 Thanks for joining me today. We're here to talk about volume, and specifically volume of a box. So remember what volume is. Volume is a numerical measurement of how much stuff we could fit inside of a box. So the idea here is that suppose all of you were to start with an eight and a half by 11 inch piece of paper. Now we're gonna make you think about this because you are older. You're not going to actually make a box yourself. We're gonna make you think through what would happen. So imagine you had a piece of paper, eight and a half by 11 inches, and we had you cut out from the corners different sized squares. And those squares could be really teeny, tiny, and only half an inch on the side, up to maybe four inches on a side, or maybe three and a half, okay? So we're gonna think about what would happen if we did that. Now I made some boxes just as visual aids for you. So the first box I made, I took an eight and a half by 11 inch piece of paper, and I cut out the square. These I did, I think we're an inch and a half on a side. So you can see where all the corners are cut out, and then I just folded it up to make my box. And you can see this one's a little shallow, not too tall, but the base is pretty big. All right, then I made another box. The other box I made, I made by taking squares that were three and a half inches on a side. So when I did that, this is what it looked like, and then I folded it up. And you can see how this box is much taller, but the base is a lot smaller compared to the last one I did. So I also have a computer animation that will show them side by side for you. All right, so this first one, the blue one, this one looks like maybe it was a box that was made by cutting out a teeny tiny square that was just half an inch on a side. And you can see how it's really shallow, kind of like a really shallow tray. The green one that's next, that one maybe was like an inch and a half, maybe almost the size I did. And then the yellow one you can see is definitely the tallest of all of them. Maybe that was a square that was three inches on a side or two and a half or something like that. But when you compare them side by side, you can see how different they look, even though all of them started out from the same size piece of paper. So what we're going to start investigating is, okay, so suppose we did cut out these squares of different sizes. What would be the dimensions, the length, the width, and the height of the box that got created? All right, so you should have on your handout a table that we're going to fill out now. So if I go to the table, and we'll start thinking through this together, but then I'm going to have you do it yourself. So I'll start you out. So suppose we started with an eight and a half by 11 inch piece of paper, and we were to cut out a square from each corner that was half an inch on a side. All right, so if you envision that, remember what happened when I made my own boxes. The height of the box comes from whatever the size of the square was. So in this case, since the square I cut out was only half an inch on a side, the height of my box is going to be half an inch. And all the heights are going to be like that. The height is always going to be the same as whatever the size of the square was you cut out. That leaves the length and the width. That's the part we have to think through. So imagine that the length is 11 inches long, and from each end we take off half an inch. That leaves 10 inches left over for our length. Now let's do the width. The width, remember, of the paper was eight and a half inches long. And from each end of that width, we take off half an inch. That leaves us with seven and a half inches for the width. So I'll do the next one with you, and then I'm going to give you time for you to complete the rest of the table on your own. So now suppose we cut out a square that was one inch on a side. When we turn up the sides to create our box, that box ends up being one inch tall. So the length, remember, started out being 11 inches, and from each end of that length, we cut off one inch. That leaves nine inches remaining for our length. And the same thing for the width. The width, remember, started out as eight and a half inches wide. We take off an inch from each end. That leaves six and a half inches, right? So hopefully you've discovered the pattern. I'm going to give you a few minutes to get the rest of the table filled out. And then you can come back to me and we'll make sure everyone got the right answers. So let's see how you did. So the next one would be one and a half inches tall for a box in which the square you cut out was one and a half inches on a side. The length would be eight, and the width, five and a half. I'm just using decimals because since we are going to be doing this in our calculator later, I thought it would just be a little bit easier to do decimals. The next one would be a height of two. So when we figure out the length, that would be seven inches and the width, four and a half. The next box would have a height of two and a half inches, making the length six and the width, three and a half. The next box would be three inches high. It would have a length of five inches and a width of two and a half inches. Next we'd have a box that was three and a half inches high. That's the pink one I had made. The length was four and the width of the bottom was one and a half inches. And then finally the tallest box that we could have would be one that's four inches high. The length becomes three of the base and