College Algebra - Lecture 18 - Equations in One Variable

now we're gonna go ahead and look at some mathematical models models are where we apply some of the techniques we've learned the mathematical techniques for solving equations and find out where these equations actually turn up in real-world type problems so this will be called setting up equations more mathematical models we've seen mathematical models when we talked about functions all right let's go ahead and start talking about a little bit of advice before we begin and sort of an outline of what we need to do the first thing we need to do with a mathematical model of a real-world problem is translate into natural lengths well translate sorry from translate from natural language natural language like English for example the language I am currently using into the language of mathematics the language you've been practicing with all along and once we have it there once we have it in mathematical form then we'll be able to apply the various techniques we know so translate from natural language into the language of mathematics the second thing is and I can put it in the phrase like this reality is messy the real world is not as clean as the problems you've seen in practicing these mathematical techniques the reality is messy which means the coefficients that you'll see in quadratic equations for example will not necessarily be integers and other things come into it we will have to draw pictures we will have to analyze the English to see what it says but what you need to do is you need to understand the real-world terminology and the situation involved but keep this in mind as we go one other point let me make this another point on this page here call it a tip be sure all your units are consistent now we haven't worried too much about units up to now because units are something that really have bearing on real world problems we haven't been doing many of those what I mean by having them consistent is when you're talking about say length be sure that everything is in terms of the same measure you're talking about length in terms of feet everything should be terms of feet for example you're talking about length or distance you have lots of choices make sure that everything in the problem is in the same set of units you're talking about miles everything should be in miles talking about feet everything should be in feet here in meters everything should be in meters etc and the same thing goes for other things for example time you've got years to worry about days minutes hours etc all right whatever your units are make sure that everything is in days everything is in minutes everything is in hours etc it's your choice which one you use but make sure that everything is consistent okay before we start with a real hard world real world problem where we want to model it and try to solve the question let me go ahead and try to pose to you a translation problem actually three of them at once and here are three little translation problems all they involve is translating into mathematics right equations or statements involving mathematical symbols the first one there are twice as many students as professors write that in mathematics the second one hundred dollars is invested X dollars in stocks the rest in bonds express the amount in bonds another mathematical translation question and finally number three if n is a number right the number that when added to end yields ten and be the positive number whose square is 3 less than n now look these over and then after you've had a chance to think about them write out what you think are the mathematical expressions corresponding to these when I come back I'll show you what I have written okay here are the solutions that I would have written now here are the problems let's look at the first one there are twice as many students as professors well it seems to me that I would probably want to use a variable name for a number of students and number of professors so let me do that first let's let s be the number of students okay and let's say P be the number of professors then what I'd like to write out is the statement there are twice as many students as professors now what did you write down here's the correct answer s equals to P whatever number of professors you have multiply it by two and you'll have the number of students there are twice as many students as professors it's very very common for someone to write to s equals P that's wrong and that comes from not thinking about what this means but by saying twice to s and P and then just writing an equation to SP that's wrong it doesn't make any sense it means that there are twice as many professors as students which is exactly the opposite of what you want to say so then the moral of this problem is once you have it labeled be sure that you think about what this means don't just write down letters and hope that you're in the right ballpark alright the second one the second one was about investments and the total invested in this second problem was $100 the amount that was invested in stocks you were told was X and you were to write an expression for the amount invested in bonds and if you remember the statement of the problem a hundred is invested X dollars and stocks and the rest there's the key word the rest of the 100 dollars is in bonds well if exes and stocks what's left well 100 minus X is what's left so the amount in bonds is 100 minus X simple as that all right let's look at the third one the third translation here there were two parts part a and let's see what it said if n is a number so n will stand for a number here right the number that when added