Calculus Mysteries and Thrillers

Calculus Mysteries and Thrillers

R. Grant Woods
Copyright Date: 1998
Edition: 1
Pages: 152
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  • Book Info
    Calculus Mysteries and Thrillers
    Book Description:

    Calculus Mysteries and Thrillers consists of eleven mathematics projects based on introductory single-variable calculus, together with some guidance on how to make use of them. Each project is presented as an amusing short story. In many of them a group of undergraduate mathematics students, formed into a consulting company called Math Iz Us, is hired to solve mathematical problems brought to them by clients. The problems solved include: helping to prosecute an accused pool shark, defending a driver accused of speeding, assisting a hockey coach in making his star forward a more effective goal scorer, and advising a pirate captain on how to divide a gold-plated goose-egg fairly among his crew. In each problem, the problem solvers are required to present to their client a detailed written report of their findings. Thus, students must produce and analyze accurate mathematical models of complex, verbally presented real life situations, and write a clear technical account of their solution. Instructors who are looking for problems that are novel, interesting, and several levels more complex than the typical text book word problem will find them in this book. It will be of particular value to instructors who wish to combine training in applications of calculus with training in technical writing. The complexity of the problems makes them suitable for use as group projects. The calculus concepts on which the problems are based include: tangent and normal lines, optimization by use of critical points, inverse trig functions, volumes of solids, surface area integrals, and modeling economic concepts using definite integrals. Although a few ideas from physics and economics are used in the problems, no prior knowledge of these fields is required.

    eISBN: 978-1-61444-114-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. The Purpose of This Book
    (pp. ix-xii)
  4. An Overview of the Projects
    (pp. xiii-xiv)
  5. Detailed Mathematical Requirements
    (pp. xv-xx)
  6. The Projects
    • 1 The Case of the Parabolic Pool Table
      (pp. 3-8)

      A hush descended over the classroom as Inspector McGee strode to the podium. After all, he had served on the Fraud Squad for over 20 years, and the talks that he gave to each year’s crop of police recruits were the stuff of legend. No one else had the fund of stories about past experiences mat he did. Each year he amazed the rookies by telling them of yet another bizarre technique mat he had used to unmask the schemes of the city’s con artists. It seemed that there was no branch of knowledge that he hadn’t exploited at some...

    • 2 Calculus for Climatologists
      (pp. 9-12)

      Jason and Anne had been friends since grade school, so each of them was delighted when they learned that the other had been accepted into Megastudent University. Mega U. was situated in a small city several hundred miles from their home town, and although it had a reputation for academic excellence, some of their older friends had returned from their freshman year mere with scary tales about how cold and impersonal the large campus could be for shy students. So that summer, as they prepared to leave their homes and friends, Jason and Anne made a pact that they would...

    • 3 The Case of the Swiveling Spotlight
      (pp. 13-18)

      Hi. My name is Friday—Joe Friday. I’m a calculus student. I want to tell you a little story—a story about parabolic boulevards, focal fountains, and a car speeding through the night. I like to call it . . . the Case of the Swiveling Spotlight.

      It all started one bleak Monday evening in early November. I’d gotten home from classes, made some sandwiches, and settled down with a tall glass of something cold when the phone rang. It was my older sister Delia. Delia’s a smart woman—she graduated from Law School a few years ago and now...

    • 4 Finding the Salami Curve
      (pp. 19-22)

      The year is 20xy. Hockey fans throughout Manitoba are following the fortunes of the Winnipeg Gliders, the city’s entry in the new, co-ed Intercontinental Hockey League. But things are not going well for the Gliders; the team is mired in last place, game attendance and revenues are down, and the owners of the Gliders are threatening to move the team to Buenos Aires, which desperately wants an ICHL franchise.

      One morning you are sitting in the office ofMath Iz Us(the small consulting firm that you and your two partners have recently opened after having difficulty finding satisfactory summer...

    • 5 Saving Lunar Station Alpha
      (pp. 23-28)

      It was the loneliest summer job that Ranjana, Bruce, and Tara had ever had, and at the beginning they had wondered whether they should even take it. For four months they would be isolated in Lunar Station 50, so named because it sat astride the 50th parallel north of the equator of Xzyqgon, the small hollow spherical satellite of Neptune discovered in 2050. Lunar Station 50 had been abandoned in 2140, when the North American Federation had launched its eighth massive government downsizing and decided that the country could no longer afford to maintain it. However, more recently the Federation,...

