Exercises
Data and tables
Note: These problems ask for plots of isolated points, not of smooth curves. You need to use ListPlot, not Plot.
For information about using ListPlot, see the section . For information about handling data, see the example on U.S. population trends in the section .
1. Percentage below the poverty line
Here are the percentages of the U.S. population who lived below the poverty level during the period 1960-1992. (To get the data, execute the button below.)
Source: Bureau of the Census, U.S. Department of Commerce, "Persons Below Poverty Level, 1960-92", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 394.
* Beginning in 1980, a revised definition of poverty was used.
Make a plot of points of the form {year, percentage} to give a graphical display of this information. Describe the general behavior of the percentages over time.
2. Comparing numbers and percentages
Here is how many people lived below the poverty level in the U.S. during the period 1960-1992. (To get the data, execute the button below.)
Source: Bureau of the Census, U.S. Department of Commerce, "Persons Below Poverty Level, 1960-92", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 394.
* Beginning in 1980, a revised definition of poverty was used.
Compare the plot of the numbers (of people living below the poverty line) with the plot of the percentages (of people living below the poverty line) above. Do they both generally go up and go down together? What can you conclude about the growth in the U.S. population during the period 1960-1992?
3. Here are some data giving the population of Louisville, Kentucky. In each pair, the first number is the year, the second is the population.
Source: Bureau of the Census, U.S. Department of Commerce, "Population of 100 Largest U.S. Cities, 1850-1990", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 381.
Make these data more readable by lining them up in two columns (only if the data did not appear in two columns already). Then, make them even more understandable by plotting the data. Describe how the Louisville population has changed over time. What do you think the population will be in the year 2000?
4. Here are some data giving the population of Los Angeles, California. In each pair, the first number is the year, and the second is the population.
Source: Bureau of the Census, U.S. Department of Commerce, "Population of 100 Largest U.S. Cities, 1850-1990", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 380.
Make these data more readable by lining them up in two columns (only if the data did not appear in two columns already). Then, make them even more understandable by plotting the data. Describe how the Los Angeles population has changed over time. What do you think the population will be in the year 2000?
5. Here are some data giving the population of Nashville, Tennessee. In each pair, the first number is the year, and the second is the population.
Source: Bureau of the Census, U.S. Department of Commerce, "Population of 100 Largest U.S. Cities, 1850-1990", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 380.
Make these data more readable by lining them up in two columns (only if the data did not appear in two columns already). Then, make them even more understandable by plotting the data. Describe how the Nashville population has changed over time. Have there been any remarkable changes in the population?
Hint: Between 1960 and 1970, Nashville adopted a "metro-style" government and combined to become "Nashville-Davidson County". That automatically caused a big jump in the population of "Nashville".
6. Millennial population
The table below estimates the world's population over several thousand years:
Source: Isaac Asimov and Frank White, March of the Millennia: A key to looking at history, New York: Walker, 1991, 154.
The first number in each pair represents the date (in thousands of years) in the usual (Gregorian) calendar. For instance, 
represents 3000 B.C.E., and 2 represents the year 2000. The second number gives the estimated world population in millions.
Here is a plot of the data.
That spectacular increase in the last thousand years makes it hard to see what was going on before. Here is the plot with the last pair removed.
There still is a rather spectacular jump in the last thousand years shown.
a. Drop the last pair of this list, and plot the rest of the pairs. Is there still a spectacular jump at the end?
b. Suppose you had this information available to you 1000 years ago. (That is, suppose you had all these data points, except for the very last one (2, 6000). How would you go about predicting the world population growth during the thousand years to come (that is, up to our actual present)? Would you say there was cause for alarm?
Applications in the sciences
7. Schwarzschild radius
Source: William A. Hiscock, "The Inevitability of Black Holes", Quantum 3(4), 1993, 26.
Hint: To help you get started, read example on escape velocity in the section . Here is the equation for the escape velocity needed to escape "to infinity" from a star or planet.
The universal gravitational constant is about 6.67 
(in units that are appropriate), with the radius measured in meters and the mass in kilograms. The Schwarzschild radius of a body is the radius that leads to an escape velocity that is equal to the speed of light. The mass of the sun is about 333,000 times the mass of the Earth. What is the Schwarzschild radius of the sun?
Estimate your own weight in kilograms. Does it make any sense to talk about the Schwarzschild radius of your own body? Pretend that it does make sense, and find that radius.
