Math Matters: Why Do I Need To Know This? 1 Surface area Home improvement projects with paint

Transcription

1 Math Matters: Why Do I Need To Know This? Bruce Kessler, Department of Mathematics Western Kentucky University Episode Six 1 Surface area Home improvement projects with paint Objective: To illustrate how area formulas are relevant to our everyday lives by examining a real-life example of their use, specifically in calculating how much paint is needed in a home-improvement project. Hello, welcome to this week s show of Math Matters: Why do I need to know this?, where we try to show you some of the math that you re learning in these entry level math courses and how they kind of apply to the real world. I ve got some neat stuff to show you today, so I m actually kind of giddy about the stuff I m going to show you today. I think it s really kind of slick. Let s get going. The first thing that I want to show you today is an application of surface area and how it applies to home improvement projects, and particularly painting home improvement projects. There s all kinds of things that, where the concept of area is kind of useful, wallpapering for example, but it takes a little more knowledge than just the area of the wall because there s other issues involved such as you know is there a design on the wallpaper and these kinds of things, and so do you have to worry about losing part of your wallpaper at the top and the bottom of the wall, things like this. Or, after you cut a section off, you can t use it somewhere else, so area is a thing to worry about when you re doing that, but it s not the only thing to worry about. However, if we talk about painting, painting, paint is one of these things that I can kind of smear around and there s very little waste involved. There s a little bit when you think about rinsing out your brushes and your rollers, these kinds of things, you do end up wasting a little bit of paint, but not a ton. So really, when we re painting, an area measure is pretty useful to us. I m just basing this on some gallons of paint that I had out in my garage. A gallon will cover from 250 to 400 square feet of the material depending on the coarseness of the surface and if it s very coarse, it will probably take closer to 250 and we also have to take into account that a second coat is usually going to be needed on newer surfaces. Now what I want to look at today is a case where suppose you ve built a storage shed and it s covered in exterior paneling that you intend to paint. (Figure 1) Now exterior paneling is very rough, so I m going to go with that 250 number in this and you may even find that we may need to scale this down a little bit, but we re going to work on the premise that I can cover 250 square feet of this paneling with paint and really, I ve got a nice picture of my shed here, I don t need to worry about the roof so I m actually going to take the roof off of this, let s go back and take a look at things. I ll take the roof off, that s aluminum roofing, I m not going to worry about that and I ll give you the dimensions of all the different things we re looking at. (Figure 2) The question I would like to try and answer with you guys is how many gallons of paint do I need to buy in order to coat this twice, in order to put two coats on the exterior paneling of this shed and it s an important question, I mean you can always go back and buy more 1

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3 paint, but if you re doing this as a contract or something like that, you may want to, you probably have to give an estimate, so you want to know up front how much paint do I need, these kind of things, or how much do I need to pay for? Alright, let s take this house apart and do things a piece at a time. There are actually four surfaces, four flat surfaces that I need to apply paint to. I m painting the outside I should say so I m going to take it apart and look at a piece at a time. This is the back wall and it is, the whole shed is 12 by 16. The base of this short wall is 12 and it goes up eight feet, let me kind of get my pointer going, up to here and that s 12 feet up to that point. But let me just deal with this 12 by eight business. That s a rectangle. It doesn t look like a rectangle that s because there s dimension involved, but it s flat, if I looked at it flat straight on, that s a rectangle and the area of a rectangle is length times width so that is 12 times eight and then I ve got this triangle on top here that I want to deal with. If the whole thing is 12 feet, I ve accounted for eight of those. It s four feet up to there and the area of a triangle is half base times height, so that s one half, the base is 12 so height here is four so that works out to be 120 square feet on the back wall. I m just painting that other surface on the back wall, that s what I m painting. Now if we look at the front wall, that s the same except that it has a door. A three by seven opening for the door so we re going to subtract that area. That s seven times three, 21 and so the front wall then is 99 square feet. So that s got the front, got the back. Figure 3, Segment 1 Now let s move on to the sides. The right hand side is simply a rectangle. There s no window there. It s 16 by eight so that is 128 square feet if I ve done my calculations correctly there. (Figure 4) And then I ve got the opposite wall that would also be 128 and then, but 3

