 Thanks for joining me today. We're here today to talk about volume, specifically volume of a rectangular box. Now, you might remember, if you think back to your middle school years, joining us for the box problem enrichment, in which each one of you would have had an 8 1⁄2 by 11 piece of paper, and you would have made boxes of different sizes. So here's an 8 1⁄2 by 11 piece of paper. And if you did do this before, you might remember cutting out from that piece of paper squares from the corners of different sizes. What we usually have our students do when they do this with us is they'll cut out squares of maybe a teeny tiny one that's 1⁄2 on a side, up to 4 inches on a side. And we talk then about what do we notice? What are the patterns? What effect does the size of that square that you cut out have on the volume of a box? So I have a computer animation just to kind of put you in the right frame of mind. In terms of what we're talking about here. All right, so the first one is the blue box. And if you take a look at that, that looks like maybe that's one that was created by cutting out a square of just a half inch, maybe an inch on a side. And you can see how shallow it is. Almost just like a little tray. The green one looks like those squares were a little bit bigger, and maybe 1⁄2 inches, 2 inches on a side perhaps. And the last one, the yellow one, that has a much bigger square that was cut out. And you can see how, even though they all started out, the same size piece of paper or construction paper, whatever we used, that in the end, they look very different from each other. So back then, what you would have done was you would have investigated what the length, width, and height of each of those boxes were. The height of the box always comes from the size of that square that you cut out. And then based upon that, you can easily calculate the length and the width. Of course, then volume is length times width times height. What we're here today to talk about, though, is the algebraic point of view. Because now that you're older and you've learned algebra, you can write equations for these things. So we're going to now take it from a very analytic algebraic point of view. We're going to come up with an equation that models the volume based upon that original size of 8 1⁄2 by 11. And we're going to investigate that on our calculator and see what we can find out. All right, so let's start writing this out and figuring out how to come up with these equations. So suppose I were to draw on my whiteboard an 8 1⁄2 by 11 type of rectangle. All right, so let's call this the side that's 11. And we'll call this side the one that's 8 1⁄2. All right, and from that, what we're going to do is we're going to represent sizes of squares that we could cut out from the corners. Now remember, we want to algebraically represent this. So we don't want to think of a specific number or anything in terms of the size. So let's just call those sides X. So obviously it's a square we're cutting out, so the two sides are the same. So we can represent each of those as being X. So when I'm drawing this, I'm trying to make these squares about the same size as I can. So all these sides of these squares that I cut out would be as X. So if the length of the paper is 11 inches and from each end of that paper, I'm cutting out a square that is X inches long. We need to figure out, all right, well, when we create the box by folding up the sides, this part right here, this is really the length of the base that remains. So we start out with 11 inches. We cut out X inches on either end. So the length that remains can be represented as 11 minus 2X. Now we're going to do the same with the width. So we start out with a width of eight and a half inches long. Once again, we're cutting out a square that is X inches long from either end. So this little part in here, that's going to be 8.5 minus 2X. All right, so from all this, we need to come up with an equation that's going to represent the volume. Now remember how we get volume of a box. It's length times width times height. Well, we have the length and the width, that's the 11 minus 2X and the 8.5 minus 2X, but envision where the height comes from. So you've taken this piece of paper, you've cut out the corners, you've then folded up the sides. The height of the box is going to be the same size as whatever the square is that you cut out. So our length, that's 11 minus 2X, our width is 8.5 minus 2X, and the height is simply whatever that square size was that you cut out. So what we have here is an equation and we could multiply the binomials together then distribute the X. When you do that, you come up with the equation 4X cubed minus 39X squared plus 93.5X. Obviously there's three X's, that's why it comes out as a cubic equation. This also explains why we measure volume in cubic units because it is a cubic three-dimensional figure. We have a length, a width, and a height. So that's why we have a cubic equation now. So what we wanna talk about next is now that we have this equation of a volume and we all know what a cubic equation looks like, hopefully. A typical cubic equation is going to sort of look like something like that, of that shape. Well, it's going to have a maximum, it's going to have a minimum. We can actually use our calculator to find those. Later on, once you've learned calculus, you will know how to get this purely from an algebraic point of view. You won't even need your calculator to graph it. But for now, we can always graph this equation and then find the maximum and minimum from there. So if you grab your graphing calculators, I'll grab mine. Now the other thing I've set up for us in here, just so you can take a peek at it, is remember that table of data, back in middle school when you would have done this and created your own boxes and measured the dimensions. I put all that information in the calculator so that we can look at a scatter plot of it first. So what we have here, under list one, I have the height of those boxes. Under list two, I have the volume. All right, so what I went ahead and did, I set myself up a scatter plot. I'm sure you've learned how to do this in your past. And let's take a look at the scatter plot. That's what the scatter plot of those points look like. Maybe this is something you did back when you did the enrichment with us before. Now looking at it though, it very much looks like a parabola. But we know from the equation that we just found, well, it's really a cubic. So what we're going to do is go under y equals and we're going to type in that cubic equation that we just found. Now what you can do on your calculators since you don't have these points plotted, all right, why don't you just go ahead, I'm gonna turn off my stat plot for a minute. Just so that my calculator can match up with what yours is