Math Matters: Why Do I Need To Know This?Bruce Kessler, Department of Mathematics

Western Kentucky UniversityEpisode Six

1 Surface area – Home improvement projects with paint

Objective: To illustrate how area formulas are relevant to our everyday lives byexamining a real-life example of their use, specifically in calculating how muchpaint 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 mathcourses and how they kind of apply to the real world. I’ve got some neat stuff to show youtoday, so I’m actually kind of giddy about the stuff I’m going to show you today. I think it’sreally 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 itapplies 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 forexample, but it takes a little more knowledge than just the area of the wall because there’sother issues involved such as you know is there a design on the wallpaper and these kinds ofthings, and so do you have to worry about losing part of your wallpaper at the top and thebottom of the wall, things like this. Or, after you cut a section off, you can’t use it somewhereelse, so area is a thing to worry about when you’re doing that, but it’s not the only thing toworry about. However, if we talk about painting, painting, paint is one of these things that Ican kind of smear around and there’s very little waste involved. There’s a little bit when youthink about rinsing out your brushes and your rollers, these kinds of things, you do end upwasting 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 basingthis on some gallons of paint that I had out in my garage. A gallon will cover from 250 to400 square feet of the material depending on the coarseness of the surface and if it’s verycoarse, it will probably take closer to 250 and we also have to take into account that a secondcoat is usually going to be needed on newer surfaces. Now what I want to look at today is acase where suppose you’ve built a storage shed and it’s covered in exterior paneling that youintend to paint. (Figure 1) Now exterior paneling is very rough, so I’m going to go with that250 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 withpaint and really, I’ve got a nice picture of my shed here, I don’t need to worry about the roofso I’m actually going to take the roof off of this, let’s go back and take a look at things. I’lltake the roof off, that’s aluminum roofing, I’m not going to worry about that and I’ll giveyou 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 paintdo I need to buy in order to coat this twice, in order to put two coats on the exterior panelingof this shed and it’s an important question, I mean you can always go back and buy more

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paint, but if you’re doing this as a contract or something like that, you may want to, youprobably 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 foursurfaces, four flat surfaces that I need to apply paint to. I’m painting the outside I shouldsay so I’m going to take it apart and look at a piece at a time. This is the back wall and itis, the whole shed is 12 by 16. The base of this short wall is 12 and it goes up eight feet, letme kind of get my pointer going, up to here and that’s 12 feet up to that point. But let mejust deal with this 12 by eight business. That’s a rectangle. It doesn’t look like a rectanglethat’s because there’s dimension involved, but it’s flat, if I looked at it flat straight on, that’sa rectangle and the area of a rectangle is length times width so that is 12 times eight andthen 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 halfbase times height, so that’s one half, the base is 12 so height here is four so that works outto 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 hasa door. A three by seven opening for the door so we’re going to subtract that area. That’sseven times three, 21 and so the front wall then is 99 square feet. So that’s got the front, gotthe back.

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Now let’s move on to the sides. The right hand side is simply a rectangle. There’s nowindow there. It’s 16 by eight so that is 128 square feet if I’ve done my calculations correctlythere. (Figure 4) And then I’ve got the opposite wall that would also be 128 and then, but

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I’ve got that window there which is a rectangle, 3 times 3.5. So if I’ve done all my arithmeticcorrectly, that works out to be one 1171

2square feet. (Figure 5) Now what I want to do is

add all that up, get the total area so this is my original kind of construction. I add all thosethings up and I get 464.5 square feet total and then I want to do two coats, so I’ll double thatand then that comes up to 929 square feet and then I need to do something I did a few weeksago and that is convert that into gallons. So I’m drawing an equivalence here between onegallon and 250 square feet. I multiply by one, I get the square feet to cancel out and so itworks out that it is about 3.7 gallons of paint to cover that. So you’d end up, you’d probablyend up making a purchase of four gallons of paint to cover up the outside of that. (Figure 6)

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Hopefully this is a good demonstration of how kind of take the known area formulas forlike 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 tokind of slice them and use a rectangle on the bottom and a triangle on the top. So that’s agood trick, that’s a great trick for calculating area. We also did the subtract trick where Ihad the area of the whole thing and then I had a portion of that I wanted removed and so Isubtract that. It’s very important when you’re doing real world applications that you’re ableto adapt your formulas to kind of fit the situations.