4 I ve got that window there which is a rectangle, 3 times 3.5. So if I ve done all my arithmetic correctly, that works out to be one square feet. (Figure 5) Now what I want to do is 2 add all that up, get the total area so this is my original kind of construction. I add all those things up and I get square feet total and then I want to do two coats, so I ll double that and then that comes up to 929 square feet and then I need to do something I did a few weeks ago and that is convert that into gallons. So I m drawing an equivalence here between one gallon and 250 square feet. I multiply by one, I get the square feet to cancel out and so it works out that it is about 3.7 gallons of paint to cover that. So you d end up, you d probably end up making a purchase of four gallons of paint to cover up the outside of that. (Figure 6) Figure 4, Segment 1 Hopefully this is a good demonstration of how kind of take the known area formulas for like rectangles, and triangles, and things like that and adapt them to find the area of things. You know those ends were pentagons. They were five-sided things, but yet we were able to kind of slice them and use a rectangle on the bottom and a triangle on the top. So that s a good trick, that s a great trick for calculating area. We also did the subtract trick where I had the area of the whole thing and then I had a portion of that I wanted removed and so I subtract that. It s very important when you re doing real world applications that you re able to adapt your formulas to kind of fit the situations. Let me give you one weird example. On this campus, we have a planetarium, the Hardin Planetarium and that is basically shaped as a hemisphere and I thought about trying to get a picture of it and squishing it in there, but I ll just do this. Half a sphere, the other half of a sphere, that s actually the shape of the Hardin Planetarium. Now, and you can check your books for this, the area for the surface outside the Hardin Planetarium or for a sphere 4

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6 is 4πr 2, and since we re doing a hemisphere just and I m just talking about this business right here, the outside curvy part. For the hemisphere, it would be half of that and now the part I m not taking into account is okay the circular base. That actually doesn t need to be painted. Suppose we measure the circumference of this thing at 300 feet and the reason I say circumference is this is solid. I can t measure through it so I have measure around it. But that s okay, the circumference I can, from the circumference I can calculate then the radius. 2πr is the circumference, the distance around is 300. So then solving for r, you get this little measure and then I can plug that into the area formula that I talked about. Do the arithmetic, you end up with a π 2 in the denominator here that will cancel out and it works out to be about 14,300 square feet. If you do that same kind of calculation it ends up being about 57.3 gallons of paint. It would take a lot of paint. I bet that is a fairly accurate measure. I didn t go measure the Hardin Planetarium, but I bet you that s pretty close to what s going on. (Figure 7) Figure 7, Segment 1 Alright, I flashed a lot of stuff by you very quickly so let me flash up some summary pages that kind of talk about the things that I mentioned. 6

