Chapter 3 - Models Everywhere

1. What Is a Model . . . ?   A model of an object, process, or system is a picture or representation of it that preserves relevant properties or relations. A quote from physicist E.W. Hughes in a September 1997 issue of Science News reflects this definition -- "We want to make sure we have the correct picture or model of what is inside the proton . . . "  Most of the charts, graphs and energy diagrams of the first two chapters are models. All the figures in the first two chapters are examples of models.
      Models can be maps, matrices, and even myths.  A map is a representation of the territory.  A table or matrix might represent the pieces of a game board such as chess. The Gorgon Medusa of Greek mythology is a good model of the octopus.
      A (mathematical) model is always relatively more abstract than what it represents. A model and its "target" form a relationship. The target is some object, process, or system.  It is not quite right to treat the term model as a stand-alone noun. We should use the phrase "model of [the target," as in "This map is a good model of the Chesapeake watershed."  Here is our definition --

Do you spot a word that is very unusual for a mathematical definition . . . ?   How about relevant?  This where judgement comes in. This is what makes applied mathematics different from the abstract studies of such subjects as algebra.

     The word model has become increasingly fashionable, a testimonial to its evocative power. We should be aware of attempts to wrap a scientific mantle around some pet notion or product in order to market it, or to make something appear to be more scientific than it actually is. Here are some alternate uses of the term --

The last example, scale model, is very different from the first four because there is a very tight correspondence between a physical model and the target object or process.   Rather than being more abstract than the target, engineers' physical models are usually smaller replicas of the target. It was noted in the Systems chapter that a change of scale can lead to a change in properties. A scale model of a ship or an airplane will have many handling characteristics unlike the target.  Engineers have a clever "dimensional analysis" method to deal with these differences. The section on units, later in this chapter, deals with related but very much simpler aspects of dimensional analysis.

2. Formulation and Interpretation   The target might be a process that we can observe directly although usually it is necessary to work from a verbal description and accompanying data. The first step is to formulate or construct a model of the target.  For example, if we observe responses to various inputs, we might plot the data points. After some mathematical transformations, such as regression analysis, we will give meaning to, or interpret our modified model and check it against the target.
     This reality check might suggest re-formulating the model, followed by a re-interpretation. This cycle makes up the modeling process. Formulation takes us from the concrete to the abstract; interpretation takes us from the abstract to the concrete. A diagram of this process appears in Figure 1.

      Like scientific theories, models are used to clarify, to suggest further investigations, and to predict. They also share with scientific theories (and almost everything else) the No free lunch principle. The trade-off for simplicity, structure and generality is the loss of complexity, information and specificity, as shown in this table --

	Model		Target
	Simplicity		Complexity
	Structure		Information
	Generality		Specificity

      In chemistry there are many ways to represent chemical compounds. Two methods use simple algebraic notation and simple graphical notation.  Let us look at the four simple hydrocarbon fuels, combinations of Carbon (C) and Hydrogen (H) atoms. These fuels burn in oxygen to form water H₂O and carbon dioxide CO₂ (a "greenhouse" gas).
     The algebraic representations are easier to display, so let's look at those first.  Subscripts are used to indicate the number of atoms in the compound.  

	Methane (Natural gas)	CH4
	Ethane	CH4
	Propane ("Bottled" gas)	C3H8
	Butane	C4 H10

We construct a graphical model by representing Carbon with a large black dot (•) and Hydrogen with a small blue dot (• ) --

CH4

CH4

C3H8

C4 H10

  Figure 2      

The graphical models tell us about structure (bonding) and also suggest step-by-step generalization.  The algebraic model is more abstract, but notice how compactly it captures the Carbon-Hydrogen pattern, C_nH_2n+2.

Exercise 3.1   Construct two models for the fifth hydrocarbon, pentane.

Exercise 3.2   Come up with three examples of objects, processes, or systems and their models.

3. The Representation of Data   In 1990 there were about 20,000 rhinos left in the world, down from 850,000 in 1910. The other figures re 500,000 in 1930, 220,000 in 1950, and 80,000 in 1970.  Below are three ways to display these data -- a table and two charts. They are logically, but not psychologically, equivalent.

Year	1910	1930	1950	1970	1990
Rhinos  (in 1000's)	850	500	220	80	20

Figure 3   Rhino data plotted as a bar chart

Figure 4   Rhino data plotted as points with connecting lines

Exercise 3.3   Predict what you think the rhino population will be in the year 2010. Which of the three above displays was most helpful in making your estimate?

Exercise 3.4   In 1986 about 130 Florida manatees ("Sea cows") died, but the figure skyrocketed to 415 in 1996. The deaths are largely attributable to human activities (especially speed boat collisions and propeller slashes). The intermediate figures are: 135 in 1988, 200 in 1990, 115 in 1992, and 180 in 1994. Construct a table and two charts for manatee mortality.

4. Units   In applications of mathematics it is usually very important to keep track of dimensions or physical units such as feet, volts, dollars, etc.  in pure mathematics the sum of 15 and 2 is 17 and that's the end of it, 15 + 2 = 17.  However, units can completely change arithmetically bizarre equations such as --

15 + 2  =  1   or   15 + 2  =  7

into correct statements. They can make sense in the context of money and ordinary measurement --

:             15 cents + 2 nickels  =  1 quarter   or   15 ft + 2 yd  =  7 yd .

