Unit III - Part C

3.5 Power Models

Exponential equations and graphs are among the most important non-linear models. But many important non-linear patterns are modeled better by equations and graphs of other types. Perhaps the second most common simple curves are those that are based on equations with the form y = A(x), for various specific values of A and b. Models of this type, where values of the input variable are all raised to some fixed power, are commonly referred to as power models.
The patterns of growth produced by power models can be used to calculate and explain patterns of change in a variety of familiar situations. The next several problems explore only a few of the many possibilities.

1. In many engineering and construction projects, the first step is building a scale model of the proposed machine or structure (either on paper, on a computer screen, or in real materials). Then plans have to be scaled up to real size. For example, the New York City parade on Thanksgiving Day, sponsored by Macy's Department Store, is famous for its display of very large balloons in the shape of cartoon characters. Suppose that a scale model is made of intended real size.

(a) If the model is 2 feet tall, how tall will the balloon be?

(b) If the model has a belt that is 1.5 feet around its waist, how long will the belt be on the large balloon?

(c) If the model has a surface area of 6 square feet, how many square feet of material will be required to make the large balloon?

(d) If the model holds 2.5 cubic feet of air, what will the real balloon's volume be?

(e) How would your answers to questions (a) - (d) change if the model was or or of real size?

2. If your answers to the various parts of question 1 are typical, you probably got (a) and (b) correct without much hesitation; you probably also puzzled over questions (c) and (d). To help make an informed guess about the answers to these questions about area and volume, it helps to do some exploration of growth patterns with a simpler figure. The sketch that follows shows a cube--a box with all edges the same length.

(a) Complete the following table of sample girth, surface area, and volume measurements and describe the different patterns of change in the rows:
Edge Length           1    2      3      4      5      
Surface Area                                           
(b) What rules will give the girth (distance around), surface area, and volume of any such cube? How do the differences in those rules help to explain the differences in rates of growth for the three size measurements?

(c) If the edges of one cube are twice as long as those of another, how are the girth, surface area, and volume of the two cubes related? What if the edges of another cube are three times as long as those of the first? What if the edges of yet another cube are 20 times as long as the first?

(d) How do your answers to (c) suggest answers to questions 1(c) and 1(d) above?

3. One of the most famous scientific examples of a power model is the relation between time and distance for objects in free-fall subject to gravity. For example, if you were to drop a golf ball off the top of one of the campus high-rise dorms, the relation between elapsed time (in seconds) and distance fallen (in meters) is given approximately by the equation d = 5t. This model does not account for the effect of air resistance, but, for modest time and distance values, that simplification is not too much of a problem.

The basic quadratic power relation between time and distance was actually known in the 15th century (if not earlier), when devices for measuring time were really very crude. To study the question, Galileo came up with an ingenious idea for an experiment that "slowed down" gravity while retaining the fundamental pattern of its effect. He suggested collecting (time, distance) data as a ball rolled down a smooth ramp, somewhat like that pictured in the following sketch:

(a) Use a stopwatch to collect (time, distance) data for a table like this:
Time in seconds                                                                    
Distance in meters  0.50   0.75   1.00   1.25   1.50   1.75   2.00   2.25   2.50   
(b) Enter the data in your calculator's stat lists and get a plot of the (time, distance) data. Then experiment with various potential modeling rules of the form
y = Ax, testing each by plotting it over the data to see how it fits.

(c) Use the STAT CALC command PwrReg (for power regression) to get what the calculator suspects is the best fitting model of the power type and compare it to your data.

(d) Try LinReg and ExpReg to see what the calculator suggests about possibilities of these models to fit the data pattern.

Think about how scientists 500 years ago might have done

this experiment without stopwatches!

Scientific research often produces patterns of data that are described well by power expressions other than the basic quadratic (y = Ax) and cubic (y = Ax) models. In fact, it often turns out that a relation is best modeled by a rule with fixed exponent that is not a positive whole number.