8. Interest rates
According to a story by the Associated Press, the Russian ruble drastically fell in value against the U.S. dollar in October 1994. The Russian government decided--among other actions--to raise short-term interest rates--from 130 percent to 170 percent.
Source: "Ruble's plunge spreads panic in Moscow", Louisville Courier-Journal, 12 Oct. 1994.
Hint: To help you get started, read the example on interest in the section .
Looking at the high interest of 170 percent, suppose you had $1500 that you could deposit in a bank at an annual simple interest rate of 170%. How would your earnings increase if you left it there for 10 or 15 years?
Display your results in a table and with a plot. Discuss what you see. How does 170% interest compare with 130% interest over a 10- or 15-year period?
9. Inflation
Here is part of a newspaper article about economic problems facing the Russian government.
"The (Russian) government--which has brought down monthly inflation from 25 percent a year ago to under 5 percent this summer--also said that it intended to pass a budget for 1995 that will be anti-inflationary, although it provided no details."
Source: "Ruble's plunge spreads panic in Moscow", Louisville Courier-Journal, 12 Oct. 1994.
Hint: To help you get started, read example on interest in the section .
Suppose you have a collection of items that cost $200 all together. Then, suppose that inflation takes over at a rate of 5 percent per month, every month for 2 years. How much will those items cost at the end of the two years? How about inflation at a rate of 25 percent per month for two years? Show graphically, numerically, and symbolically how the prices change over that time.
10. Target heart rate
The target heart rate is the heartbeat rate a person should have during strenuous exercise to get the full benefit for cardiovascular conditioning. Here is how the American College of Sports Medicine says you should calculate your target heart rate.
1. Subtract your age from the number 220.
2. Multiply the result by the desired intensity level of the workout (0% up to 100%).
3. Divide that result by 6.
Source: "Finding Your Target Heart Rate", in The World Almanac and Book of Facts 1995, Mahwah, NJ: Funk & Wagnalls Corp., 1994, 701.
What you just computed represents your target heart rate for a 10-second pulse count--the number of heartbeats you would count in 10 seconds by feeling your pulse at your wrist or neck.
Come up with a function, say mytarget, that will take a desired workout intensity level level, and return to you the target heart rate. Make a table of values of the form {level, mytarget[level]}, for several possible intensity levels. (The next time you exercise strenuously, see how the table matches up with your actual heart rate after the exercise.)
Hint: For information about handling data, see the example on U.S. population trends in the section .
11. Pat Summitt's coaching pay
"Pat Summitt, whose Lady Vols fell to Connecticut in the NCAA championship game this past season, has won a contract extension into the next century...Summitt's base pay jumps $7,000 to $125,000, giving her the biggest base salary of any Tennessee coach...Summitt, who has guided the Lady Vols to three national championships, earned $8,900 a year when she began coaching at Tennessee 21 years ago.
Source: "Lady Vols' coach gets $27,000 raise", Bowling Green (KY) Daily News, 2 May 1995.
Hint: For information about handling data, see the example on U.S. population trends in the section .
a. Suppose that Pat Summitt had received her pay raises in equal dollar amounts each year, starting 21 years ago at $8,900, and ending with $125,000. Make a table showing how her income would have changed over the years.
b. Suppose that Pat Summitt had received her pay raises in equal percentage amounts each year, starting 21 years ago at $8,900, and ending with $125,000. Make a table showing how her income would have changed over the years.
Note: This is a difficult exercise. One way to go about this is to set up , relating the salary of one year with the salary of the prior year.
12. Titius-Bode Law
The 18th-century German astronomer Johann Titius thought he saw a pattern in the orbits of the planets and the asteroid belt between Earth and Mars. Johann Bode agreed and popularized the pattern. They thought that the distance of planet number n from the sun was given by
Source: Cesare Emiliani, The Scientific Companion, New York: John Wiley and Sons, 1988, 95.
The radius was in astronomical units. One astronomical unit is the mean distance between the Earth and the Sun (149,597,870,700 meters). Here was their numbering system.
Here are the mean distances from the Sun as we have been able to calculate them nowadays, in astronomical units.
Compare these figures with what the Titius-Bode Law gives. How well does that law do?
Note: You can use the built-in Mathematica command Infinity just like a real number.
To be fair to Titius and Bode, nobody knew that Neptune or Pluto even existed then. Compare with figures for only the other planets. How well does that law do?
