 Imagine a box with a square base has a length plus girth of 104. Now, before we go on, some of you might be like girth. What is that? Is like girth Brooks or something like that? No, no, no. This is a dimension that's often very important for like mailing and logistics and things like that. Like if you went to take your back, you took a package to like UPS or FedEx or some of these other companies, they might measure the girth of the box, right? So the girth is the distance around the box. So if we have, so we have a box with a square base. So let's start off with that. So we have like some square base, whoops, square base, and then some height to it. We don't necessarily know what that is yet. The bottom is a square and maybe, maybe I'll add some little dashes here so that you feel like this is transparent now and then like this. So we have this square base. So let's say it's X by X right there. And then the height is something which we'll call it Y for the moment. So when I talk about the girth of this thing, you're asking like how far around, how far around it's the distance around the box. So, and they always pick the largest one. So it's like, if you went around, take your measuring tape and went around this, this is what we're talking about with the girth. It's like the perimeter of the other dimension. So you have a length, right? You have a length and then the girth will be the perimeter around the box that's perpendicular to that length, all right? So this is often, and like I said, this is often used for shipping costs, right? If you're mailing something, the girth matters. It's not just master volume. If you have a really long thing that has a huge girth, then of course that's gonna be calculated in the volume, in the shipping cost there. So what is the length of the box if the volume is supposed to be 2100 cubic inches? So what do we understand about this problem so far? So volume makes sense to us, right? We're talking about the length of the box. What should the length of the box be if the volume is gonna be 2100 cubic inches? So volume, so it's equal 2100 cubic inches. But the volume is also equal to length times width times height. And so without label, without saying which one is which, right? I have a square base, I have a length Y here. This is gonna be equal to X squared Y. It's just some variables introduced into the problem. You could call them different things if you want to. It doesn't really matter. As long as it's very clear to you what these variables mean. And a picture can be a very useful thing to illustrate what do we mean by these dimensions here. So okay, so that's the volume. So we have, but we have two variables. How do you deal with the two variables? Well, that's where the first sense comes into play. So first of all, since the base is a square, that helped us out a lot knowing that the length, that the width and the height were the same number here. Although I've oriented so the length looks like the height, whatever, you can always rotate the box. And so, but we also know that the length plus the girth is equal to 104. So we have that 104 is equal to the length, right? The length, let's call that, that's the longest side on our box. The length is gonna be Y plus the girth. The girth here, if we go with the longest side, the girth would be how much we go around it like that. So it's gonna be X plus X plus X plus X. So our girth would turn out to be 4X right here. And so we can use this to substitute, whoops, we can use this to substitute. I'm just gonna draw the girth back there. We can use this to substitute out one of the variables here. So looking at this equation, it has two variables, X squared and Y here. We could solve for Y. That would actually be pretty cheap. Y equals 104 minus 4X. We could then substitute that in there. That would be great. I like that approach. Because then our equation would look like 2,100 is equal to X squared times 104 minus 4X. That gives me a cubic polynomial equation I could try to solve. If you went the other way around though, if you try to solve for X, you would get X equals instead 104 minus Y over four. That has a fraction that already gives me some issues. And then you have to plug it in there for the X square. And so you have to square the fraction. I think solving for Y is gonna be much more economical approach to this one here. And so we try to solve this equation for X. We need to multiply out the right-hand side. So that's gonna give us 104X squared minus 4X cubed. This equals 2,100. If we then move, then we subtract the 2,100, we're gonna get negative 4X cubed plus 104X squared minus 2,100 is equal to zero. And personally, I like when my leading coefficient is positive. In fact, actually all of those numbers, 104 and 2,100 are divisible by four. So to make life much easier for us, we're just gonna divide both sides by negative four. And this then gives us the equation X cubed minus, let's see, four goes in there how many times, yeah, 104 is divisible by four. That should be 26. So we had negative 26X squared. And then let's see, you're gonna get 2,100. Well, if