When variables are related by equations like y = Axor y = Axit is customary to say that y varies directly with x or with the square of x. In cases where y is related to x by a rule like y = or y = we say that y varies inversely with x or with the square of x. These relations are also considered power models because, using negative exponents, they can also be written in the general form y = Ax. For example, y = is equivalent to y = axand y = is equivalent to y = ax. Some very important relations between variables fit this sort of inverse power variation pattern.

4. Use your calculator to explore the patterns of change implied by inverse and inverse square variation and compare them to models of linear and exponential decay.

(a) y = (b) y = (e) y = 10 - x (d) y = ()

The next three problems involve situations that are modeled by equations of inverse square variation. Use the what you've learned about the pattern of change implied by such variation in (4) to draw some conclusions about how the application situations will behave.

Hearing and sight--whether by humans, animals, or robots--both depend on our ability to find patterns in sound and light energy that hits our ears and eyes. To avoid damage to our ears and eyes, we are advised to keep away from the loud noises of machinery or radios and close to good lights for reading and detailed work. The intensity of sound and light energy is related to our distance from the source--the more distant the source, the lower the sound or light intensity.

5. The illumination from a light source is inversely proportional to the square of distance of the light from its target. For example, if a flashlight is listed with a rating of 160 lumens, it provides illumination at a distance of d feet would be given by B = lumens per square foot.

(a) How will intensity of the flashlight change as distance increases from 1 to 10 feet? How is that pattern shown in tables and graphs of (distance, illumination) data? How is it predictable from simply looking at the equation?

(b) What does the inverse square pattern of light intensity say about the use of flash lighting in photography?

6. The intensity of sound is measured by several different scales. One that uses units of power is watts-per-square-meter. It is a measure of the pressure that a sound forces on your ear. For a stereo system with its volume turned up very high, the intensity of sound is a function of the hearer's distance from the speakers that can be modeled by an inverse square relation. For example, if a radio has its volume turned up very high, its sound intensity as a function of distance might be given by an equation like I = where distance is in meters from the speaker to the listener's ear.

(a) How will the intensity of sound vary as distance increases, and how is that pattern shown in tables and graphs of the (distance, intensity) relation? How is it predictable from simply looking at the equation?

(b) What does the inverse square law for sound say about choice of seats in a large university lecture hall? How will acoustics of a room alter the pattern of the simple inverse square law?

7. The gravity that pulls flying objects toward the surface of the earth is the same force that holds the moon and NASA satellites in their orbits around the earth. The force of attraction (F) between any two objects is a function of the distance (d) separating the centers of those objects, with rule in the form F = .

(a) What does the form of the rule tell you about the way gravitational force on the astronauts in a space shuttle changes as that shuttle moves into higher and higher orbits around the earth?

(b) How does your answer to (a) explain the apparent "weightlessness" of astronauts in space?

(c) Why doesn't gravity simply pull the satellite back to earth once its booster rocket engines cut off?

3.7 Periodic Models

Linear, exponential, and power models are the fundamental algebraic tools for describing and reasoning about patterns of variation in a wide variety of scientific, technical, business, and social situations. But there is another very common family of patterns that require new sorts of tools for mathematical modeling--the phenomena that exhibit periodic or cyclical variation. The most common places where such patterns of change occur are in physical or biological systems where some variable of interest changes over time. The most common shape of those patterns is depicted in the graphs referred to generically as sinusoids.

Think About This Situation

You've all undoubtedly seen graphs with this general shape many times before and you probably know that the mathematical theory that provides a basis for those graphs is trigonometry.

1. What patterns in the graph make the labels periodic or cyclic variation seem

2. How does the pattern of change in this graph differ from linear, exponential,
and power models of change studied earlier?

3. What different situations can you think of where some variable changes over
time in a periodic or cyclic pattern?

The mathematical ideas used to model periodic or cyclical patterns of change have very ancient roots in topics of geometry that might seem at first to have very little to do with the phenomena that produce sinusoidal graphs. The next three problems illustrate, in very brief form, some of the key ideas behind the subject that is the basis of models for periodic change.

1. One of the oldest, but persistently impressive, applications of geometric ideas is the use of sun shadows to deduce distances of otherwise unmeasurable objects. One classic problem is often posed with stories like this: How could you estimate the height of one of the light towers at Byrd Stadium, without climbing the tower?

