 Welcome back to our lecture series, Math 1050, College Outfit for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misledine. In this lecture number 34, I wanna talk about, well, without a better way of saying it, story problems, right? This sometimes scares people all the time. Oh no, story problems are so hard. Which of course, if we never do story problems, then you sometimes beg the question, why would we bother learning math if we never use it for real life problems? It's sort of like this give and take when it comes to story problems. Yes, they can be challenging, but they really are necessary and motivate a lot of what we do in this series here. So for this lecture, I wanna go through four different story problems involving rational functions of some kind. Now be aware that a lot of these examples actually will be polynomial functions, because every polynomial function actually is a rational function. It's just a special subset. And oftentimes it happens that as we set up some things, they do lead to these polynomial expressions, some rational expressions as well. The one we see at hand, well, it's probably a polynomial equation we have to solve. We'll see some proper rational expressions showing up in a future story problem in this lecture, lecture 34 here. So let me just read the prompt for us and see how we can approach this. So imagine a new bakery offers decorated sheets of cakes for children's birthday parties and other special occasions, maybe weddings or whatever, people. People like to get together and eat cake. The bakery wants the volume of a small cake to be 351 cubic inches. That's gonna be sort of a relevant thing. So if anything, as I'm reading through this, because sometimes when you read a story problem, you have to read through it more than once. Sometimes just read it once, just kind of get all of the information at once. And then you go again through it the second time to kind of look for this pertm information. I have someone at the disadvantage, I should say the advantage, maybe disadvantage for the viewer here, that I've already worked through this problem prior to the video, so I kind of know what the pertm information is. And so that might be a little bit misleading if you're trying to do a story problem on your own. Again, if I have a story problem I've never seen before, I probably read through it once, get all the information, and then go through it again and pick out what information actually is relevant. So our baker wants the volume of a small cake to be 351 cubic inches. Okay, that's a statement about volume right there. The next sentence, the cake is in the shape of a rectangular solid. Okay, that seems useful because that connects somehow volume with shape there. Maybe I can employ some type of volume formula I know from geometry, but is the volume even relevant? What's the question at hand? They want the length of the cake to be four inches longer than the width of the cake and the height of the cake should be one third of the width. So there's some specific dimensions that kind of make the cake more, maybe it's easier to bake or maybe there's an aesthetic artistic appeal to it. This cake looks great. I don't know, but essentially the question comes down to what should the dimensions of the cake pan be, right? So how should we make, how should we shape the cake? All right, let's go back to some of this information we discovered here. The cake itself is supposed to be a rectangular solid, some type of rectangular prism. So if I were to just draw it, we anticipate something like the following, just a rectangular prism of some kind. And so we wanted the dimensions of the cake, all right? So we're gonna have to have some type of length and width and height, which honestly, I never know which one is which, which one's length, which one's width. I just know that width is the other one from length, whatever. Height usually means go up and down. So we have some rectangular prism. We're trying to find the dimensions that we have to find this L, W and H. So we knew some information on volume, right? We knew the volume, well, the volume of a rectangular prism is gonna be length times width times height, but it should also equal 351 cubic inches. Now, when it comes to story problems, I think it's very important to keep track of the units in play. We're measuring volume with a cubic inch. Each of our dimensions length, width and height should also be measured in inches in some kind. And so at the moment, we don't have enough information to determine what the individual length should be, but we know the total volume should be 351. Okay, we're gonna find out that that 351 was a generous choice by the question writer right here, we'll see how to approach that in just a second. So what can we find out about these? We need some more information here. So we want that the length, okay, so the length, that's my L value, right? The length of the cake should be four inches longer than the width. Okay, so if we try to digest that a little bit. So they want the length of the cake to be. So when you see things like this to be, like forms of the word is, when I say the length is four inches longer than the width, that typically means an equal sign of some kind. So L is, L is what? L is four inches longer. That suggests to me I'm gonna take four plus something. Four inches longer, I go four plus the width. Okay, the width is W. So I put that together and I get the