    • 6 An Income Policy for Mediocria
      (pp. 29-34)

      The summer of 20xy promised to be just as difficult as the previous summers for university students seeking employment. Heather, Sasha, and Li had approached hundreds of local businesses to attempt to convince them that they could benefit from the services ofMath Iz Us.But replies were slow in coming, and the three students were beginning to be afraid that they would have to dissolve their consulting firm and accept jobs waiting tables at the campus restaurant instead.

      As a last resort, Heather had suggested to the group that they place an ad inThe Campus Clarion.So in...

    • 7 The Case of the Cooling Cadaver
      (pp. 35-40)

      It was a dark and stormy night. The three members ofMath Iz Ushuddled in their sleeping bags and listened to the cold rain lashing against the outside of their tent. When Heather had first suggested it, the idea of using their very generous fee from the Government of Mediocria to finance a vacation spent cycling across Britain had appealed to them all. But the soggy British weather was not cooperating, and now they found themselves marooned in a tent on the grounds of a large old country estate. When they had arrived bedraggled at the door of the...

    • 8 Designing Dipsticks
      (pp. 41-46)

      The “Great White North” holds a romantic appeal for many people, and Anne was no exception. So when she saw a posting at the Campus Employment Center advertising a summer job at a remote fishing lodge in northern Manitoba, she immediately filed an application and was delighted when she was hired. She dreamed of a summer spent catching 10-pound trout, reading Robert W. Service poems by moonlight, and listening to the haunting howl of distant wolves.

      Reality proved to be somewhat different. She spent her days cleaning one-pound trout, reading outboard motor repair manuals by flashlight, and listening to the...

    • 9 The Case of the Gilded Goose-Egg
      (pp. 47-50)

      “Hello. You have reached the farm of Silas and Elsie Friday. We can’t come to the phone right now, but if you leave us a message after the tone, we’ll get back to you as soon as we can.”

      “Hello Mom, hello Dad, it’s Delia. I’m sorry that I haven’t phoned you before now, but I’ve been really, really busy at work. Yes, I know you’ve been worried about me, and yes, I did get all those messages that you left on my answering machine. It’s just that Per—that I ended up staying longer in the Bahamas than we...

    • 10 Sunken Treasure
      (pp. 51-56)

      I knew that you two would be worried about Joey and me, so I decided that I better try to have a letter smuggled out to you. The pirates have been very insistent that Joey and I won’t be allowed to communicate with anyone until we finish the project. But one of the crew seems really nice, and I think he kind of likes me, so maybe I can persuade him to fax this to you. I sure hope so—I hate to think about you guys sitting at home fretting and not knowing what’s going on. And could you...

    • 11 The Case of the Alien Agent
      (pp. 57-64)

      It had been a long week for agent Muldie, and she was looking forward to a quiet Friday evening at home with a fascinating new book, “Conspiracy Theories throughout History.” Unfortunately the New Age Network had postponed its special about how Atlantis was really a Martian colony peopled (aliened?) by pyramid builders who had fled to Egypt in an ark after a giant asteroid plunged into the Pacific and caused worldwide flooding, but her book would be a suitable substitute. As she read, though, she found herself growing drowsy, and soon the book slipped from her hands.

      She was awakened...

  7. The Solutions
    • 1 The Case of the Parabolic Pool Table—Solution
      (pp. 67-70)

      In this brief we shall show that Mr. Luigi McTavish did keep in the establishment known as Luigi’s Lizard Room, owned by Mr. MacTavish, two pool tables of similar design but different dimensions to aid and abet a fraudulant gambling scheme. We shall further show that the table used by Mr. McTavish was designed so that McTavish could win a bet with his co-gambler, while the table used by the co-gambler was designed to render it impossible for said co-gambler to win. This submission shall constitute evidence corroborating the charge of fraud against Mr. McTavish.

      Consider a pool table constructed...

    • 2 Calculus for Climatologists—Solution
      (pp. 71-74)

      When Anne arrived at Thirsty’s the following Friday afternoon, she saw that Len and Jason had gotten there before her and had chosen a quiet table near the windows. Len was slouched back in his chair, absorbed in the demanding task of cleaning his fingernails with the blunt end of a toothpick. Jason was systematically tearing a paper napkin into thin strips. As there was no food or drink on the table, they apparently had not yet been served.

      Jason looked up. “What happened to you?” he asked, giving Anne a rather fierce look. “You’re late.”

      “I’m sorry,“ replied Anne....

    • 3 The Case of the Swiveling Spotlight—Solution
      (pp. 75-80)

      The defense proposes to show that if the defendant were traveling at the speed that the police witnesses claimed to have measured, then he could not have been under observation for as long as the 20-second minimum that the law requires. We will establish this by deriving an equation whose rootTis the time (in seconds) that would have elapsed from when the defendant’s car was first observed until the time when the spotlight on the police cruiser would have been pointing directly sideways. According to the testimony from Sergeants Preston and Renfrew,Tis larger than the time...