(a) Generate as many different ideas as you can for solving this problem.

(b) Find at least one method that requires only a ruler and a sunny day.

2. Another classic problem related to (1) is often posed with stories like this: How could you estimate the length of a support wire that is to be attached to a radio tower and the ground and so that it meets the ground at an angle of 70at a point 50 feet from the base of the tower?

(a) Generate as many different ideas as you can for solving this problem.

(b) Find at least one method that requires only a ruler and protractor.

3. The simplest methods for solving problems (1) and (2), except for asking someone who already knows the answers, involve some elementary facts about similar triangles. What are the assumptions required in each case?

The two problems involving inaccessible points on towers both involved right triangles and the general principle that in two similar right triangles the ratios of corresponding sides will always be the same. Ancient scientists and mathematicians knew very well that the shape of a triangle depends on the measures of its angles. So if we know the angle measures in a right triangle, we know the ratios of its sides. Furthermore, in a right triangle one angle must be 90and the sum of the other two must be 90, so knowing the measure of one of the acute angles we can deduce the ratios of the sides to each other.

For the problem on attaching support wires to a radio tower, we could draw a right triangle with one leg 50 and the adjacent acute angle 70. We could then measure the hypotenuse and scale up our answer to the real situation.

But mathematicians, being uneasy about scale drawing as an accurate method of calculation (imagine the effect of errors if the scale drawing is only a very small replica of the real thing), have chosen to produce tables that give very accurate ratios of corresponding sides in right triangles for any possible acute angle combinations. To serve the different kinds of information that might be available in different situations, they focused on three key ratios:

These three ratios are now available in nearly every scientific and graphing calculator at the press of a button. Knowledge of one acute angle and one side in a right triangle permits deduction of the other sides and angles. Using the inverse sine, cosine, and tangent buttons, one can deduce angle size from side ratios.

4. To see the ways that one can use the trigonometric ratio information in reasoning about triangles, solve each of the following problems. All information is given with reference to right triangles labeled in the standard notational convention of geometry.

(a) If angle B is 40and side c is 30 cm, what is the measure of angle A and the lengths of sides a and b?

(b) If angle B is 70and side a is 20 cm, what is the measure of angle A and the lengths of sides b and c?

(c) If angle A is 65and side a is 75 cm, what is the measure of angle B and the lengths of sides b and c?

(d) If the length of side a is 25 cm and side b is 40 cm, what are the measures of angle A and angle B and the length of side c?

(e) If the length of side a is 35 cm and the length of side c is 50 cm, what are the measures of angles A and B and the length of side b?

5. Ancient scientists very cleverly used trigonometric ideas to do some calculations about distances from Earth to really inaccessible places like the Moon and the Sun. In one technique, an Earth-bound observer waits for a time when the sun and moon were both visible in the sky and exactly half the moon was illuminated by the sun, a situation like that pictured at the top of the next page.

(a) What could you deduce about distances from Earth to Moon and Sun in such a situation?

(b) Suppose you knew that the distance from Earth to Moon was about 260,000 miles (there is some evidence that this was estimated early from eclipse data and other methods). How could you use that information and some other measureable facts to estimate the distance to the Sun from Earth?

(c) Using what we now know about Earth ... Sun ... Moon distances, what angle measures would be observed in the situation? (Earth to Moon 260,000 miles and Earth to Sun 93,000,000 miles)

The use of sine, cosine, and tangent ratios to calculate angles and distances in right triangles probably seems a long way from graphs of periodic variation. But the connection can be made by considering distances and angles in circular motion. The following sketch is intended to depict something like the sort of images you see on weather or air-traffic radar screens.

The ray from screen center to the right edge of the screen provides a base line for measuring angles along which radar blips are found. Each blip then can be located by a pair of numbers giving angle and distance. The angles range from 0 to 360.

6. How can the sine, cosine, and tangent ratios be used to calculate (x,y) coordinates (on a map these will be directions east-west and north-south of the center) for the blips located by angle and distance on the preceding diagram?