doing. All right, if you have the equation under y equals, just go ahead and do zoom six for now. This way at least we're all looking at the same graph. And you can see it definitely has that shape. Doesn't look like we can see the top too well. We can see the bottom there, the minimum point down in the fourth quadrant, but we can't really see the top up in the first quadrant. So let's adjust our window a little bit. Let's go under window and let's change our y max. I think if we make it 70, I think that'll be high enough for us to see the top of it. And go ahead and hit graph. That looks much better. We could even adjust it a little bit more because it looks like there's really no part of the graph over in the second and third quadrant. So if you wanted to, if you go back under window, we could make the x min zero. Maybe that'll give us a little bit better picture to look at. And hit graph. Ah, much nicer. So you have to remember what this graph represents. Our x coordinates, if you look at the equation we wrote from our picture we drew, the x represents the height of the box. The y's represent the volume. All right, because under y equals, we put in the volume equation. Hence, every y value on this curve is going to represent the volume for a certain height box. Now this is specifically for an eight and a half by 11 size piece of paper we start with. If we were to change the size of that paper, we're going to have a different equation. All right, so each equation is tailor made to the size of the sheet of paper that we begin with. So if you take a look at this, and let's just sort of investigate what we have. Well, if we hit trace, all right, mine happened to go to the very bottom of the cubic equation. All right, take a look at what the coordinates are down there. All right, it has an x value of five, which represents that the height of our box would have been five. The volume though is a negative seven and a half. Well, we all know you can't have negative volume. So that's really a part of the graph that we really wouldn't need because we cannot have a negative volume. So let's trace a little bit to the, let's go left. All right, once we start getting over here, let's see, well, there's around three and a half right there. So if our height of the box is about 3.51-ish, we know that our volume's going to be about 20.6546 cubic inches. So what we have here is a really great way that we can come up with a lot of different possibilities. Suppose we want to have a certain volume. We could figure that out, how big our box, how tall our box should be. What if we had, we knew our height was a certain height? Well, what's the volume going to be? There's a lot of different ways we can investigate this. So let's talk about some of those different things we can do. The other thing we might want to think about if we go back to our window, well, remember what our x's were and remember the size of the piece of paper we started with. This was an eight and a half by 11 size piece of paper. So in terms of our height, we're going to get to a point that, well, we can't have a box that tall because of the limitations on the size of the paper we started with. All right, remember our paper was eight and a half wide by 11 inches long. Well, we could cut out a square that was four inches. Some of you might remember doing that back in this activity before. That's going to be a box that's pretty tall, but the base is pretty small. Well, let's see. Cutting out a square of four inches on a side, that's eight inches right there of the eight and a half. So chances are we really can't go much bigger than that. So let's change our x max to four. You could probably go a little bit bigger than that, but not too much. So let's hit graph again. Now this looks much more like the scatter plot. All right, let me switch back to the scatter plot really, really quick. Just so you can see and kinda, you'll have this on your calculators. Let me switch back to the scatter plot just to show you what that looked like. There's the scatter plot that I already had set up and there's the curve you should be looking at on your calculator. All right, and you can see how well that curve goes through all those points. It very much models very well what these points are telling us. All right, so this is what you probably learned about regression equations. If we were to have run a cubic regression on these points that I happen to have in my scatter plot, we would probably come out with pretty darn close to the equation we just got and wrote by hand. All right, so let's keep investigating, though, that equation. Let me turn my scatter plot off so I'll get back to looking at a graph like you have set up. So there's the graph you're looking at right now. So suppose I wanted to figure out what would be the volume if the height of my box and therefore the size of the square I would have cut out was maybe 2.75 inches. Well, we can use the calculators feature to easily do that. If you do second trace, you'll get all the calculate functions that you see here. The first one value will allow us to type in whatever x we want, and it will tell us what the volume corresponding to that height would be. So if our height of the box, also the size of the square we would have cut out was 2.75 inches, hit Enter, and we get a volume of 45.375 cubic inches. And you could really make up any x value to do that with. There's certain x values that we know we can't have. We already talked about have, you really can't have an x much bigger than four. What if we did do four? Let's see what happens there. Oh, you can see it all the way over there on the right. You get a volume of only six cubic inches. Kind of interesting that the largest square you'll cut out will give you the smallest volume. Well, what if we also wanted to maybe go the other way? What if we wanted to have a box that was a specified size in terms of volume? What we could figure out is, well, what are the different possible ways we could come up with a box of that size? So let's go back under y equals. Suppose I wanted to have a box that specifically was 40 cubic inches big. So hit Graph, and you'll notice that there's two different places that the line intersects the curve. So what that's telling us is we have two possible ways that we could come up with a box that has a volume of 40 cubic inches. So let's go find those really quick. If you hit Trace, and remember, you always use your right and left arrows to move your spider. So I'm going to go find that intersection point there on the left first. So if you move over to the intersection on the left, now your numbers might not be the same as mine, which is fine. And to find the intersection, remember you do second trace, and it's down at number five. Hit Enter. Remember that you need to tell the calculator which two