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9 2 Complex numbers Video-game design Objective: To illustrate how complex numbers can be used in a real-world situation, specifically to manage the translation and rotation of points in the plane is a computer animation. The next application I would like to show you involves complex numbers and I ve got to tell you, this was a real challenge because complex numbers, by definition, are not real values. They re kind of imaginary in the sense that you can t have a bank account that has a complex number in it, for example. So tying this into the real world was kind of a challenge. Now it does come into play. There are a lot of applications where you can use complex numbers to kind of sidestep difficulties that you have with real numbers and you get a real number answer and voila! There you go, you ve solved your problem. But I really wanted to show you some way that we could apply this to the real world and I m going to do this by showing you an example from video game design, okay? The idea behind imaginary numbers is that you ve got this crazy thing that you could square and you get a negative one, which is pretty bogus. I mean you can t take a real number, square it and get a negative number. But let s imagine that you can, so here s this i for imaginary and sometimes you ll see this notation. It s not technically correct, but you know we ll say i is equal to 1. The technical part is that you re not supposed to have negatives under that radical but that s okay, I won t call the math police on you or anything like that. Complex numbers, we actually take a real number and we add a multiple of this i together and call that a complex number. The reason you learn it in college algebra is because that is something that pops up when you start using the quadratic formula and that certainly would be an application of this. But I don t want to get into that. (Figure 1) I want to get into the equivalence of complex number to points in the plane. Now you can do a lot of things in the plane where we more around and so forth. Well, a complex number, if you think about plotting the real number part of the complex number along the x- axis, and the imaginary part along the y, there s an equivalence between complex numbers and points on the plane. One plus three i would be the point (1, 3) in the plane and two minus i would be the point (2, 1), and so forth. So I can go back and forth between the two. (Figure 1) The advantage of complex numbers is I have operations. I can add, subtract, multiply, and divide them in an obvious way. You just simply collect your like terms. If I m adding these two complex numbers, I collect the real parts to get four and I collect the two i and the four i to get six i. If I m subtracting, well distribute your negative and collect your like terms. So 1 3 and then 2i 4i and then do the same thing. (Figure 2) To multiply them, you just use the FOIL method and the same kinds of things you ve always done. First would be 3, outer would be 4i, inner would be 6i, and then last is 2i times 4i, which would be 8i 2. But now remember the i squared business is 1. So that is 10i in the middle and 8 right here, so that s a i. Now that s stuff that I can t do with points in the plane. I can t do the operations on them. So there s advantages to doing things in complex numbers and then applying them to points in the plane. (Figure 2) That s what I m going to do with you guys when I start talking about video game design. The thing I ve got to convince you on is that these operations move points around. So for example, when I add things, I move points around. Translations: left and right, up and 9

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11 down. Now if I start with three i, 3 + i, excuse me that s (3, 1), the point, and I add two to that, it takes me to 5 + i, which is two units over. That s a translation of two to the right. If I then add 3i to it, well that s 5 + 4i and that moves me four units, excuse me, I mean three units up and then if I subtract 4, that s going to move things four units to the left and if I subtract 7i, that s going to move me seven units down. That goes to 1 3i and that is right there. So this is a method, this simple addition and subtraction of complex numbers to move points around. If I add, if I add real numbers, I move left and right, let me do it so that you understand it. Right if it s a positive number, left if it s a negative number. If I add and subtract purely imaginary numbers, then I go up and down. It s a good way to move stuff around. (Figure 3) Figure 3, Segment 2 Alright, same thing with multiplication. If I multiply by a complex number that is on the unit circle, and that just simply means that if I square the first and the second part, I get one. I want that so that I don t scale things. I just want to move things. If I multiply by i, i here is 90 degrees above this horizontal. So if I multiply two by i, that constitutes a 90 degree, counterclockwise rotation. If I do it again, I get another counterclockwise rotation of 90 degrees. It takes me over to negative two. If I change the thing I m multiplying by, and then go to this thing which is 45 degrees, that has coordinates 1 2 and 1 2. If you square this and you square that then you get half plus a half is one so I think you believe me there. If I multiply this out, I get this point which is right there. And so that is a 45 degree rotation. So, again, I can rotate things by multiplying by a complex number that s on the unit circle like that. (Figure 4) Now, here s the thing that happens with video games a lot of the time, you ll have what 11