Your reasoning in solving problem 6 should lead to the conclusion that it is possible to define the sine, cosine and tangent ratios for any angle from 0 to 360. Furthermore, if we graph those ratios, we get patterns like this:

7. Follow the rotating radar beacon to explain why the shapes of these graphs are what one could reasonably expect for patterns of change in cosine and sine ratios.

8. Now imagine that the rotating radar beacon keeps on going and the angles are measured in angles from 360to 450and 540and so on. How will the graphs of cosine and sine ratios extend as time passes and the radar beacon passes through greater and greater angles?

If one uses the image of a rotating radar beacon to think about cosine and sine ratios for "angles" beyond 90, the original notion of ratios in right triangles leads in a natural way to periodic graphs that you recognize as sinusoids. In particular, if we think of increasing angle measure as a measure of time passing, we get the desired models of periodic change over time.

In many scientific applications of periodic models, the rotation is measured not by degrees of angle, but by radians (distance around a circle of unit radius). Thus to make use of cosine and sine functions in your graphing calculator, you will have to first specify Radian or Degree on the settings and then adjust the window settings accordingly.

9. Use your graphing calculator to study the graphs of these variations on y = sin x . In each case, explain why the result is what you could reasonably expect.

(a) y = 2 + sin x (b) y = 2(sin x) (c) y = sin (2x)

Conclusions and Connections

This very brief sketch of the foundations of periodic models only scratches the surface of the issues involved. However, we hope you have some glimpse of the subject!

1. How would you describe to someone not familiar with trigonometry what

periodic or cyclic variation is?

2. What situations would you mention as examples of periodic variation?

3. How would you explain the connection between right triangle trigonometry
and periodic graphs?

4. What are the most striking differences between periodic models and linear, exponential, or power models?

3.8 Combinations and Systems of Models

Linear, exponential, power, and periodic models capture the essential patterns of change in a huge number of important situations. However, to accurately describe and reason about the specific conditions in any given problem, it is often necessary to modify or combine those primary models to give more complex and accurate representations of the relations among variables. The required combinations often involve addition, subtraction, multiplication, or division of two basic model forms.

For example, Newton's Law of Cooling from physics predicts that when a hot liquid at 90C is poured into a cup or pan that is exposed to a cooler ambient temperature like 20C, the (time, temperature) graph will have the following shape.

You've encountered at least two different kinds of algebraic models that look similar to this graph--for example, exponentials like y = 0.5and inverse powers like y = or y = . But neither case quite fits the bill since the graph looks to be approaching a lower limit of y = 20.

The basic strategy for building a model in this case, and for many others like it, is to look for a modification or combination of things we know. Since the given graph looks like a standard exponential or inverse variation model shifted upward by 20, it makes sense to try either y = 20 + 0.5or y = 20 + . If you test these two possibilities, you'll see that we're on the right track. Only some fine-tuning is needed to get y = 20 + 70(0.5). The inverse variation models don't quite fit, because they are not defined for x = 0.

Other combinations of linear, exponential, power, and periodic models are very commonly encountered. In each case you can try again to find ways to piece together a combination of basic rules to give the needed model. For example, consider the way that a linear and a quadratic power model can be added to give a parabola that might be useful to fit a basic quadratic pattern with additional conditions.
y = 2x + 6                           y = -x 2

When combined by addition, these two rules give the quadratic model y = 2x + 6 - x2 with the following graph.

While it is probably not critical for you to develop great skill in building algebraic rules for all such combinations of basic models, it is important to realize that the world does not always display patterns that can be modeled with simple rules. Furthermore, experimenting with combinations of models can be a good way to review the patterns that characterize each basic family.

In the following problems you are given graphs that tend to be typical of several different situations. Use what you know about linear, exponential, power, and periodic models to devise combination models that seem to fit the more complex conditions given. Then test and refine your ideas with graph experimentation on your calculator.