This appears to be a pattern without any particular good reason for why it should work as well as it does.
13. Titius-Bode Law: Another version
The 18th-century German astronomer Johann Titius thought he saw a pattern in the orbits of the planets and the asteroid belt between Earth and Mars. Johann Bode agreed and popularized the pattern. Here is another version of the Titius-Bode Law.
Start with this sequence of numbers:
This version of the Titius-Bode Law says to take each term of the sequence, add 4 to it, then divide by 10. The resulting list of numbers was supposed to give the distance of each planet from the Sun, in astronomical units. One astronomical unit is the mean distance between the Earth and the Sun (149,597,870,700 meters).
Source: J. Donald Fernie, "The Neptune Affair", American Scientist 83(2), 1995, 116-119.
The planets Titius and Bode knew about were:
a. Find the list of distances predicted by this version of the Titius-Bode law.
b. Neptune and Pluto were unknown at the time. Look for the pattern in the Titius-Bode sequence and decide what you think the next two numbers in the sequence ought to be. Then find the distances for those two.
c. Here are the mean distances from the Sun as we have been able to calculate them nowadays, in astronomical units.
Compare these figures with what the Titius-Bode law gives. How well does that law do? Be fair; should you treat Neptune and Pluto differently since Titius and Bode did not know they existed?
14. Counting calories or maintaining present weight
How many calories do you need to maintain your present weight? Here is a three-step process suggested by the Kellogg's Company to estimate that number, along with an example they provided.
1. Current weight times 10 = BMR (Basal Metabolic Rate)
2. BMR times 0.30 = Activity Calorie Level (if you are sedentary)
3. BMR + Activity Calorie Level = Calories used daily to maintain your current weight
Source: Kellogg's Special K cereal box, Battle Creek, MI: Kellogg Company, 1994.
You are sedentary if you do not get much exercise.
Example: Adult female, 140 lbs.
(140)(10) = 1400 (This is the BMR.)
(1400)(0.30) = 420 (This is the Activity Calorie Level.)
1400 + 420 = 1820 (This is the number of calories used daily to maintain current weight.)
a. Use this information to come up with a one-step process that will do the same thing. That is, come up with a function that takes weight as input and gives the appropriate number of calories as output.
b. If all other things stay the same, would an active person need more calories, or fewer calories, to maintain current weight, compared to a more sedentary person? The Kellogg's process says to multiply BMR by 0.30 for people who are sedentary. For more active people, should that 0.30 be changed to something bigger, or to something smaller? Why?
c. The Kellogg's ad says:
"...to lose weight, cut down on your 'Calories used daily', but do not go below 1200 calories per day. A better method is to cut your calories down to your BMR."
What should a person who weighs 110 pounds do? Is there any conflict between these two methods?
15. Space shuttle spacewalk temperatures
A newspaper article of 9 Feb 95 reported that two astronauts spent some time outside the space shuttle Discovery while it was in orbit 240 miles above the Earth. They were testing the ability of their spacesuits to keep them warm. The temperature, with the shuttle craft blocking the Sun, was reported to be between 
and 
degrees.
Source: "Spacewalkers brave cold, frigid darkness", Bowling Green (KY) Daily News, 9 Feb. 1995.
Was the reported temperature in degrees Celsius or in degrees Fahrenheit? In which case would it be colder? Whichever you think it was, what would the temperature range have been in the other temperature scale?
If you want to use the Mathematica Convert command, you need to execute the command below. This will load the Convert command into memory.
To find out how it works, we can ask Mathematica.
As an example, we convert 5 degrees Celsius to Fahrenheit.
16. Low-flow toilets
Federal regulations require that all new toilets use 1.6 gallons of water, or less, per flush. This compares with the 5 gallons that most existing U.S. toilets use per flush. (Those made in the U.S. after the late 1970s use at most 3.5 gallons per flush.) These new toilets are more expensive than the old ones, but they will save on water and sewer bills.
If you have to buy a new toilet, you will have to get a low-flush one. Some people might think of buying one even if they do not have to. How long would it take before the savings on water and sewer bills let them break even with the cost of the new toilet?
The answer is, of course, "it depends." Consumer Reports estimates that a new low-flush toilet would save about 7500 gallons of water in a year, for a household of three people. Here are the combined water and sewer rates for three locations in 1994.
Source: "Low-flow toilets", Consumer Reports 60(2), 1995, 121-124.