to n yields 10 well what number can you add to n to get 10 the answer is 10 minus n what's the proof the proof is take 10 minus n add it to n and what do you get 10 there you go Part B this one's a little trickier you may have had to fiddle a bit now let me put this at the top all right the positive number whose square is 3 less than n what is 3 less than N and minus 3 is 3 less than n we want a number whose square is equal to that in other words we want this number the square root of n minus 3 and what's the proof that that's the right number well this is the number whose square is equal to 3 less than n so there's my proof so there you are with translations for those three mathematical statements okay now with that out of the way let's go ahead and look at some particular models I told you we'd be looking at mathematical models we're gonna look at several very important ones here's the first one interest and as always in a real problem there's a whole series of actual real definitions that need to be here so definitions plural here's the first one principle principle which will denote by P in our work is the total amount total amount borrowed by well it goes one way or the other principle can be borrowed from you or you can borrow principle from a bank so borrowed by in the one case an individual from a bank that's called a loan this is equal to a loan that's when principle is borrowed by an individual from a bank or a bank can borrow principle from an individual from an individual and that is what's called a savings account the bank is borrowing the money from you and paying you interest on that account speaking of interest let me define interest be continuing with definitions interest which will denote by I is money that is paid for what it's paid for the use of other money to make it really simple so the interest you make on a savings account is the bank paying you for the use of your money and the interest you pay the bank on your loan is you paying the bank for the money that you're using that they loan to you finally there is something called rate of interest so we have principal interest and rate of interest this will be denoted by our for us and this will be measured in percent now what is percent it the word is per cent now what does that mean Sam stands for centum which stands for a hundred so it's per 100 so let me give you an example so you know what I'm talking about 6% is by definition 6/100 which is 0.06 and we'll use that calculation again and again so what is the rate of interest rate of interest are which will be measured in percent is the amount charged for the use of principal over a given period of time and actually it is the percentage of the amount and I'll show you an example what I mean by this so there are our definitions and then putting these all together we can have a model for simple interest if a principal of say P dollars is borrowed by someone for two years at a rate of interest and see again these word problems these real problems take awhile to lay out at a rate of interest per year the interest is over a given period of time of are written as a decimal then the interest on that principal is I equals P times R times T and this is recall is called the simple interest formula as opposed to other more complex formulas for interest so this aunt it's very simple simplest is you take the principal you multiply it by the time and the rate of interest and you will end up with the interest so for example we're talking about this equation now separated from everything else P times R times T okay just keep it separate there from everything else and the kind of units involved would be say here's an example the unit's the interest will be a dollar amount and how would you get that from here well principal is also in dollars and the rate of interest is in a percent number that will be as a decimal for us per year and then times the number of years so you can see from the units that years will cancel dollars then will equal dollars and this is just a pure number multiplied by dollars okay that's the basic outline of how simple interest is done now here's a problem which will take this into account and unfortunately this problem takes me to pages I couldn't get it all on one page two pages to lay out so I'll try and give you enough time to read this an investor has seventy thousand dollars and places part of it in bonds paying 12% of course that means point one two as a number and paying what interest per year and the rest of the money other than the part and bonds the rest of the money is in a CD CD stands for a certificate of deposit but that doesn't matter to us so much is just CD paying eight percent which is 0.08 interest per year so this investor has $70,000 and places it in two different areas bonds and a CD paying respectively 12% and 8% interest per year what's the question the question is she wishes to receive an overall return that's the phrase that's often used another one of those reality terms of course that means interest she ERISA has returned an interest of 9% on her original seventy thousand dollars which is 0.09 per year so the question is here is the question we are posed how much should she place in each investment how much of the $70,000 should go into bonds and how much should go into the CD so this is a problem that you need to think about try to organize your thinking realize you'll be using the interest formula I equals prt and put this all together in some fashion and take the time to do this now let's look at this interest problem first of all as in all of these problems I recommend that you do a translation now you may not be formal and