1. Although Newton's Law of Cooling seems to refer only to the way hot objects cool to lower ambient temperatures, there is a comparable law of warming to cover the case when a cool object is introduced to a warmer environment. The graph will look something like this:

(a) What does the shape of this graph imply about the pattern of temperature change over time?

(b) What types of basic algebraic models have shapes somewhat like this?

(c) What combination model can you design to match the pattern here?

2. In businesses that manufacture products, there is a cost factor called a learning curve. The idea is that as production proceeds on an item, the company gets smarter and reduces its average cost of producing each item. A typical pattern looks something like this:

(a) What does the shape of this graph imply about the pattern of production cost change as the company gets experience in manufacturing?

(b) What types of basic algebraic models have shapes somewhat like this?

(c) What combination model can you design to match the pattern here?

3. In psychological studies of learning and memory, if a subject is given the challenge of memorizing a list of words, the typical pattern of learning follows a graph like this:

(a) What does the shape of this graph say about the pace of learning as time is spent on the task?

` (b) What combination of basic model types might be involved in this situation?

(c) What single equation gives a graph that looks like this?

4. When a diver jumps off a high diving platform, his or her height above the water is a function of time into the dive. The graph of that (time, height) relation is going to be quite close to that pictured here:

(a) What does the shape of this graph say about the rate at which the diver falls toward the water?

(b) What combination of basic model types might be involved in this situation?

(c) What single equation gives a graph that looks like this?

5. The basic graph of sin(x) or cos(x) gives the familiar sinusoid pattern. But many periodic phenomena need models that are modifications of those simple cases. For example, if one graphs the number of daylight hours in the day here in Maryland through the seasons of several years, you get a graph like this:

(a) What does the shape of this graph say about the pattern of change in daylight hours as time in a year passes?

(b) What variation on sin(x) might be a good match for the pattern in this situation?

In addition to combining basic algebraic models to match specific problem conditions, it is often the case that a problem or decision-making situation involves the interaction of several relations among variables. In those cases it is often helpful to study tables or graphs of the variables simultaneously by graphing two relations on a single screen or tabling several different y lists as a single common input variable x changes. The next two problems involve just this kind of situation.

6. When summer approaches, most high school and college students start thinking about finding summer jobs.
              Want Ads -- Help Wanted                   
Fast Food --Restaurant     Natural Lawns -- Summer    
seeks summer help;         help needed.  $5 per       
cashiers, cooks,           hour. No prior             
cleanup; 20 hours per      experience required.       
week all shifts            Call 24 hours              
available.  Call 234-5678  1--800-314-1589.           
Camp Staff--Summer         Child Care -- Tiny Tots    
play-ground and camp       Day Care Center seeks      
counselors age 15 or       summer help for child      
higher.  Good pay and      care positions.  Hours     
lots of fresh air.  Call   7-5 four days per week.    
987-6543                   Send references to Box     
In some cities and towns there is work available in shops and restaurants and in seasonal service jobs like lawn mowing or life-guarding. But there often aren't enough jobs to go around.

To help students find summer work--both to earn money and to get work experiences--many city and county governments have special summer job programs. Students are hired to do cleanup and building jobs in parks or other community facilities. As with most government projects, there is seldom enough money to go around.

Think About This Situation

In Kent County, the government budget sets aside $100,000 each year for a youth jobs program. The officials who run the program have to make some decisions before announcing their plans and inviting applicants for the jobs.

(a) What choices can you imagine that the jobs program planners have to make in setting up the program each year?

(b) What factors would they have to consider in making those choices?

(c) How are the choices related to each other; that is, how does one decision

affect the other options?

One plan for the Kent County summer jobs plan focused on the connections among pay for each student worker, hours worked, and number of students who would be interested in jobs at various levels of pay. The jobs program staff were unsure how much they should offer as pay. Recalling that the county has $100,000 to spend on student salaries:

(a) How many student workers can be hired if the county pays $2000 per worker for a summer contract covering 8 weeks? How many if the county pays only $1500 per worker?

(b) If the number of students who could be hired is represented by H and the offered summer pay rate by P, what equation gives H as a function of P?

(c) If the planners wanted to display their options in a graph, what type of graph shape would the relation between H and P give?