12 Figure 4, Segment 2 they call I think a first player game and it looks like you re moving through the scenery, but you re not. You re standing still and the scenery is moving around you. You re sitting still, your gun or whatever it is right here and things are moving around you. So that involves a lot of translations and rotations to shoot things or whatever you re doing. Now that can be kind of tedious and they have software to do this, but I m saying that you can actually do this with complex numbers. I ll show you an example. Here is my game field, which is pretty dull, I know but you know I ve got to sleep sometime, I can t spend infinite time on these things, but this is the player and here is the game field. It has a rectangle, it has fence, a circle, and a triangle and what I d like to do is take this player through a walk and what I ve done is calculated, I ve moved actually the scenery around this player by using repeated applications of either adding or multiplying complex numbers. (Figure 5) So here we go, I m going to take you through this real quick. We re going forward, I m actually adding a negative complex number, that s where I want the scenery to come down. Then I m rotating to my left, I m rotating the scenery actually to the right, so this is the number that I multiply by. Now I m sliding to the left, okay, so I added so I moved the scenery over to the right. Now I m traveling in an arc. I m moving the scenery back and then rotating the scenery to the left a little bit and then I m alternating between those things and the last thing I ll do is I m rotating 120 degrees to the right. What I ve done actually is I ve applied this mapping to just points, to just the corners of things, moved them around. Let me just show you that again because I think it s cool. You know all I ve done here is I ve taken the corners of things and moved them around by taking the complex numbers to kind 12

13 Figure 5, Segment 2 of do those complications quick and dirty and then animated it. I ve taken several slides and then you could do it with pencil and paper where you kind of flip through a notebook to get the slides. (Figure 5) So there you go. A nice real world application of complex numbers and how you can use them for something I think is pretty cool. I know that a lot of the kids I talk to that want to be computer scientists want to be video game designers. Well, there you go, there s an application of it. Let me slide, put up some summary slides that kind of talk about the things that I mentioned very quickly and let you absorb some of that information. 13

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15 3 Inequalities Optimization problems with constraints Objective: To show how linear inequalities can be used in a real-life situation where we want to find the optimal solution to a problem subject to linear constraints. The last thing I d like to talk to you today are inequalities and how we can use them to solve some real world kind of problems as far as finding optimal solutions. A lot of times we like to do things in such a way that we for example, maximize profit, minimize waste and we have to work within certain constraints. That can be a difficult thing to figure out by trial and error. I mean you can just try everything that is possible, but never know that you ve got the best answer. Well here s a way using inequalities that we can go through and find the best answer to get the optimal answer to problems like that. Here s the example I m going to work with you. Let s say you re helping the band boosters or some other civic club and you re helping out by baking cookies or brownies for them to sell at their next event. And you ve told them Well, I can devote ten hours to the project and then I ve got to do my other stuff and so that s a constraint, you re only going to spend ten hours on the baking. And then the other constraint that we have to kind of worry about is that you only have eight trays that you can provide full of cookies or brownies, whichever it is. Now, there s certain things to take into account here. Let me kind of point to them. I can cook, I can bake cookies faster than I can bake brownies. I can fill a tray of cookies in about 45 minutes, although it takes me about two hours to fill a tray with brownies. So you know the question here is, which is better? Should I try to just make, in two hours I could make five trays of brownies, should I do that? It turns out I could profit by 18 dollars on each tray of brownies. That s more than ten dollars than I profit on each tray of cookies. Maybe I should just do that. On the other hand, and I could definitely fill all my trays here with cookies in the ten hours allotted. So the question then becomes okay what mix of trays of cookies and trays of brownies would be best to produce the most profit and you re going to want to work on this, that s what you re going to do. The biggest bang for your effort, you want to get the money out of it. (Figure 1) So let s talk about how we set this up. We re actually going to set up the constraints. We re going to use a graph now and we re going to graph, I m going to let x be one thing, the number of trays of cookies. I ll let y be the number of trays of brownies and let s actually kind of graph the set of valid answers that I could look at. So my constraints, the obvious constraints on x and y, x for the number of trays of cookies, y for the number of trays of brownies is that they both have to, I can t have negative trays of cookies or brownies. So I m talking about working here in this first quadrant, definitely things have to be here. (Figure 2) But if I start to look at the other constraints, the time constraint is that I have a maximum of ten hours to spend on this project. I can spend, it takes me 45 minutes to fill a tray with brownies, so that s three fourths of an hour. I m doing things in terms of hours, and it takes me two hours to construct, or put together, bake a tray of brownies. So depending on what x is, however many trays of cookies I have, I m going to multiply by three fourths, however many brownies I have, I m going to multiply by two. And now let s put this on our graph. I actually need to solve for y so I ll bring all the other stuff over, divide by two real quick, and then I ll graph that and because this is y less than this 15