the width of the base is just half an inch. So I'm sure you noticed some patterns to the numbers here and what we're going to calculate next would be the volume. And of course volume we get by doing length, times width, times height. So take a few minutes to calculate the volumes of all these boxes we would be creating and then we'll come back and compare answers. All right, so the first box, the one that was created by cutting out a square, half an inch on the side, that should have a volume of 37 and a half when you calculated it. The next one, the one that was created by cutting out a square that's one inch on the side would be 58 and a half. The next one should be 66 followed by 63. That's the box where the square you cut out was two inches on the side. Then 52.5, 37.5, 21 and six. Interesting that the tallest box has actually the smallest volume. So what we want to do is see, well exactly what kind of pattern do we have here. So what we're going to do is, in our graphing calculators, we're going to graph the data and the points we're going to plot in a scatter plot are going to be the X coordinates as the height of the box. Also you can think of it in terms of the sides of the square you cut out. And then we're going to go ahead and graph the data that we cut out and the Y coordinate is going to be the volume. So we'll talk about how to set that up in your calculator, how to create the scatter plot, and then we're going to take it a step further and actually use the graphing calculator functions to come up with what we call a regression equation that will model the path that you'll notice these points create. All right, so let me get my calculator and we'll go ahead and set up our graph. So on your calculator in the second row, if you hit the stat button, you should have a menu that looks like this and we want to edit because we want to enter the points. So we can just enter them all in one list. Under list one, we're going to put the height of the box and under list two, we're going to have the volume. So you can just type in the number and hit enter to get it in there. All right, once you have your Xs in, your heights of the box, just use your right arrow and that should take you to the top of list two. And now we're going to enter in all the volumes that corresponded to these heights. So when the size of the square, also the height of the box was only half an inch, remember our volume was 37.5 and we're just going to enter them down one by one. So when you're done, you should have eight points. All right, so now the date is in the calculator, but now we have to set up the graph. So the way in which we do that, if you hit the second button on your calculator, then Y equals, we need to tell the calculator to set up a scatter plot for us. So we're just going to do it in plot one. So if you just hit enter on plot one, you'll want to make sure that plot one is turned on. So you'll notice right now it's blinking on on. So just hit enter. And you'll notice the next line down, we have six different graphs that the calculator will make for us. And we want that very first one, the one that you see mine is blinking on right now. That's a scatter plot. All the other graphs that you see there, you'll learn about those in other math classes. The X list, we want that as L1, because that's where we put all our X coordinates. And yours should be defaulted to this right now. Y list we want is L2. And down on the bottom, you can decide for yourself what type of mark you would like your points to look like. I usually like the first one. You can pick whatever you'd like. All right, once you have that set up, simply hit Y equals for me. And what you'll want to double check, make sure you don't have any equations there under your Y1 and maybe kind of scroll down just to make sure there's none hiding down there. Because if we did, that could screw up our graph. So the easiest way maybe to get this graph, if you hit the zoom button and go down to it's number nine, you'll notice it says zoom stat. That's a really great one any time you've entered the data yourself. Zoom stat will decide a really nice window for you. So just hit enter on that one. And hopefully you get a graph that looks like this. All right, and what this is, this is a graph of all those points we just determined. Now what a lot of people find interesting, when they do a project like this, they think that they're gonna just keep getting a box that's of bigger, bigger and bigger volume. Well, when you look at the points though, what really happens is there is a maximum volume. If you hit trace on your calculator, you'll be able to jump from point to point. So right now you'll notice the little blinking spider. We actually do call it a spider. Right now it's sitting on that box that was half an inch high and had the volume of 37.5. And if you just use your right arrow, you can jump from one point to the next. It looks like it's that third one, the box that was one and a half inches high and had a volume of 66. That looks to actually be the biggest. And it really just goes down from there. So what happens is, even though you might expect that the volume could just simply get bigger and bigger and bigger, that's not what happens at all. There's actually a maximum volume that you're going to top out at. So the next thing we can do is, okay, well what if we wanted to come up with an equation that would maybe allow us to figure out, well what if we did a box that was one and three quarter inches high? You know, how could we figure