curves you want the intersection of. So right now I'm sitting on y one. Hit Enter. Now I'm sitting on y two. Hit Enter again and again. And we find that a volume of 40 cubic inches we can get by cutting out a square that's about 0.545 inches on a side. Well, there is another possibility because there is another intersection point. So let's hit Trace again, and we'll go over to that intersection point on the left, on the right rather. And again, you just have to get kind of close to it. Your numbers might not be the same as mine. Do second trace down to intersect. And if we were to cut out a square that's about 2.922 inches big on a side, we'll still get a volume of 40 cubic inches. So these are just some of the very introductory basic things you could investigate. Well, let's figure out what the maximum volume is. All right, taking a look at this cubic equation, obviously there's a highest point up there. So let's hit Trace. Let's trace all the way to the top. If your equation's in the way like mine is, let's fix our window maybe and go up a little higher than 70. Let's make it 80 perhaps. If your equation's in the way like mine is, you can't see your spider too well. So let's hit Trace again. Ah, much better. All right, so if you go to the very, very top of that cubic curve, and what you wanna do, keep an eye on your numbers. Again, your numbers might be slightly different from mine if you have a different TI calculator than I happen to have. What you wanna do is try to start out at the biggest y value you can. All right, now again, your biggest y number might be a little few decimal places off from what mine is. All right, so let's see, mine's about, think right about there. All right, so go ahead on your calculator and just move to the highest y value that you can. From that point, we're going to run the maximum feature in your calculator. All right, so if you go under second trace again, you might've seen it when we were finding the intersection. It's down at number four, and you'll notice that it's going to ask you two questions. It's going to ask you for a left bound and a right bound. My recommendation would be just go a little bit to the left, a little bit to the right, of where you're starting out. That's why I recommend that you start out at the highest y value. So for my left bound, from where I'm sitting right now, I'm just going to go one hit to the left with my left arrow, and I'm going to hit enter. And you'll notice then that the question changes to right bound. So I'm going to go one hit back to the right where I started, and then go one more to the right, hit enter again, and you'll notice if you look really closely at your calculator there, there's two little triangles that are marking the interval in which that maximum point's going to be located. Hit enter once more, and you find out what the actual maximum volume is. So it's about 66.148 cubic inches. Now when you did this back in middle school, you might have remembered finding that the maximum volume was 66, which given those numerical methods we were finding, that was pretty darn close. This is even a more exact answer because we've used it finding the algebraic equations. That maximum volume comes when we have a height of about 1.585 inches. So the interesting thing is what a lot of people think when they're younger, and they maybe see this problem for the first time, is that the volume can just get bigger and bigger and bigger when really it doesn't. There's a maximum to it, all right? And we can see that very, very easily from our calculator. Now there's a lot of other things that we can investigate as well, all right? And these are some of the things that you start getting into, especially if you were to take calculus. Because one big idea behind calculus is that of rate of change. Well, we have an equation that models the volume. So what we could find out is for any one point on that curve, how fast is the volume changing? And that's something that we can investigate very easily without really knowing the calculus by using our calculator, all right? So let's go back to the graph that we had, and I'll show you how we can do that very easily. All right, I'm going to go under y equals and just get rid of that 40, just so I'm not looking at it anymore. And this is where you really want to keep in mind what this curve represents. Remember, this is representing the volume, or y values are the volume. For any given x, and the x in this case represents how large of a square we would have come out, cut out, and that turned into what the height of the box was. So maybe we want to figure out, well, when the size of the square that we cut out is a certain size, how fast is the volume changing at that point in time? So if you do second trace, the one that's going to tell us, if you look at number six, you'll notice it says dy dx. Once you take calculus, you will become very familiar with that notation. That represents a rate of change. In calculus terms, it's the derivative of y with respect to x, which is always a rate of change. Specifically, it's an instantaneous rate of change. So if you hit dy dx, and you'll notice that your graph looks like this, we're always looking for the instantaneous rate of change at one particular point in time. So suppose we wanted to know how fast the volume was changing when the height of the box was two inches. So just type in two, and you'll notice it automatically changes to x equals two for you. Hit enter, and you get dy dx. Now take a look at where your blinky spider is on the curve. That is the spot at which x equals two on the curve. And notice how we get a dy dx value of negative 14.5. So think about what that represents. When x equals two, which means the height of our box is two inches, or you can think of it that the size of the square was two inches, we find that the volume is changing at a rate of 14.5 cubic inches per inch that the x value would be changing. The negative represents that it's decreasing at that point in time. If you were to find that dy dx at different places on the graph, and you'll notice on your handout, there's the opportunity to do that, to investigate that, you'll start getting a sense of what these different rates of change tell us. All right, what does it represent when the rate of change is negative, like that one we just saw? What happens that the rate of change is positive? What does that represent? Or what if that dy dx, that rate of change comes out zero? What does that represent? So you'll notice on your handout, there's the opportunity to do that and discuss it among your own classmates. And that's some of the stuff that sort of gives you a glimpse of what you'll be doing once you get to calculus. So thanks for joining us today. It's been great being with you and good luck as you move forward in your math classes.