16 Figure 1, Segment 3 amount, I ll graph this line and then I ll take the bottom half or the bottom part of my graph, so I ve actually eliminated a lot of the solutions I had earlier, okay? Definitely things have to fall within this triangle. The valid answers have to fall inside this triangle. (Figure 3) The other constraint is the number of trays. Well x plus y cannot be bigger than eight. I ve got eight trays, that s it, that s all I ve got to work with so I ll graph this as well. y is less than or equal to eight minus x. I ll graph eight minus x which is actually right through here and again I ll take things under it, but again I want the overlap, what s already shaded so watch this, I ll put the graph. Here you ve got some overlap, yeah, we ll probably go back and show this. So I ve eliminated now, let me show you the part, this part was in there, but now it s eliminated. So what we have here, this light blue business, that s the set of valid answers. (Figure 4) My answer has to come out of this and what I would claim to you folks is that to find the optimal answers that will maximize profit, I just need to look at these corners of the polygon. It s the quadrilateral, okay? And let me show you why that s the case. If you think about the profit that we re going to make on this, it would be a certain amount for every tray of cookies, and a certain amount for every tray for every tray of brownies. Well, that, if you graph that in three space, you ve got x is something, y is something and you take that amount and graph that on the z-axis. You kind of think about here s everything happening in the plane and then the amount I get for x and y, put that on the z-axis, so you get a three-dimensional graph. Well, there s different things that could happen. You can have things where it only grows in the x direction. In this case, that would be the highest point. That s directly above this corner of the polygon. (Figure 5) Or it could only grow in the y 16

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18 Figure 4, Segment 3 direction, okay, in which case this would be the highest point on the polygon. (Figure 6) Or, as is the case in our problem, it could go grow a little bit in each direction and it s kind of slanted there in which case this is the highest point on the polygon and it s kind of hard to see that the way things, let me move the graph up just a little bit so you can see, change the perspective on it, see that that is now the highest point on it, on the graph, okay? (Figure 7) So what we ve got to do now is take our profit function, our profit ten dollars for every tray of cookies and 18 dollars for every tray of brownies. And you know, I don t really want to worry about what the graph looks like, but I just want to check the corners. I know that this is a plane in three space, whatever, but I just have to check the corners. Even if it s just along, you know if one of these lines happened to be less than level, I could still check the endpoints and get the same values, so it s sufficient just to check the corners. So that s what I m going to do. The only constraint that I really have here is that the corners may not be whole numbers and I kind of need to think about doing whole number trays of things. So instead of (4.8, 3.2), that s where these things actually kind of hit each other, I m going to look at whole number answers just inside the polygon, like (4, 3) and (5, 3) are inside the polygon (4, 4) is not. (4, 4) would be about right here and that s actually outside the blue region, so we just check these amounts. At (0, 0), I get zero, that makes sense. If I don t do any trays, I don t get any money. If I do all cookies, I get $80. If I mix them up and do four and three, I get 80, I mean $94 profit. If I do five and three, that is this point closest to that, I get $104 and if I do all brownies I just get $90. So the maximum answer, the maximum mix is do five trays of cookies, do three trays of brownies. That will maximize the amount of profit that I can get from this endeavor. (Figure 8) 18

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21 Neat little application of inequalities and just then just plug it into a formula just to get your maximum answer. I m going to flash up some of the graphs that kind of show you the kinds of things that I was doing as a summary and we ll come back in just a moment and wrap things up. Summary page 1, Segment 3 Closing Oh, we re back! I hope you ve enjoyed the things we ve talked about today. I m really kind of proud of some of those things, especially the complex number thing. I know that I talked about things very fast and I have to, it s kind of a short program. But these, all these episodes are downloadable as computer files on our web page and we ll be flashing that up in just a few minutes. With that, I am done. Thanks for watching us this week and join us next week, thanks. 21

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