expression that the length is supposed to be W plus four. Okay, so I want the length to be a little bit longer than the width, which if I wanna satisfy my picture right here, maybe I switch locations here and call this the width and call this the length. Clearly don't need to draw your picture to scale anything like that, but you can do that. Let's see, what else do we need to know? So then continuing that sentence, and the height of the cake is to be, well again, that's an is type word there. So H is equal to, equal to what? One third of the width, one third of the width right there. And so when you put that together, the height is equal to one third of the width. What I notice here is that I can express the length with respect to the width and I can express the height with respect to the width. So if I revisit my volume formula, volume is equal to the length, well that's W plus four, the width, well that's just W, and then the height is one third W. This is supposed to equal 351. So then if I focus on this part of the problem right here, right here, then it's like, hey, this is a polynomial equation. If I solve this polynomial equation for W, then I can use that to determine what is the, what should be the dimensions of my cake. So I've now set up my equation right here, I wanna solve it. Now, honestly in practice, I would use some type of computer software calculator to solve these polynomial equations for us, because oftentimes, I mean, to solve this equation, we have to find essentially the x-intercepts of a polynomial function, which that often turned out to be something irrational. And so like I said, a calculator can be very helpful here, but to practice skills, we've learned previous in this class, let's use the algebra that we've learned and I've necessarily given us numbers that we can solve without any numerical analysis whatsoever here. What I'm gonna do is I'm gonna multiply out the left-hand side because when we have a polynomial function, we want it to look like P of x equals zero, right? That's our goal, so we can start factoring the left-hand side, we have to get there. I first wanted to, let's see, let's times the W's here together. This is going to give us x, well, I'm gonna stick the one third out in front. We're gonna get x plus four times W squared equals 351. So one thing I'm gonna do is I'm gonna distribute the W squared here. So we get one third times, where did that x come from? That should be a W. W cubed plus four W squared is equal to 351. Now personally, I mean fractions are great numbers and all, but if you had a choice between a fraction and not a fraction, I think we all agree that we don't want a fraction here. Now we can clear the denominators that I multiply the left-hand side by three, that'll cancel out the one third, but what's good for the goose is good for the gander. I have to do it on both sides of the equation equally so that equality is preserved. So we get now W cubed plus four W squared. And so then we have to do 351 times three, which I'll give you 1,053, like so. And so then if I move the negative, well, I should subtract 1,053 from both sides and we end up with the equation W cubed plus four W squared minus 1,053 equals zero. Like so, great. And so now I have this polynomial equation. I'm gonna highlight this. We have to solve this polynomial equation. Now find the, notice we have a polynomial equal to zero. This basically means we're looking for the roots of this polynomial. We need to find the roots of this thing, which is something we did in the past. So one of the first things I'd asked myself is, you know, certainly there's gonna be some domain concerns like what answers are actually considered acceptable. I can't have a negative width of a cake. So I really don't care about negative answers here and I can't have an imaginary length of a cake. The answers have to be real. You could have a square root. You know, you can have a square root of two length of like inches or something, that is possible. So irrational numbers are acceptable but I need to have something positive. And so when you look at this right here, we're gonna take the factors of 1,053 and that's, we're gonna use like a rational roots test to kind of work through that. So we look at these possible P's and Q's. Well, you always get one as a possibility. I know three is a factor, right? If you try to start factoring the 351 a little bit more, we could do that and feel free to use a calculator, you know, to help you out here. Which case 351, let's see that factors. Well, you know, honestly, it's like, we, again, you could actually write all the factors that we wanted to. And so just so you're just clear from the rational roots test, you would end up with like 9, 13, 27, 39, 81, 117, 351 and then 1,053. So the idea there is like 350, 351, it's 13 times nine, I believe. That seems possible. Anywho, you get all, whatever those turn out to be. One thing I want to point out to you is that we knew three was a divisor immediately because remember, we took 351 in times of I3, that's where we got 1,053. So instead of searching every divisor of 1,053, we might just start with the divisor that's kind of obvious. What if we tried three in this situation? Okay, so then that's when we're gonna start running synthetic division. So using our coefficients, we have one, four, there's no linear term, so we need to put a zero there and then a negative 1,053 in that situation. So if we tried running