write down translation when you do it but you need to do it regardless first of all there is the principle involved in this problem principle we're denoting by P and the P of course you remember is $70,000 now you notice I'm writing out dollars in English with the actual word I do that because the symbol for dollars appears in front and we like to have our units at the end so it's probably better to just write out dollars all right we want what we want X to be the dollars in bonds and we know that 70,000 minus X because we're putting everything into both of these is going to be the dollars in the CD so the entire $70,000 is going into bonds and CD if X goes into bonds everything else 70,000 minus X goes into the CD and then what we're going to do is we're going to do it we're going to write what I call an interest equation now what am i doing here this is something I'm gonna do in every one of these word problems you find a quantity that you can write an equation in the interest equation we'll say that we want a certain amount of interest out for the amount of interest that we're going to be getting off of each of these investments and then we'll be able to solve for the X that we're interested in so you have to find a quantity in each of these problems that you can write an equation in and I'll try and be very clear on that in each one so here's my interest equation first of all I'm going to write it out in English total interest a desired over one year okay that is what I want out of this and that will consist of the interest from the bonds which is one of my investments plus the interest from the CD that certificate of deposit okay so now I've written out an equation in English that's a good way to begin that is totally an interest interest equals interest plus interest okay now how much further can I go well I know specifically that the total interest I want is 9% I want 9% from my principal that is the $70,000 and that's supposed to come from 12% from bonds plus 8% from the CD now let me be clear on this the principal will generate 9% of it as interest there will the Barnesville jetan do generate 12% interest in the CD generates 8% somehow some combination of these will yield 9% I hope let me let me recall that this is about interest and then interest is equal to P times R times T and here T equals one year here we're just talking about a single year so each of these interests on either side of the equation needs to be broken down now into a p x and r times a T so I can do that the principal the total principal is $70,000 times the rate that I want is 9% so that's point O 9 times the time which is one year and that is the left side of the equation the total interest over one year seventy thousand nine percent over one year what will I do for the rest I do the same thing the amount being put in bonds is X the rate at which the bonds pay interest is 0.12 and the time period is one year and then plus what is the amount being put into the cd70 thousand minus X paying at 8% that's 0.08 times one year now please note this is a linear equation we've done lots of linear equations there's an X here and an X here and they're only to the power one so let's take this to the next page and start simplifying things well let me put it at the top of the page here let me be quite frank about this I'm gonna put this in my calculator to save me some time although you can do this in your head 70,000 times 0.09 the point O 9 will cut off two of the zeros so it's 700 times nine that's 6300 that's not hard to do but don't feel bad if you do that in your calculator so I get 6300 on the left and I'm suppressing the units the units we know are dollars I'll put them in at the end point 1 2 times X is just just that 0.12 X plus and then here 70,000 minus X times 0.08 again I can do that calculation the point of wait times 70,000 cuts off two zeros and the eight times seven hundred will give me 5,600 minus 0.08 times X so there I am with a an equation the same equation I had before and I'm on my way to solving it so what I'm gonna do are the obvious things I'm going to combine the two terms involving X and I'm going to subtract the 5,600 and bring it to the other side that will then leave me with 700 on the left and on the right I'll have 0.12 minus 0.08 is 0.04 X so reversing the order and then dividing I have X is equal to 700 over 0.04 and then I put that in my calculator and I get 17,500 and I now put the unit's on because I know what they are and since X was in bonds I'll go ahead and put dollars into bonds so seventeen thousand five hundred dollars into bonds and that means that the other part seventy thousand minus X is fifty two thousand five hundred dollars and that will go into the CD now those were the questions I was asked how much did she invest in each part to get nine percent out and now I'm done so there's one way to analyze this problem now your steps your individual steps may not be exactly the same as mine however the logic and the reasoning have to be there somewhere now let's go ahead and look at another type of problem this one comes under the title of a mixture model this is a model in which there are mixtures and you're trying to come out with a total mixture of some kind so let me read this to you and again it takes two pages I'm sorry I couldn't get it on one page in a chemistry lab the concentration of one solution is 10% HCl HCl stands for hydrochloric acid and of a second solution is 60% HCL so you have two solutions you don't know how much of each right now and that's probably what we're gonna need to find but you have one solution is 10% hydrochloric acid and another the 60% okay here is