you're ever trying to do these calculations by hand, things you wanna look for is like 2,100 is 2,100 times 100. Four goes into 100, of course, 25 times. So you stuck with 2,100 times 25, which is not the worst calculation in the world. We could probably do it if we had to. But you're gonna end up with 525 equals zero. And of course, you don't have to be a hero. You can use a calculator to help you out here. It's not a big deal whatsoever. So we need to solve this equation. I wanna try to do a factoring. Can I find factors of negative 525 that would work here? In which case, I mean, some of them I know immediately. I mean, 525 is 21 times 25, right? So we could try some of those numbers. Both of them kind of seem a little too big for me, right? I wanna try something a little bit smaller. I mean, look at 525 clearly that's divisible by five since the last digit is five. If I tried synthetic division with five, what would I see there? One negative 26, don't forget the zero, negative 525. If you try dividing by five here, bring down the one, you get five, right? Which negative 26 plus five is negative 21, which we get right there. For which case, you're gonna get five times negative 21. That's equal to, let's see, negative 105. Add that to zero, you get negative 105. Then if you times that by five, that's where you're gonna get this negative 525 from, for which then you get zero right there. So five was actually a pretty good guess to start off with. So we end up with x minus five times the quadratic x squared minus 21x minus 105, like so, equals zero. We can continue to try to factor this thing. Factors of 125 that add up to be 20. Those kind of seem like big numbers for me. I might try the discriminant approach here, right? To see if this can be factored. The discriminant remember is b squared minus four ac. For which case, if we take 21 squared, you get 441. And then you're going to add to that four times 105, which is 420, which when you add that to 441, that gives you 861. And we wanna take the square root of that. That's a irrational number. It gives you about 29 something, right? So if you continue with the quadratic formula, you would get 21 plus, again, approximately, 29.3 over two, which if we try to compute that together, that's giving us something like approximately 25, we'll say 25.2 inches. So just because we have an irrational solution does not mean that this isn't feasible, right? It could be a potential answer, right? So we discovered here five worked and we have also like this plus 25. The other one would turn out to be negative. So if you took the negative choice there, that would give you something negative. So we're looking around five or perhaps 25. So let's come back and see which of these actually seem reasonable. I perfectly like the answer of five, right? Cause that's a nice whole number there. And if you check this out here, if you take five times five, what would be the length in that situation? Come back to this formula right here, this one right here. If we were to take 104 minus four times five, four times five is 20 and so that would give us 84. So the length is supposed to be 84 inches and then the base is supposed to be five inches. So we're looking at something that looks like 84 by five by five, which that does give us 2,100 volume, 84 times 25 is 2,100. So that seems feasible. And then with the girth we found out that works out. So five inches of these dimensions right here do seem very feasible. The length being 84 inches seems quite fine. The other possibility we still have to consider would be if X was approximately 25.2, which admittedly if you times that by four, that gives you about, it gives you about 100.7 or something like that. I'm rounding of course, for which case if you subtract that from 104, that is an acceptable girth and that gives you a length that's gonna be about four inches, right? That is a possibility, which I'm gonna throw this one out from considerations because not because I don't like it, but mostly when you look at the story problem it doesn't seem to make any sense for the following reason. When it comes to measuring the girth of a box, like I said, the length is for these shipping purposes, the length is the longest dimension of the object and the girth is in the perimeter around the other way. So that's why I drew my box the way I did. The length was the longest dimension and the girth has to be something like that. So in order for this second model to work, we would have to have the X coordinate be, the X value having 25, so it's like 25 by 25 and then the other one's about four or something like that. Irrationality is not the problem here. The issue is that that doesn't match up with what girth means. The length should be the longest dimension. So if we want Y to be bigger than X, I guess that's what I'm trying to say here is that in order to get the answer there's sort of like this subtle inequality in the background X needs to be less than or equal to Y. In order for that to hold, we have to throw out this possibility and this that gives us that our box is 84 by five by five.