it with three, bring down the one, one times three is three plus four is seven times three is going to be 21 there plus zero is 21. Times that by three, you get 63 and notice that's not gonna equal zero. So this kind of shows us here that really our number, the number three was gonna be, it's just too small, right? That I mean, this specific number here, I mean, if we need to, we got what, negative 990. That seems about right there. So the point is you got something way, way too small because that negative is so big there. So three isn't gonna work but we basically need to try something bigger in order for these coefficients to balance out with negative 1,053. And that's when we start investigating then divisors of 351. And so that's when I'm like, okay, okay. So I need something bigger than three. How do I factor 351? Well, 351, if you start checking it, notice the digits, three plus five plus one adds up to be nine. That means it's divisible by about three and nine. If you divide 353 by three, you end up with 117, which was also on our list. If you then divide, if you try to factor 117, notice one plus one plus seven is also nine. That's still divisible by three. You can take out another three, you get 39 and 39 factors as 13 and three. So there's actually a lot of factors of three in there. So I think I misspoke when I said earlier, did I say nine times 13? It should be 27 times 13, whatever. The point is this number, the 1053, not that one, this one right here, it does factor as three to four times 13. So we can start to try to find other ones. We can try bigger factors if you want to. If you try like 27, right? If you have this list, 27, how does 27 work with this? I'm gonna erase these numbers right here. So in that situation, if we try 27, we end up with, we'll bring down the one. One times 27 is 27 plus four is equal to 31 times 27. That's gonna be 837. Add that to zero, you get 837. Times that by 27. You end up with 22,599. That's gonna be way too big now, right? You end up with still like 21 and a half thousand. So we tried three, it was too small. We tried 27, it's way too big. Notice everything at the bottom is a positive number. That says that everything bigger than 27 is gonna be way too big. They're just gonna exacerbate the problem. So now we're done to something like nine or 13, right? So if we tried 13, we'd see that one also turns out to be too big. If we try nine, that's actually gonna be the magic one right here. If you try nine, bring down the one. One times nine is nine plus four is 13 times nine is 117 plus zero is 117. Times nine is 1053, which that then gives us the root we were looking for. And so that gives us a factorization that's helpful. We get that our polynomial factors as w, w minus nine times, then looking at the quadratic and play right here. We're gonna get w squared plus 13 w plus 117, for which then if we could try to factor that, we would do so. Now it is a quadratic, I mean factors of 117. We saw those a little bit above, right? So three and 39, it has to add up to be 13. That seems a little bit challenging trying to add things together. My temptation to think is there's not gonna be a magic pair of factors of 113 that have to be 13. Sorry, factors of 117 that have to be 113. I don't know if we can do it. So we might run to the quadratic formula, look at the discriminant for example. So we're supposed to take b squared minus four ac, right? And so in that situation, you would get 13 squared, which is 169 minus four times 117. That turns out to be 468. And you can see it's like that's gonna be a negative number. So the quadratic formula tells you there's no real roots to this number. So the only thing we get here is the width has to be nine inches. So if we come up here and summarize what we found, then we see the following. The width should be nine inches. That's what solving the equation told us. If we plug that into the other equations, then the length, the length should be nine plus, nine plus four, which would be 13 inches. And then the height should be one third of that, should be three inches. And then you can check that out there. If we take nine times three times 13, that does give us the 351 that we had before. So those should be the dimensions of the cake. Now again, this question, we ended up finding a root using these techniques we learned from the previous unit, number four there. But like I said, in practice, if we learn what's trying to solve this quadratic equation, it's very likely that, well, the answer might not be a pretty little integer like it was. This is sort of like the Mickey Mouse version of this problem I confess. Oftentimes you get some irrational number as the solution. And so we have to get an approximation without delving into how one solves these problems numerically, that takes us beyond the scope of college algebra. Just want to be aware that the real kicker for this one, the real thing I care about the most is understand the viewer understanding where this equation came from. The technique of solving it by all means, people could critique me in the comments all day long. I'm perfectly happy with that. But the point of story problems is not necessarily about how you solve it, but how you set it up. If a student can understand how to set up a story problem, then they're probably good at solving it. And so I don't have much of a worry. That's the real thing about story problems. How do you set these things up?