the question how many milliliters which are denoted this way and then the leaders is thousandths of the leader of each solution should be mixed to obtain 50 millimetres of a 30% hydrochloric acid solution so you want to take parts of the 10% in the 60% solutions mix them together and come up with 50 milliliters that's your amount of 30% hydrochloric acid give this some thought logically work it out try to set up an equation I set up an interest equation before try and set up an equation here for some quantity and I'll be back in a moment all right on this solution first of all again you may not be this formal but I'll go ahead and do it here I'll write out a translation we want what we want some amount I'll use the letter X but you can pick your own letter we want X to be a certain amount and we're measuring our amounts totally in milliliters of the 10% hydrochloric acid solution and so that means that the rest of it the 50 minus X because the total is supposed to be 50 milliliters the rest of it should be a certain number of milliliters of the 60% hydrochloric acid solution now remember these were the two solutions given to us we had to take a mix of them and get 50 milliliters and if one of them is X the other one has to be 50 minus X so that part is easy what I'm now going to do is I'm going to write and HCl equation a hydrochloric acid equation I'll have total hydrochloric acid at the end equal to the sum of the hydrochloric acids from the two amounts of solution just like I wrote an interest equation before so as long as you pick a quantity and write an equation you're fine so total hydrochloric acid in the 30% solution now that will equal the hydrochloric acid in the 10% solution however much that is that we need plus the amount of hydrochloric acid in the 60% solution and just for the record let me mark something about the units here the amount of hydrochloric acid will be hydrochloric acid per milliliter that's a small M times milliliters and you see the milliliters cancel and you're left with hydrochloric acid percentage so total hydrochloric acid and 30% solution well it's 30% 0.30 of how much of their of the solution is there supposed to be at the end 50 milliliters then that will be 10% of the amount of hydrochloric acid which we called X plus 60% of the leftover amount which is 50 minus X that's the amount of the 60% solution so this is a hydrochloric acid equation this is linear and since we've done many linear equations before you know what we're going to do we'll multiply things out point 300 times 50 will leave me with 15 point 100 is the same as point 1 X might as well simplify that 0.60 times 50 is 30 minus 0.6 times X all right we will now take the 30 over to this side and we'll combine the 2 X's as we've done before moving that 30 over gives me minus 15 equals minus 0.5 X so again solving for x and reversing the order because I like my X's on the left that's minus 15 over minus 0.5 and that is let me put that down here that will be 30 and now I'll put the unit's milliliters of the 10% solution and that is how much I wanted for the 10% and then the rest 50 minus X is going to be 20 milliliters of the 60% solution now if you combine these two together 20 milliliters of 150 of the other of course you end up 30 of the other you end up with 50 of course and with these 10% and 60 percent ratios you will end up with a final solution of what was it that we wanted 30 percent solution so that's how I would have attacked that problem let's look at the third type and this is a type that occurs everywhere all of the time and is very very useful we'll call it the uniform motion model but you'll have a shorter name as soon as I finish explaining it and let's let some letters stand for some standard things let's let s this is very standard stand for distance traveled and let's let V stand for average velocity now there are other words used here instead of average and instead of velocity instead of average you will see also constant or uniform as in the title of this model the velocity is sometimes referred to as the rate and velocity equals plus or minus the speed and I'll talk a little bit more about that later T will be time elapsed so putting these all together we come up with a very famous formula involving distance velocity and time which everyone who's ever driven anywhere knows s equals V times T distance is velocity times time this is uniform motion because the velocity is uniform and the units and words work out like this distance equals distance per time that's philosophy times time and you might see units like miles equals miles per hour times hours etc okay and you remember that velocity is measured in miles per hour or meters per second or any of a number of other things so s equals V times T distance equals velocity times time now what about that business about velocity and speed that I said something about well let me make a remark just so we can clear that up right now velocity for us will be plus or minus speed because velocity by definition has a magnitude which is a size plus a direction whereas speed only has a magnitude that is to say we could say five miles per hour east and over here we could say five miles per hour five miles per hour does not give you a direction it just says how fast you're going five miles an hour east gives you a direction in addition to the speed now for us because we're not going to be dealing with directions so much usually velocity because we're only dealing with with algebra not looking calculus or vectors velocity will be either forward or backward and forward or backward can be conveyed by plus or minus in a given situation so I wanted