out what the volume would be for that? Well, the calculator will do that for us as well. Go across the top to calc. Now there's several different options here. These are all different types of regression to create equations, many of which you'll learn about in the future. The one we want is the one that says on number six, cubic regression. That's because, and you will find out later on once you've had algebra, that the equation that would model these points is a cubic equation. So bring that up and just hit enter on your calculator so you should be looking at this. And we want to take that equation and put it under y equals because we want to be able to see it on our scatterplot. So next to where it says cubic regression, hit vars. It's a button on your calculator. It says var s. Go across the top to y vars. And the first option for function. And you'll notice there there's all your y equal two options. So we just want to put it in y one. And then just hit enter after that. And this is the equation that would model those scatterplot points. It's a cubic equation, as you can see from the x cubed variable in the equation. So now go ahead and hit the graph button. And you can see how nicely that curve goes right through those points. So what this allows us to do, now that we have an equation and a curve, we can ask ourselves a whole lot of other questions that we couldn't really just do from the numerical data. So for example, what if we wanted to know the volume of the box if the square we had cut out was maybe two and a quarter inches long? So remember what that means is the height of the box would be two and a quarter inches. So to do that on your calculator, do second trace and you should see a menu like this. The first one is value. That's the one we want, so just hit enter. And next to where it says x equals, let's do 2.25. And hit enter. And you can see the cursor go right to where the curve is, up there sort of in the middle of the screen. And you'll notice that it says y equals 58.5. What that tells us is that when the size of the square that you cut out was two and a quarter inches big, the volume of the box ends up being 58 and a half cubic inches. And you can do this for any x value you want. Let's see, let's make up another one. How about four, let's do 3.75. And hit enter. And there you can see it down on the lower right, the volume would end up being 13.125 cubic inches. Now another thing we can do is what if we wanted a box that was 40 cubic inches big and we wanted to figure out kind of going backwards how big the square would have to be in order to give us a box of that volume. So we can easily do that on the calculator as well. If you hit y equals again, you'll notice there that long equation, that's the regression equation that the calculator figured out for us. So use your down arrow and go down to y2. And in there, type in 40, because that's how big the volume we want to be. Now just hit graph, and you can see that there's two places that that y equals 40 line intersects with the regression curve. What that tells us is that there's two different ways we can come up with a box that has a volume of 40 cubic inches. So let's find the first intersection point. If you hit trace, right now the blinking spider is sitting on the first of the scatter plot point. So just hit your up arrow and you'll notice that the spider jumps to one of the curves. Now use your left arrow and we'll go find that intersection point over there on the left. So all you need to be is close to it. Your numbers might not match up with mine. Exactly, that's okay. So once you're kind of close to the intersection point, do second trace, and we're going to go down to intersect. Hit enter, and it's going to ask you what two curves you want to find the intersection of. You'll notice at the top of your calculator screen, right now it's telling us the spider is sitting on the y1 curve. So hit enter, now it's jumped to the y2 curve. So hit enter again, and then enter one more time. So what this tells us is that one way we could come up with a box that has a volume of 40 cubic inches is if the side we cut out for the square was about .545 inches big. But there's another option because there's a second intersection point. So hit the trace button again. You'll have to use your up arrow to jump the spider back to the curve. And let's go over to the right and find the other intersection point. And again, you just have to be close to it. The decimals you have for your x value might not be the same as mine. Do second trace down to intersect again. And once again, it's going to ask you what two curves you want the intersection of. So just hit enter through those. And you should find that the second option for getting a box that has a volume of 40 cubic inches is about 2.922 inches on a side. So this is a little introduction as to how you can use your calculator features to really investigate different things and different characteristics about not only scatter plots and quadratic equations or other types of regression that you're going to learn later on. But you get to play with different things. And in this case, we investigated what effect different squares that we cut out from corners of a set size piece of paper are going to have on the volume. So later on, as you move on in mathematics, you'll learn how to take those equations and actually find the maximum rate from the equation. So thanks for joining me today. And hopefully I'll see you later when you get to those other levels of math classes.