For a new toilet that costs $320: (a) estimate the number of years it would take to save enough to cover the cost in each of the cities listed; (b) find a function that will give the number of years it would take, as a function of the combined water/sewer cost per 1000 gallons.
Make some estimates for you own situation. How long would your break-even time be? Are there any other considerations that would be important if you actually were thinking about buying a new toilet?
17. Farm numbers
Here are some data on the number of American farms over time, according to the U.S. Census Bureau, as reported in a newspaper article.
The points are in the form 
, where 
is the year, and 
is the number of farms, given in millions.
Source: Randolph E. Schmid, "Number of American farms is at its lowest since 1850", Bowling Green (KY) Daily News, 4 Feb. 1994.
Here is a plot of the data.
a. According to the newspaper article, from 1987 to 1992 the number of farms went down but the average farm size went up, from 462 acres in 1987, to 491 acres in 1992. Did the total number of acres of farmland go up or go down from 1987 to 1992? Explain.
b. Describe how the number of American farms has changed over the last 150 years.
c. Here are some things that interact: the size of the population, the number of farms, the average size of the farms, and the efficiency of farming methods. (Other things may have an influence too, of course.) How do these things interact? For example, if population and average farm size stay the same, but the number of farms goes down, what does that say about efficiency of farming methods? Discuss this and the other types of interactions.
18. Caller ID and saving time
An Associated Press article of 6 Feb 95 on the telephone feature of "caller ID," reported that businesses are using that feature to save time. When a repeat customer calls, the name, address, and other details immediately appear on the computer screen.
Here is what the article said about one company:
"Memphis, Tenn.-based FedEx handles about 260,000 calls a day in the United States. A shaving of just 10 seconds off every call comes to more than 722 hours saved on the phone daily. 'It does not take a mathematician to realize if you're handling that many calls and you're saving a few seconds, it amounts to a lot of dollars,' said Dave Barnwell, FedEx's managing director of customer service."
Source: Brian S. Akre, "Caller ID becoming mini security guard for many companies", Bowling Green (KY) Daily News, 6 Feb. 1995.
Is the article correct about the number of hours saved assuming that it is correct about the number of calls? If the company did save 722 hours a day, how could you find out how many dollars it would save? What other information would help you answer the previous question?
19. Consumers Union surveys
"Sending the Annual Questionnaire to over 4,000,000 subscribers costs $829,000 for paper, printing, and postage. ... It is the single most expensive research effort in our reporting program. ... This year we are asking you to enclose, if you can, a voluntary contribution of $6 or more to help cover the cost of this questionnaire."
Source: Flyer enclosed with Annual Questionnaire sent to subscribers of Consumer Reports, Apr. 1995.
a. Suppose that Consumers Union wants to cover the entire $829,000 cost of the questionnaire with $6 donations. How many donations will they have to receive? About what percentage of all their subscribers would this say they are hoping will contribute?
b. Suppose that Consumers Union wants to cover half of the $829,000 cost of the questionnaire with $6 donations. How many donations will they have to receive? About what percentage of all their subscribers would this say they are hoping will contribute?
c. Come up with two functions 
, and
. The output of 
should be the number of donations that Consumers Union will have to receive in order to take in enough contributions to cover the number 
percent of the $829,000 cost of the questionnaire. The output of 
should be the percentage of all their subscribers who contribute $6 when Consumers Union takes in the number 
percent of the $829,000 cost of the questionnaire. Plot both functions.
d. Do problem c above and use the functions you came up with there. Come up with a way to write 
in the form 
, where 
. (This amounts to writing 
as a function. We will study composite functions in more detail later.)
Recurrence relations
To make the command RSolve available, execute the command below.
20. 
Hint: Before attempting these exercises, read the section .
Here is a recurrence relation.
a. Create a table of the first dozen or so terms of this sequence. Use this table to guess an explicit formula for the general term.
b. Prove that your explicit formula is the right one, by checking that it satisfies the recurrence relation. (Do not forget to check the starting points too.)
21. What do the odd positive integers add up to?
Hint: Before attempting these exercises, read the section .
Here is a sequence that gives the perfect squares.
Here are the first few terms.
One way to find out what we need to add up in order to get the perfect squares is to see what the difference is between consecutive perfect squares. Compute the difference between two consecutive squares symbolically. This amounts to discovering a recurrence relation for the perfect squares.