So do not neglect dimensions or physical units.  Units are rarely mixed in scientific formulas.  For example, each of the above equations would be written in the same units --

            3 nickels + 2 nickels  =  5 nickels   or   15 ft + 6 ft  =  21 ft

so that the units on the left side of the equation match the units on the right, and we end up with our usual arithmetic.  For example, in the  speed = distance/time   formula, it is expected that the units will match --

78 km / 3 hr  =  26 km/hr.

Both sides of the equation have the unit kilometers per hour.  We get a different take on the formula by solving for distance --

                45 miles/hr x 3 hr  =  135 miles   ( distance = speed x time ).

Notice how the unit hr "cancels" in this equation so that we have the same unit on both sides of the equation, miles.
      Do not be fooled by abbreviations such as mph and mpg. They need to be expanded so that the units are explicit:  mph = miles/hr and mpg = miles/gal.  If scientific units do not match, you can be sure an error has been made somewhere. A satellite was lost because someone had mixed English and metric units.

Exercise 3.5   Correct the equality by associating units with the numbers  2(    ) + 5(    )  =  1(    ).   [Hint  Try money.]

Exercise 3.6   In the section on scale, 14 trillion dollars of US national debt was expressed as 50,000 dollars of debt per person. Add units to the below arithmetic to justify this, assuming there are about 280 million people in the US --

                                                                           14x10¹² / 280x10⁶  =  14x10⁶ / 280  =  .05x10⁶  =  5x10⁴.

Ask Marilyn column (Parade Magazine, 31 AUG 97).   "Suppose it takes one man 5 hours to paint a house, and it takes another man 3 hours to paint the same house.  If the two men work together, how many hours would it take them?  This is driving us nuts. (Joyce Deck, Ypsilanti, MI)"  This does not sound as if it has anything to do with units, yet that is one way to deal with it.  But first, Marilyn's solution of the "Ypsilanti" problem:

      No wonder.  If the two men work together, it will take them one hour, 52 minutes and 30 seconds to paint the house. You can prove the answer like this: Say instead that it takes one man 6 days to paint a house, and it takes another man 3 days to paint the same house. Working together, it would take them 2 days, because the first fellow would finish 1/3 (2/6) of the house in that amount of time, while the second fellow would finish the other 2/3 of the house. So, in your problem, the first fellow would finish 1.875/5 of the house while the second fellow would finish the other 1.875/3 of the house. (One hour, 52 minutes and 30 seconds equals 1.875 hours.) And 1.875 fifths of a house plus 1.875 thirds of a house equal one whole house.

Got it?  Marilyn is usually much better. The first thing to notice is that if the fast painter can paint a house alone in three hours, the job will be completed sooner than three hours if the slow painter (five hours per house) joins him.  We might estimate the time by first averaging the painters' hours, (3+5)/2 = 4 hrs.  Then, since the two of them are both painting, divide this average by 2.  So the answer to the Ypsilanti should be about two hours.   Now let's do it via units, a sort of dimensional analysis:

Fast painter --   3 hrs / house or 1/3 house per hr = 1/3 house/hr

Slow painter --   5 hrs / house or 1/5 house per hr = 1/5 house/hr

Combined --     1/3 house/hr + 1/5 house/hr = (1/3 + 1/5) house/hr = 8/15 house/hr.

This translates to 15/8 hrs/house or 15/8 hours per house.  So it will take them 15/8 = 1.875 hours, or one hour and 52 minutes to paint the house (1:52:30, if you think 30 sec is significant!)..

Exercise 3.7   Visualize the Ypsilanti house as  3x5 = 15  sub-jobs and then see how many of the sub-jobs the two painters can finish in an hour.  Can you figure out how long it takes them to paint the rest and thus solve the problem?

Exercise 3.8   A company has won a bid to clean up a toxic dump in one year and three months.  It sends in two teams. The A team could clean it up alone in two years, the B team in three.  If they are assigned to the dump and the teams neither help nor hinder each other's efforts, will the company get the job done on time?

4 Mexican Free-tail Bats   A million Mexican free-tail bats are estimated to consume 10 tons of moths, mosquitos and other insects each night (G.F. McCracken, "Bats Aloft," in Bats International, Fall 1996).  Numbers this large should be scaled down so as to be a little easier to comprehend.  Bats and pounds of insects are a lot more manageable than millions of bats and tons of insects.  It is much safer to do this scaling by starting with the dimension of each quantity and then keeping track of the dimensions. There are two ways to look at this.

The final numbers will just be reciprocals of each other, but the calculations look different. The first calculation is a little harder, so we will do it.  We want lbs/bat:

So the answer is about 1/50 of a pound of insects per free-tail bat.  The answer is mathematically correct but does it convey much information -- does it give you a feel for how much a bat consumes in a night . . . ?   Let's change pounds to ounces: 1/50 lb x 16 oz/lb = 16/50 oz = 32/100 oz = .32 oz . This is about 1/3 oz, so the answer is:
                                A free-tail bat consumes an average of about 1/3 of an ounce of insects each night.
Notice that the answer is given in common language – no esoteric abbreviations, no scientific jargon, no snow-job.  However, the answer could be improved by being even more down to earth.
      One-third of an ounce does not suggest any ordinary object that we can use as a bench mark.  Exercises 9 and 10 will suggest other ways to phrase the answer.

Exercise 9   A raisin weighs about a gram. Convert the free-tail's consumption to grams and then express the answer in down-to-earth form.

Exercise 10   Estimate the weight of a single peanut in ounces (Buy a small bag and count them!) Then express the free-tail's consumption in a down-to-earth form.

Exercise 11   Reduce the bat data to bats per pound of insects. You already know what the answer is because of the above work. The important thing is to carry out the calculations step-by-step, being sure to include units at each step. Express your answer in common language.