Let me give you one weird example. On this campus, we have a planetarium, the HardinPlanetarium and that is basically shaped as a hemisphere and I thought about trying to geta picture of it and squishing it in there, but I’ll just do this. Half a sphere, the other halfof a sphere, that’s actually the shape of the Hardin Planetarium. Now, and you can checkyour books for this, the area for the surface outside the Hardin Planetarium or for a sphere

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is 4πr2, and since we’re doing a hemisphere just and I’m just talking about this businessright here, the outside curvy part. For the hemisphere, it would be half of that and now thepart I’m not taking into account is okay the circular base. That actually doesn’t need to bepainted. Suppose we measure the circumference of this thing at 300 feet and the reason Isay 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 theradius. 2πr is the circumference, the distance around is 300. So then solving for “r,” youget this little measure and then I can plug that into the area formula that I talked about. Dothe arithmetic, you end up with a π2 in the denominator here that will cancel out and itworks out to be about 14,300 square feet. If you do that same kind of calculation it ends upbeing about 57.3 gallons of paint. It would take a lot of paint. I bet that is a fairly accuratemeasure. I didn’t go measure the Hardin Planetarium, but I bet you that’s pretty close towhat’s going on. (Figure 7)

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Alright, I flashed a lot of stuff by you very quickly so let me flash up some summary pagesthat kind of talk about the things that I mentioned.

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2 Complex numbers – Video-game design

Objective: To illustrate how complex numbers can be used in a real-world situ-ation, specifically to manage the translation and rotation of points in the planeis a computer animation.

The next application I would like to show you involves complex numbers and I’ve got totell 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 complexnumber in it, for example. So tying this into the real world was kind of a challenge. Nowit does come into play. There are a lot of applications where you can use complex numbersto kind of sidestep difficulties that you have with real numbers and you get a real numberanswer and voila! There you go, you’ve solved your problem. But I really wanted to showyou some way that we could apply this to the real world and I’m going to do this by showingyou an example from video game design, okay?

The idea behind imaginary numbers is that you’ve got this crazy thing that you couldsquare and you get a negative one, which is pretty bogus. I mean you can’t take a realnumber, 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 youknow 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 anythinglike 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 isbecause that is something that pops up when you start using the quadratic formula and thatcertainly 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 youcan do a lot of things in the plane where we more around and so forth. Well, a complexnumber, 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 numbersand points on the plane. One plus three “i” would be the point (1, 3) in the plane and twominus “i” would be the point (2,−1), and so forth. So I can go back and forth between thetwo. (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. IfI’m adding these two complex numbers, I collect the real parts to get four and I collect thetwo “i” and the four “i” to get six “i”. If I’m subtracting, well distribute your negative andcollect 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’vealways done. First would be 3, outer would be 4i, inner would be 6i, and then last is 2i times4i, which would be 8i2. But now remember the “i” squared business is − 1. So that is 10i inthe middle and − 8 right here, so that’s a − 5 + 10i. Now that’s stuff that I can’t do withpoints in the plane. I can’t do the operations on them. So there’s advantages to doing thingsin 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 forexample, when I add things, I move points around. Translations: left and right, up and

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down. Now if I start with three “i”, 3 + i, excuse me that’s (3, 1), the point, and I add twoto 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 meanthree units up and then if I subtract 4, that’s going to move things four units to the left andif I subtract 7i, that’s going to move me seven units down. That goes to 1 − 3i and that isright there. So this is a method, this simple addition and subtraction of complex numbers tomove points around. If I add, if I add real numbers, I move left and right, let me do it sothat you understand it. Right if it’s a positive number, left if it’s a negative number. If Iadd and subtract purely imaginary numbers, then I go up and down. It’s a good way to movestuff around. (Figure 3)

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Alright, same thing with multiplication. If I multiply by a complex number that is on theunit circle, and that just simply means that if I square the first and the second part, I getone. 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 90degree, counterclockwise rotation. If I do it again, I get another counterclockwise rotation of90 degrees. It takes me over to negative two. If I change the thing I’m multiplying by, andthen go to this thing which is 45 degrees, that has coordinates 1√

2and 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 Imultiply 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 circlelike that. (Figure 4)

Now, here’s the thing that happens with video games a lot of the time, you’ll have what

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they call I think a first player game and it looks like you’re moving through the scenery, butyou’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 alot of translations and rotations to shoot things or whatever you’re doing. Now that can bekind of tedious and they have software to do this, but I’m saying that you can actually dothis 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 thisplayer through a walk and what I’ve done is calculated, I’ve moved actually the scenery aroundthis 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’mactually 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 thenumber that I multiply by. Now I’m sliding to the left, okay, so I added so I moved thescenery over to the right. Now I’m traveling in an arc. I’m moving the scenery back andthen rotating the scenery to the left a little bit and then I’m alternating between those thingsand the last thing I’ll do is I’m rotating 120 degrees to the right. What I’ve done actually isI’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’vetaken the corners of things and moved them around by taking the complex numbers to kind

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of do those complications quick and dirty and then animated it. I’ve taken several slides andthen you could do it with pencil and paper where you kind of flip through a notebook to getthe slides. (Figure 5)

So there you go. A nice real world application of complex numbers and how you can usethem for something I think is pretty cool. I know that a lot of the kids I talk to that wantto be computer scientists want to be video game designers. Well, there you go, there’s anapplication of it. Let me slide, put up some summary slides that kind of talk about the thingsthat I mentioned very quickly and let you absorb some of that information.

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3 Inequalities – Optimization problems with constraints

Objective: To show how linear inequalities can be used in a real-life situationwhere we want to find the optimal solution to a problem subject to linear con-straints.