to just point that out because sometimes those are confused and there it is very often written down and you'll see this in the problem coming up that's why I'm looking at it that the velocity and speed are confused in the problem and you have to abstract out what the velocity is by putting the correct plus or my sign okay let me pose now a problem to you this is a familiar type of problem here's the situation a motorboat heads upstream a distance of 24 miles on a river whose downstream current current is by definition downstream but I thought I'd remind you of that downstream current is three miles an hour so the boats going upstream against the current so the current is tending to push it back at three miles an hour you know that the trip up and back the 24 miles up in the 24 miles back takes six hours okay that's all given information assume and this is the assumption that makes this a uniform motion problem we will assume that the motorboat maintains a constant or uniform speed relative to the water and the relative to the water will be where the velocity comes in where we pick it's whether it's going with the water downstream or against the water upstream we'll assume that that's uniform okay what was that constant speed of the motorboat given all this information how fast is the motorboat going in order to go that amount of 24 miles upstream and then 24 miles downstream over six hours the question again is what was that constant speed of the motorboat now give this some thought set this up carefully draw pictures you will need to draw pictures and diagrams to help you understand this and when you're done with that I'll come back and show you how I attacked well let's see about the river in the motorboat first thing you note is that we're talking about distance and we're talking about speed and time so that should bring to mind I'll put that up here is a recall that should bring to mind the uniform velocity or uniform speed equation s equals V times at t which we're going to use quite a bit here now what is it that we want and it's always good to start out your problem with what you want and later on then you'll know exactly what it was you were looking for we want the constant speed of the boat well I don't want to use V here because I'm gonna use V in my structural equation here so I'm gonna modify the V a little bit and then later on I'll simplify let's call this beat V sub boat okay so we can keep track of it and what's that going to be well that's the speed which remember we're assuming to be constant or uniform the speed of the boat now this is what we would like to discover that by the way is in miles per hour okay well that's what we're looking for and now we'd like to analyze the problem and see if we can draw a picture to maybe line up the ideas in some way that's visually clear well half the battle on these problems is really getting a reasonable picture so that you can clearly think about what you're doing so I'm gonna draw a river now so here's here's my river this is one side of the river and way down here is the other side of the river okay in case there's any doubt river there okay now the river has a current and I will just assume that my current in my picture is moving from left to right downstream okay so this is the current and we are given the information that it is three miles per hour now somewhere along this river a boat let's say here at this dot a boat starts and then what does it do it travels upriver 24 miles well s is the letter we're using the stand for distance so let's call that s equals 24 miles and that's what the boat first does now you notice it's moving against the current then it goes 24 miles downstream so putting the drawing a little bit below to keep it clear it then goes s equals 24 miles downstream and this is where the boat hence of course these two will probably be closer together but that's schematically what's happening and remember the first one is upstream I even write that down here upstream and the bottom one is downstream and downstream is the same direction as the current of course well we've got miles per hour mentioned for the current we are looking for the speed of the boat we have two distances is there's a time involved here and we are told that the time elapsed which will be for a round-trip now round-trip remember is up and then back that will be six hours okay that's the information we were given in the problem drawing out in a kind of schematic way now I want to pull out of this some sort of equation probably involving this s equal V T so that I can solve for visa boat all right let's see if we can do that now let me bring that picture back first of all I want to note something I mentioned this earlier about speed and velocity when we say speeds here then you might want to note that speeds in opposite directions in opposite directions have opposite plus or minus signs that's the way we'll keep track of going upstream and downstream well rather I should say with respect to the direction the boat is going if you're going with the boat will say that's positive if you're going against the boat we'll say that's negative okay with that in mind let us figure out what the upstream as the boat goes upstream what its net speed so the upstream net speed of the boat now that is the actual speed of the boat as someone would measure it say from the shore this is the put from the point of view of the boat what's what do we have well the boat is traveling upstream at a velocity from the boats boats point of view of the boat whatever that is which we don't know then working against that velocity is this three miles an hour of the current