The jobs program staff figured that if they offered pay that was too low, then few students would be willing to take the jobs. After doing a survey of current high school students and recent graduates now in college, they came up with the following estimates of the relation between pay rate P and number of students who would actually apply for the jobs A.
Pay rate offered (P)  0  800  1200  1600  2000  2400

Job applicants (A)    0   80   120   160   200   240

(d) What relation between pay rate offered and number of job applicants
is suggested by these data? Explain why the pattern is or is not reasonable.

(e) What equation is a good model for the relation showing A as a function
of P?

(f) If they wanted to display their options relating offered pay P to number
of student applicants A in a graph, what type of a graph would they get?

Depending on how the job pay rate offer is made, different results could
occur. Explore the relations between possible pay rates, number of students
who could be offered jobs, and number of students who would actually apply
for the jobs. Find the pay rates that would lead to each of these three
possible outcomes: 

(g) The number of applicants would exceed the number of jobs.

(h) The number of applicants would be under the number who could be hired
at the specified rate.

(i) The number of job applicants would just equal the number who could be
paid at the specified rate.

Based on the results of your analysis, what pay would you recommend offering?

When the Kent County job program officers decided on a pay rate to offer,
they presented their plan to the County Board with a graph to illustrate
their reasoning. On the same diagram they plotted the relation showing
the number of students who could be hired (H) as a function of pay rate
(P) and the number of expected job applicants (A) as a function of pay rate

(j) Make such a graph showing the two relations.

(k) Explain how the graph shows the answers to questions (g), (h), and (i).

7. For people running businesses that sell products, there are many decisions
to make to guarantee profitability. Quite often those decisions involve
deciding what prices to charge for their products. 

Consider the case of concert promoters who have a contract to bring a music
program to a summer theater. They have to estimate costs of putting
on the show, income from ticket sales and concessions, and the profit
that can be made. When all these factors are considered, they have to decide
what prices to charge for tickets.

The first step in deciding how to set ticket prices for the concert was
to do market research on what prices people would probably pay. A market
research survey produced predictions of ticket sales (S) that could be expected
for various possible ticket prices (P). They got this data relating price
and probable ticket sales:

Proposed Price $          5       10      15    20      25      30        35    
Probable Ticket Sales    2300    1950    1600   1400    1100    900      700    

(a) What equation relating P and S fits this pattern well?

(b) Re-express this relation in a form that shows S as a function of P.

After estimating the connection between ticket prices and probable ticket
sales, the concert promoters wanted to see the probable connection between
ticket prices (P) and dollar income (I) from ticket sales. 

(c) Income from sales of concert tickets will be the product of number of
tickets sold and price per ticket. What equation gives I as a function of
P, based on your result from (b)? Sketch a graph of this relation between
ticket price and income from ticket sales and explain what the shape of
that graph says about the way income projections change as proposed ticket
price varies.

(d) What price is likely to give the maximum ticket income and what would

income be? Explain how you could find your answer using a calculator-generated
table or graph of the price-income relation.

The next step in making a concert business plan was to estimate operating
costs for the concert. The planners knew that some costs were fixed (for
example, pay for the bands and rent of the concert space) but others would
vary depending on the number of tickets sold (for example, number of security
guards and ticket takers). After estimating all the possible operating costs,
they came up with the equation C = 17500 + 2S, giving operating costs as
a function of number of tickets sold.

(e) According to that rule, what are the projected fixed operating costs
and what is the projected operating cost for each person who buys a ticket
for the concert?

(f) How will total operating costs change as the number of tickets sold

(g) Since ticket sales seem to depend on price charged, it is possible to
express operating cost as a function of price. What equation would express
that relation?

[Hint: Substitute the expression for ticket sales in terms of price into
the cost equation.]

(h) Operating profit for the concert will be the difference between income
(I) and operating costs (C). Using the equations that express I and C as
functions of price P, write an equation giving projected profit as a function
of ticket price and find the price that seems to project maximum profit.
Explain how you could find that optimal situation using a calculator-generated
table or graph of the price-profit relation.