Chapter 1 MATHEMATICAL MODELS 1.1 …

1 Chapter 1 MATHEMATICAL MODELS INTRODUCTION All of mathematics has its roots in attempts to describe certain aspects of the worldin which we live. We begin with a very simple example. In my apartment, garage,and workshop, I have a large number of metal cabinets with plastic drawers. I usethem to store small items such as nuts, bolts, buttons, ink cartridges, and pegboardhooks. There are some extra drawers here in the kitchen, while in the bathroom isa cabinet that is missing a few drawers. Are there enough drawers in the kitchen tofill out the cabinet in the bathroom? One way to answer this question is to bring thedrawers from the kitchen to the bathroom and put them into the cabinet. Anotherway, a way that uses a simple MATHEMATICAL model , is to count the drawers and thenumber of empty spaces in the cabinet, and compare the two numbers obtained.

2 Inthis simple situation, it probably doesn t matter which method we use. On the otherhand, suppose I did not have the extra drawers, and wanted to order some. It wouldbe very cumbersome to order a few , then return the extras if I got too many, orreorder if I got too few. In this case, a trial and error manipulation of the actualphysical objects is much more difficult than the mental manipulation involved incounting the number of empty spaces in the cabinet. This is one of the majoradvantages of MATHEMATICAL MODELS . The importance of this advantage is evenclearer in larger projects, such as landing someone on the moon. Here thecalculations that go into the planning are much more sophisticated than simplecounting, and attempting to obtain the same results by trial and error is so muchmore difficult and expensive (and wasteful of astronauts) as to be THE MODELING PROCESS Throughout this book we will be studying many different MATHEMATICAL MODELS .

3 Inthis section, we pause to consider the process by which such MODELS are built. Thedesire to build a model of some aspect of reality usually comes when we haveenough information to be aware of that aspect, but not enough information to feelwe fully understand that aspect. For example, at the time this is being written, ithas been observed that there are changes in the composition of an astronaut s bloodfollowing a space flight. Before that observation had been made, no one would havethought to try to model the blood of astronauts as distinct from, say, the blood ofplumbers, physicians, or deep sea divers. On the other hand, if existing MODELS ofthe behavior of blood were able to explain (as they are not) the observed changes inastronauts blood, there would be no need to build a special model for this we do try to build a model , it may or may not be MATHEMATICAL .

4 That depends onwhether the important aspects of the situation we are modeling can be treatedquantitatively. Since Sir Isaac Newton invented MATHEMATICAL physics about 300years ago, MATHEMATICAL MODELS have been spectacularly successful in the physicalsciences and engineering. This success has led scientists in other areas to try toapply quantitative methods to their own fields. As a result, the range of things thatcan be treated quantitatively has been steadily increasing for centuries, andcontinues to increase the typical reader of this book probably has not had much prior experiencebuilding MATHEMATICAL MODELS in any field, our discussion will begin with what isprobably the closest thing to a MATHEMATICAL model the reader has seen: textbookword problems (also called story problems, or even applied problems).

5 Theseexercises are an attempt to introduce the student to at least a small part of themodeling process. We will examine an atypical word problem, and later return to amore general consideration of have already noted that real modeling usually begins with some, but notenough, information. In textbook word problems, we are usually handed all theinformation we need in a sentence like:John has 17 coins worth $ exercises usually contain no extraneous information. In fact, we are oftenadvised that we must use all the given information in their solution. This may begood advice for getting through school, but it is poor advice in any real modelingsituation. In a real problem situation, a major step is often sifting through theavailable data to select what is really relevant and eliminate what is not. (Forexample, what aspects of space flight might be relevant to an astronaut s bloodcomposition?)

6 No model can faithfully represent every aspect of a real of the information has to be sifted out just to make the modeling processmanageable. Indeed, a model serves no useful purpose unless it is in some waysimpler than the situation modeled. Usually, this simplicity is attained by selectingonly some of the characteristics of the real situation for inclusion in our model . In areal problem, this selection process may help us in the next step in the modelingprocess: refining our definition of the textbook exercises, the problem is defined for us and never altered or refined:Find how many coins of each type John real problems, however, this step may require a great deal of effort. Forexample, an engineer working on engines for an automobile manufacturer mightsift through tons of information on current and past engines built by his or hercompany or its competitors.

7 Perhaps, in examining the data, the engineer finds thatone of the company s engines appears to be below average in efficiency. This mightlead to a rough initial formulation of a problem to be solved: how can the engine bemade more efficient?- 2 -Once we have roughly defined the problem, we need to reconsider the availableinformation. Having a specific problem in mind may give us a new perspective onwhat information is relevant. We may also find that certain crucial bits ofinformation are missing. This happens rarely in textbook exercises, but almostalways in real problems. In real problems, the process of gathering the additionalinformation we need may be very time consuming and expensive. Some informationmay be impossible or expensive to obtain. In that case, we may have to makereasonable guesses or assumptions.

8 The adequacy of these can be checked at a laterstage of the modeling process. If they prove inadequate, we will have to revise ourassumptions or gather more we ve gathered or guessed the information we need, we may find that we haveto refine our definition of the problem. Perhaps our engineer finds that theinefficiency of the engine being studied is in fact due to the carburetor used on theengine rather than the engine itself. This may lead to further refinement in ourdefinition of the problem and the gathering of still more data. This bouncing backand forth between data analysis, data gathering, and problem definition may berepeated many times during the modeling we have settled on a definite problem to attack, we isolate certain aspects ofthe problem situation and represent them with MATHEMATICAL symbols.

9 In order tokeep the model simple, we represent only aspects of the situation which we thinkare essential. We choose as few of these as possible in order to keep the modelsimple. We usually leave out aspects for which MATHEMATICAL representation isdifficult. In the engine example, we might want to consider other engineers opinions as to why our problem engine is inefficient, but it would be hard toincorporate opinions into our MATHEMATICAL model . This process of isolating aspectsof interest to us, while ignoring other aspects, is a process of abstraction. It is akey part of the modeling mathematically modeling an automobile engine is far beyond the scope of thisbook (and the abilities of its author), in the remainder of this section, we will modelonly the coin problem:THE COIN PROBLEMJohn has 17 coins worth $ how many coins of each type John , we have not been told what types of coins are involved (dimes,nickels, quarters) or even how many types are involved.

10 Such ambiguities aretypical of real problems. In the absence of any further information, we have nochoice but to play around with the problem and see if we can find any reasonablesimplifying assumptions that might help us to obtain a solution. This step of playing around with the problem is rarely a part of algebra textbook exercises, butit is almost always a part of solving real begin with, the table below- 3 -Type of Coin Value of 17 Such CoinsPenny $ $ $ makes it clear that more than one type of coin must be involved. (Note that there isno need to consider quarters or any higher denomination of coins once we see thatthe value of 17 dimes is too much: $ > $ )Let us see what happens if we assume the next simplest situation: two types ofcoins.