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Making Decisions

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.

Informed Consent. A patient with an aggressive cancer is faced with two options for surgery. With Option A, he has a 40% chance of surviving for a year, but if he makes it that long then his chance of surviving a second year is two out of three. With option B, he has a 50-50 chance of surviving each of the first and second years. Survival rates beyond the second year are similar for each option. Which choice should he make?

Tape Packing. A high school band is making a tape and a CD to sell to raise money for a trip to Disney World. The tape will be 60 minutes long, 30 minutes on each side; the CD will be 45 minutes long. The band has recorded 15 pieces ranging in length from 1:32 to 6:45 minutes [a specific list is provided]. Choose which songs to put on the tape (sides A and B) and on the CD to provide the most music. What should the choices be if one could order the tape of any requested length? (In this case the goal to minimize waste would be to be sure that the two sides of the tape were nearly equal in length.)

Standing in Line. It seems as if every trip to the supermarket requires as much time standing in the checkout line as in selecting groceries. You complain to the manager, who explains that she really cannot hire any more checkout clerks, but would be glad to modify the way lines are managed to minimize the time people wait for checkout. There already is an express lane; might an intermediate lane be helpful? Or perhaps a single line could feed two or three checkout lanes. What about a cash-only line? What would you advise?

Emergency 911. A city that is served by two different ambulance companies is being sued because of delays in response to 911 emergency calls. City logs record the day, date, time of call and response time for each 911 call. Analyze this data and write a report to the city council (with supporting charts and graphs) advising them on how 911 operators should decide which ambulance company to dispatch for particular calls.

Dividing Assets. Negotiators often have to find a fair way to divide assets among individuals with competing claims. This is not so easy, and not only because the competing individuals may be uncooperative. Some assets are essentially indivisible; other may have sentimental value unrelated to market-based value; and often the individuals involved cannot agree on a fair value of particular assets, much less on a fair division. A successful solution requires not just that the division be objectively fair, but that it be perceived as fair by each of the parties--even if they use different criteria in making their judgments.

Dividing Land. Your grandfather died and left a piece of undeveloped and somewhat hilly land to be divided equally between you and your brother. Given a surveyor's drawing, find appropriate locations for lines that will divide the property equally (a) in an approximate North-South orientation, and (b) in an East-West orientation. Explain how one would locate the lines on the ground from the map that you have prepared.

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Last Update: 12/28/98