The last thing I’d like to talk to you today are inequalities and how we can use them tosolve some real world kind of problems as far as finding optimal solutions. A lot of times welike to do things in such a way that we for example, maximize profit, minimize waste and wehave to work within certain constraints. That can be a difficult thing to figure out by trialand error. I mean you can just try everything that is possible, but never know that you’vegot the best answer. Well here’s a way using inequalities that we can go through and find thebest 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 boostersor some other civic club and you’re helping out by baking cookies or brownies for them tosell at their next event. And you’ve told them “Well, I can devote ten hours to the projectand then I’ve got to do my other stuff” and so that’s a constraint, you’re only going to spendten hours on the baking. And then the other constraint that we have to kind of worry aboutis that you only have eight trays that you can provide full of cookies or brownies, whicheverit is. Now, there’s certain things to take into account here. Let me kind of point to them. Ican cook, I can bake cookies faster than I can bake brownies. I can fill a tray of cookies inabout 45 minutes, although it takes me about two hours to fill a tray with brownies. So youknow the question here is, which is better? Should I try to just make, in two hours I couldmake five trays of brownies, should I do that? It turns out I could profit by 18 dollars oneach 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 herewith cookies in the ten hours allotted. So the question then becomes okay what mix of traysof cookies and trays of brownies would be best to produce the most profit and you’re going towant to work on this, that’s what you’re going to do. The biggest bang for your effort, youwant 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’sactually kind of graph the set of valid answers that I could look at. So my constraints, theobvious constraints on “x” and “y”, “x” for the number of trays of cookies, “y” for thenumber of trays of brownies is that they both have to, I can’t have negative trays of cookies orbrownies. So I’m talking about working here in this first quadrant, definitely things have tobe here. (Figure 2) But if I start to look at the other constraints, the time constraint is that Ihave a maximum of ten hours to spend on this project. I can spend, it takes me 45 minutesto fill a tray with brownies, so that’s three fourths of an hour. I’m doing things in termsof 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 multiplyby three fourths, however many brownies I have, I’m going to multiply by two. And nowlet’s put this on our graph. I actually need to solve for “y” so I’ll bring all the other stuffover, divide by two real quick, and then I’ll graph that and because this is “y” less than this

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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 haveto 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 rightthrough here and again I’ll take things under it, but again I want the overlap, what’s alreadyshaded so watch this, I’ll put the graph. Here you’ve got some overlap, yeah, we’ll probablygo back and show this. So I’ve eliminated now, let me show you the part, this part was inthere, but now it’s eliminated. So what we have here, this light blue business, that’s the setof valid answers. (Figure 4)

My answer has to come out of this and what I would claim to you folks is that to find theoptimal 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 aboutthe profit that we’re going to make on this, it would be a certain amount for every tray ofcookies, and a certain amount for every tray for every tray of brownies. Well, that, if yougraph that in three space, you’ve got “x” is something, “y” is something and you take thatamount and graph that on the z-axis. You kind of think about here’s everything happeningin the plane and then the amount I get for “x” and “y”, put that on the z-axis, so you geta three-dimensional graph. Well, there’s different things that could happen. You can havethings 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”

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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 ofslanted there in which case this is the highest point on the polygon and it’s kind of hard tosee that the way things, let me move the graph up just a little bit so you can see, change theperspective 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 everytray of cookies and 18 dollars for every tray of brownies. And you know, I don’t really wantto worry about what the graph looks like, but I just want to check the corners. I know thatthis is a plane in three space, whatever, but I just have to check the corners. Even if it’sjust along, you know if one of these lines happened to be less than level, I could still checkthe endpoints and get the same values, so it’s sufficient just to check the corners. So that’swhat I’m going to do. The only constraint that I really have here is that the corners maynot 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 goingto look at whole number answers just inside the polygon, like (4, 3) and (5, 3) are inside thepolygon – (4, 4) is not. (4, 4) would be about right here and that’s actually outside the blueregion, so we just check these amounts. At (0, 0), I get zero, that makes sense. If I don’t doany trays, I don’t get any money. If I do all cookies, I get $80. If I mix them up and do fourand 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 maximummix is do five trays of cookies, do three trays of brownies. That will maximize the amount ofprofit that I can get from this endeavor. (Figure 8)

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Neat little application of inequalities and just then just plug it into a formula just to getyour maximum answer. I’m going to flash up some of the graphs that kind of show you thekinds of things that I was doing as a summary and we’ll come back in just a moment andwrap things up.

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Closing

Oh, we’re back! I hope you’ve enjoyed the things we’ve talked about today. I’m reallykind of proud of some of those things, especially the complex number thing. I know that Italked about things very fast and I have to, it’s kind of a short program. But these, all theseepisodes are downloadable as computer files on our web page and we’ll be flashing that up injust a few minutes. With that, I am done. Thanks for watching us this week and join usnext week, thanks.

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