英文——历届美国数学建模竞赛赛题(2001-2009).

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目录

2001 MCM A: Choosing a Bicycle Wheel .............................................................................. 2

2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...) .................................... 3

2002 MCM A: Wind and Waterspray ............................................................................................ 5

2002 MCM B: Airline Overbooking ............................................................................................. 5

2003 MCM A: The Stunt Person .............................................................................................. 6

2003 MCM B: Gamma Knife Treatment Planning .................................................................. 7

2004 MCM A: Are Fingerprints Unique? ..................................................................................... 8

2004 MCM B: A Faster QuickPass System .................................................................................. 8

2005 MCM A: Flood Planning ...................................................................................................... 9

2005 MCM B: Tollbooths ............................................................................................................. 9

2006 MCM A: Positioning and Moving Sprinkler Systems for Irrigation .................................. 10

2006 MCM B: Wheel Chair Access at Airports .......................................................................... 10

2007 MCM A: Gerrymandering .................................................................................................. 12

2007 MCM B: The Airplane Seating Problem ............................................................................ 12

2008 MCM A: Take a Bath ......................................................................................................... 13

2008 MCM B: Creating Sudoku Puzzles .................................................................................... 13

2009 MCM A: Designing a Traffic Circle .................................................................................. 13

2009 MCM B: Energy and the Cell Phone ................................................................................. 14

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2001 MCM A: Choosing a Bicycle Wheel

Cyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike. Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for

information to help determine what kind of wheel should be used for a given

course.

Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right.

The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear.

Task 1. Provide a table giving the wind speed at which the power

required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments. (Road grade is defined to be the ratio of the total rise of a hill pided by the length of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road grade.

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A rider will lose about 8 kph for a five percent grade over 100 meters.

Task 2. Provide an example of how the table could be used for a specific time trial course.

Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision.

2001 MCM B: Escaping a Hurricane's Wrath (An Ill Wind...)

Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare.

The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned.

A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of the coastal strip. (More accurate data, if sought, are widely available.)

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