which means that the speed of the boat visa boat minus three as seen from the shore is visa boat minus three and that is in miles per hour because it's trying to go up steep upstream at that visa boat speed but the current of course is forcing it backwards and so the net speed upstream is that number and this makes sense when you think about it you should you should explore what the variables could be at a problem to see if they make sense for example if the boat had a speed of zero what do you think would happen well then it would simply drift downstream with the current if the boat was zero then you'd have minus three here and the boat would be traveling downstream negative at three miles an hour which makes sense if the boat went upstream at three miles an hour it would exactly counter the current and wouldn't go anywhere so if we had three minus three here we have zero and that makes sense also so see by analyzing what the variable is you can see whether your argument makes sense okay well if we have upstream we can now write down downstream and come up with a similar result downstream net speed of the boat and in the downstream direction we have the velocity of the boat visa boat is now going downstream and it is going with the current so that means the to add together and you have visa boat plus three miles per hour as the net speed the speed that someone from shore would notice that the boat is traveling on this river going downstream okay good we have really made progress here we've got a lot of sense out of this now let's continue a little bit further and use finally that s equal V times T equation so using s equals V times T okay well the way I analyze this is there's the distance upstream so that'll be my first equation the distance upstream upstream and I know that that's 24 miles okay but it's the same distance downstream but it's 24 miles upstream and what was the velocity upstream recall it was visa boat minus 3 that was the upstream velocity of the boat times the time it takes to go upstream now I don't know what that is so I'm just gonna put time upstream for the moment now I may have to solve for that I may not but for the moment that's correct equation let me write another one down collecting information here distance downstream and distance downstream is 24 same distance but now the velocity as you know has changed these sapote plus 3 because we're going down with the current and then the time is again a time I don't know so I'll just write down that time downstream now what else do I know I know that the total time involved in this entire trip this entire round-trip is 6 hours I'm suppressing the unit's here but this will be in hours and of course that of course is the time upstream plus the time downstream added together now I have here all the information I think from the problem and I written it down in equation form now from this I want to what do I want to do I want to solve for visa boat okay that is what I want to look for which means all the other variables somehow have to disappear so looking at what I have here I've got an idea what I will do in the first equation is solved for T upstream get T upstream equal to something the second equation T downstream equal to something and then I can add those two something's which will involve only visa boat here and set it equal to six and I'll have an equation that only involves visa boat the variable I'm looking for now these are the kind of things you begin to see as you start writing things down in a neat fashion so I will now do a little rewriting following the same pattern on the previous page this is the upstream or the distance upstream equation now I'm solving for time so this is 24 over V sub boat minus 3 equals T upstream and the downstream one I do similarly downstream that's 24 over V sub boat plus 3 equals T downstream so putting those together into that last equation which recall was total time equals is 6 equals the upstream and the downstream times well we now have the upstream and downstream times written in terms of V so now I have an equation this will be a time equation and what do I have I have 6 is equal to 24 over visa boat minus 3 plus 24 over visa boat plus 3 and that equation is entirely in terms of the single variable V sub vote the velocity or the speed if you like of the boat which is what I am looking for so now I just need to solve good I'm in very good shape at this point now before we go any further let's look at the algebra involved here notice that algebraically these two equations two terms here are undefined when visa boat is three because that would be division by zero or visa boat is minus three well luckily that coincides with the physical requirements so let me write that down algebra requires that visa boat is not equal to three or minus three because if it's equal to three then that means it's not going anywhere because it's going upstream at the same rate as the current is going downstream and if it's going minus three that means it's heading downstream which is the opposite of the direction it wants to go now algebraically of course these correspond to division by zero which we don't allow and as I just said this agrees with the physical situation so our mathematical model is sensible okay now use in this place of visa boat visa boat is going to be a little complicated to carry this particular symbol around I'll just use V in place of visa boat now that everyone knows that we're solving for the velocity of the boat I think we can just use that simple letter okay let's do it here is the equation from the previous page six equal twenty-four over V now minus three plus 24 over V