    • 4 Finding the Salami Curve—Solution
      (pp. 81-86)

      Please find below our report giving the equation of the Salami curve relative to a suitable coordinate system, together with a scale diagram. We have included our mathematical derivation of the equation of the curve. In order to accommodate the different sizes of rinks found in the ICHL, we have denoted relevant distances with letters; you can substitute the appropriate numerical values for your rink. We trust that this is satisfactory.


      Math Iz Us

      LetLdenote the perpendicular distance between the goal-mouths. (All distances are measured in feet.) Letwdenote the width of the portion of the...

    • 5 Saving Lunar Station Alpha—Solution
      (pp. 87-94)

      This report addresses the calculation of the launch velocity of the rocket fired from Lunar Station 50 at 3:00 PM on July 21,21xy, to intercept the meteor on a collision course with Lunar Station Alpha.

      LetRdenote the radius of Xzyqgon (all distances will be measured in km). Letddenote the distance from Lunar Station 50 (henceforth denoted LS50) to the meteor at 3 PM. Letudenote the speed of the meteor (all speeds in km/sec), and let v denote the maximum possible speed at which the research rockets can be launched.

      The situation at 3 PM...

    • 6 An Income Policy for Mediocria—Solution
      (pp. 95-102)

      As requested, we present below an analysis of some aspects of an income policy for Mediocria. The basic task is to investigate the way in which income distribution will affect the fraction of their earnings that Mediocrian citizens spend. To this end we establish some preliminary results. In what follows, we shall use data provided by the Mediocrian Census Commission and the Mediocrian Bureau of Statistics. For the moment we denote the relevant quantities by letters.

      LetPdenote the population of Mediocria. The total amount of money available for the Mediocrian government to pay out in salaries each year...

    • 7 The Case of the Cooling Cadaver—Solution
      (pp. 103-106)

      We present below our estimation of the time of death of Lord Boddy, who passed away sometime on the night of June 23-24,20xy, as the result of a severe blow to the back of the head administered with a heavy brass candlestick. Since only one suspect failed to have an alibi at the time of death, we can therefore conclude who committed the murder.

      We accuse Professor Prune of murdering Lord Boddy in the study with the candlestick at approximately 11:28 PM on June 23, 20xy. The reasoning that led us to this conclusion follows.

      LetTdenote the temperature...

    • 8 Designing Dipsticks—Solution
      (pp. 107-112)

      Here is my report on how to calibrate the dipsticks for the cylindrical and spherical fuel tanks. In each case I have worked out an algebraic expression giving the fractionF(d) of the tank capacity that is filled with fuel as a function of the lengthdof dipstick that is wetted by the fuel. I have also provided a table of values forF(d) for values of d ranging from 0 toD(the value ofdwhen the tank is full) in increments ofD/32. For the cylindrical tank I have derived the algebraic expression in two ways....

    • 9 The Case of the Gilded Goose-Egg—Solution
      (pp. 113-120)

      Your captain has asked me to show you how to slice your gold-plated spherical “goose-egg” in such a way that each crew member receives the same amount of gold. He has asked me to denote bom the radius of the egg and the number of crew members by letters, in case division of a spherical egg of a different radius among a different number of crew members should become an issue at some time in the future. In what follows, distances can be measured in any convenient units; I will not specify any.

      A sphere of radiusRis generated...

    • 10 Sunken Treasure—Solution
      (pp. 121-126)

      First of all, we want to tell you that we are very disappointed by your conduct. Even pirate kings ought to know the basic rules of etiquette, and one of the most fundamental is that well-bred hosts do not knock out their guests and kidnap them.

      However, enough of that. You have asked us to calculate the exact length needed of a roll of material in which a barge with a parabolic cross-section will be slung and lowered to the ocean floor. By measuring certain dimensions of the barge, we can calculate the shape of the parabola. When the barge...

    • 11 The Case of the Alien Agent—Solution
      (pp. 127-131)

      As you may know, one of your members anonymously sent us top secret documents giving the calculation by the IPP of the volume of freutcaquium fallout in the Earth's crust. He (or she) pointed out that this calculation did not take into account the curvature of the Earth, and might therefore have overestimated the actual amount of freutcaquium that settled on our planet. If this were true, the amount of freutcaquium might have been insufficient to wipe out the ancient aliens that you (and we) believe may have populated the Earth at that time.

      However, our calculations seem to indicate...