lot of companies are investing a lot of money into going into smaller things, miniaturization, right? The smaller the whatever it is that you're creating, the less raw resources you need, the lower your cost, right? Hence, well, not your profit. This isn't profit. Revenue might increase because it's smaller. People like the smaller, but your profit increases because your expenses are less because you need less resources, right? And there's different models of this, different crafts. We could create a graph of profit, right? Revenue minus expenses becomes profit. And we'll talk a lot about that in the future as well as we build content for economics. But what I want to do now is sort of for the economic stuff, for math, we're going to jump around a fair bit. There's a lot of different concepts that you can cover. And what I want to do is jump to a concept where it's a graph that I use where I, you know, when I'm talking with people, maybe related to politics and economics, or when I'm teaching this stuff to students to try to make them appreciate how graphs can be used to visualize things. What we're going to do is we're going to do a little jump and we're going to talk about differential accumulation, we're going to talk about disruptive innovation, and we're going to talk about mergers and acquisitions, okay? And we're not going to get into quantifying it. What we're going to do is we're going to draw a graph and sort of try to visualize how our economic system works and what we really have to keep in mind regarding the job market, regarding what careers we choose, regarding how our society's growing and expanding and which direction it might be going, okay? And some of the options available to us and how investing, you know, the graph basically, I use this, you know, the following graph in the next video and many different things to try to get certain points across. And I just want to show you guys that. And it's a good visualization that I really like. And it's something that is embedded for me, okay? And that's it. And this is, you know, simple revenue problem, maybe not simple if you don't know, if you don't know polynomials and graphs and equations of lines, this might, you know, be a little bit too much. But when it comes to mathematics, this is a simple model. It could be a simple model, let's use them a lot of places, okay? That's it for now. I'll see you guys in the next video.