plus three I would like to get rid of the fractions but before I even do that I noticed that I can divide by six and any time I see a simplification I'm going to do it it'll save me work later so I have 1 is equal to 4 over V minus 3 plus 4 over V plus 3 then I'm going to multiply through by V minus 3 times V plus 3 because that will clear the frak and that's allowable because I have already said that V won't be equal to 3 or minus 3 so multiply by the minus 3 times V plus 3 on the left that's multiplying times 1 that's easy and then V minus 3 times V plus 3 on the right multiplies by the entire side now 4 over V minus 3 and 4 over V plus 3 now on the left of course there's nothing to be done here except to multiply this out in simplification but luckily this is exactly what you need to get V squared minus 9 it is V minus 3 and V plus 3 so there isn't any middle term that's Pleasant on this side if we multiply through the V minus 3 will cancel in the first one I will not draw a line in there as I said before because that's dangerous however then I will have 4 times V plus 3 left over plus 4 times likewise here the V Plus 3 will cancel but I won't write it and they'll be V minus 3 now please note at this point the problem is no longer a rational problem as it was at the beginning it is now reduced to a simple quadratic so it's good to note these things okay since it is a quadratic this is V squared minus 9 on this side let's multiply out the right hand side and combine terms 4 times V here and 4 times V here gives me 8 V 4 times 3 is 12 4 times minus 3 oh that's nice minus 12 which of course adds to 0 so now I can take this and I can write it in standard form for a quadratic which is V squared minus 8 feet bringing that 8 v / - 9 equals 0 and now I have a nice quadratic to solve which I know how to do and it turns out that I'm lucky this happens to factor so I'm going to take advantage of it if I didn't see a factorization I could have used the quadratic formula I'd get the same answers the factorization V - 9 + V + 1 seems to do the trick the minus 9 times 1 is - 9 + -9 v + 1 V is - 8 V so we're okay there well that immediately means that so V is equal to 9 or V is equal to minus 1 and remember this is visa boat so I will translate it so this is visa boat is equal to 9 miles per hour as one solution or of course this one when you think about it doesn't make any sense it means the velocity is negative the the velocity of the boat will not be negative because it's not going backwards so we'll discard that and we discard it due to the physical situation mathematically it is perfectly correct keep in mind that the answers you get are mathematically correct they simply may not mean anything in the problem because this is physical and minus 1 doesn't mean anything for us so this is visa boat the speed we were looking for and if you want you can do graphical confirmation I'll leave that to you if you like and you might see that although this is the solution there could be other solutions because anything above 3 miles an hour will allow the boat to move 3 miles an hour balances the current but if you're above 3 miles an hour you can move and if you want to go other distances and other times you'd have other velocities but if you want to look at the velocity actual graph for this I will leave that to you to check so there we are we've analyzed this problem in some depth you may not follow exactly this pattern but you'll follow something like it and remember what I said earlier you want to think about the kinds of questions I'm asking myself in addition to the things I'm actually writing down okay let's look at another problem here is an area problem it's slight variation on a traditional area problem from calculus we won't do the calculus part but we can do the algebra part from each corner of a square piece of sheet metal you've got a big square of sheet metal now you remove a square from each corner and the square has side 9 centimeters okay we don't know the size of the sheet metal that'll be something we're going to want to find but we're going to remove a square of side 9 centimeters from each corner then we'll turn up the edges of what's left to form an open box a box with no top if this box that we have created now this open box must hold 144 cubic centimeters that's volume what should be the dimensions which are the sides of the square piece of sheet metal that you started with so you start off with a square piece of sheet metal you cut squares out of each of the corners 9 centimeters on a side and you turn the the rest of the edges that are left upward and create a box with an open top I'll draw that when we come back and the question is if you want to enclose 144 centimeters cubed of volume what should the square piece of sheet metal have in the way of dimensions to begin with now you try that problem and we'll come back and see how I would have done it all right we're back and it's time to look at this box now first of all I want it to give you a picture to work with what I've done here is I've taken a square piece of paper and I've cut little squares out of the corners so what you end up is this cross shape and then if you see I'll try and do this here if I bend up the sides I end up with you see a box a box with an open top now if I want this box to enclose a certain amount of volume then this sheet that I started with has to be a certain size and that's what we want to find out in this problem so if you have a little piece of paper like this and you can create your own little box you see exactly what I'm doing okay now let's get some of this on paper here's what we want again we want to start off asking the question what is it you want we want and we need to name the side of the box let's call it X for no good reason perfectly good letter x equals the side and that of course is measured in centimeters if you remember the statement of the problem centimeters of the sheet metal square the square of metal that we begin with okay that's what we want to look for well I probably should draw a picture and let me do that right now here's a square and this is a square of the original sheet metal so we want the side of this to be X and I'll mark it on both sides even though it's a square we know that the sides are all the same I'll mark it anyway so that's what we want now what we're doing here is we are cutting out of each of the corners a square all these squares are the same size try to make them look the same here more or less and all of these squares that we've cut out have side 9 so these are 9 centimeters squares that I'm cutting out of this larger square okay I might note while I'm here since the length of the side is X and the length of each of these squares sides is nine that means this part right here in the middle must be the total amount X minus two times nine which is 18 so this little part in here must be a size eight X minus 18 as is this part up here okay I think with that information I'll even try now to draw a three-dimensional picture of the box I showed you a moment ago and let me draw that this way it'll be a little bit longer maybe then it's tall the bottom is a square because the bottom comes from this piece in the middle doesn't it and you just draw this out here so there we have the box it is an open top box so that's why you can see inside there the square bottom which is this piece in the middle here which comes around to be the bottom square is of course the length X minus 18 on a side so this side here is X minus 18 as is this side and then the height of the box comes from folding up these flaps and they're all of height 9 so maybe I'll put that back here okay also one other thing before I move onward see all I've done now is draw the picture and translated the English into mathematics as far as sketch goes now I wanna observe something else I'm going to be asking for a way of finding this particular variable X now this X stands for a certain physical measurement distance we can see immediately that X cannot be less than zero because distance is no less than zero but we can see something else from this picture since X has to include these two squares here on all of its sides X must be bigger than nine plus nine which is 18 observe that physically X must be bigger than nine plus nine which is 18 now that puts a restriction on the domain of the variable X and that may have some bearing on the problem later when we do our calculations so I think we've got a lot of information now from this pick we want to talk about the volume involved here so we'll need to know the two the three sides here X minus 18 twice and nine and with that let's go ahead and write down what we know we know that volume in general the general formula for the volume of a box like that is going to be height times width if I use these words times length and of course that means that centimeters cubed just to show you that the units are going to work out the centimeters times centimeters times centimeters now we are given the volume we are told that the volume must be 144 centimeters cubed that will then equal the height which is nine times the width which was eight X minus 18 times the length which is also X minus 18 so now we have of course a quadratic and we know how to solve quadratics so we're on our way let us continue this is 144 equals nine times the quantity X minus 18 squared and I will now divide the 9 into the 144 I will also reverse the direction because I prefer to have my X minus 18 squared on the left that's just a matter of taste it doesn't make any difference mathematically so that's 144 over 9 on the right and now I'm going to remember that form that we studied earlier box squared equals P means that box is plus or minus the square root of P remember that using that here treating X minus 18 is the Box I now have X minus 18 then is equal to plus or minus the square root of 144 over 9 well we recognize that 144 is 12 squared 9 is 3 squared so this is plus or minus 12 over 3 and 12 over 3 of course gives us 4 so X minus 18 is plus or minus 4 if we then break this into 2 equations we get X minus 18 is equal to 4 or X minus 18 is equal to minus 4 just separating the plus and minus cases that means X is equal to 4 plus 18 is 22 and that of course remember is centimeters or X is equal to 18 minus 4 is 14 that is also in centimeters however remember we discard this because recall that X must be greater than 18 so it can't be 14 so that means there is only one solution here X is 22 centimeters so I will finally be able to answer the question always remember that your goal in all of this is to answer the original question and not just to finish the work when you think you should be done dimensions of the sheet metal square the dimensions of the sheet metal square are 22 centimeters by 22 centimeters and that is the answer to the original question so you've seen a few examples now I hope they've given you some insight and how to attack these modeling problems now we're going to move on to another type of equation to be solved

Channel: UMKC

Views: 20,003

Rating: 4.8032789 out of 5

Keywords: Interest Model, Principal, Interest, Rate of Interest, Simple Interest, Mixture Model, Uniform motion model.

Id: D9pYSbUzaVM

Length: 63min 45sec (3825 seconds)

Published: Tue May 05 2009