1999年美国大学生数学建模竞赛题目返回首页1999年美国大学生数学建模竞赛题目
Problem A: Deep Impact
For some time, the National Aeronautics and Space Administration (NASA)
has been considering the consequences of a large asteroid impact on the
earth.
As part of the effort, your team has been asked to consider the effects of
such an impact were the asteroid to land in Antarctica. There are concerns
that an impact there could have considerable different consequences than
one striking elsewhere on the planet.
You are to assume that an asteroid is on the order of 1,000 m in diameter,
and that it strikes the Antarctic continent directly at the South Pole.
Your team has been asked to provide an assessment of the impact of such an
asteroid. In particular, NASA would like an estimate of the amount and
location of likely human casualties from the impact, an estimate of the
damage done to the food production regions in the oceans of the southern
hemisphere, and an estimate of possible coastal flooding caused by
large-scale melting of the Antarctic polar ice sheet.
Problem B: Unlawful Assembly
Nany facilities for public gatherings have signs that state that it is
"unlawful" for their rooms to be occupied by more than a specified number
of people. Presumably, this number is based on the speed with which people
in the room could be evacuated from the room's exits in case of an
emergency. Similarly, elevators and other facilities often have "maximum
capacities" posted.
Develop a mathematical model for deciding what number to post on such a
sign as being the "lawful capacity." As part of your solution, discuss
criteria-other than public safety in the case of a fire of other
emergency-that might govern the number of people considered "unlawful" to
occupy the room (or space). Also, for the model that you construct,
consider the differences between a room with movable furniture such as a
cafeteria (with tables and chairs), a gymnasium, a public swimming pool,
and a lecture hall with a pattern of rows and aisles. You may wish to
compare and contrast what might be done for a variety of different
environments: elevator, lecture hall, swimming pool, cafeteria, or
gymnasium. Gatherings such as rock concerts and soccer tournaments may
present special conditions.
Apply your model to one or more public facilities at your institution (or
neighboring town). Compare your results with the stated capacity, if one
is posted. If used, your model is likely to be challenged by parties with
interests in increasing the capacity. Write an article for the local
newspaper defending your analysis.
Problem C: Ground Pollution
Background
Several practically important but theoretically difficult mathematical
problems pertain to the assessment of pollution. One such problem consists
in deriving accurate estimates of the location and amount of pollutants
seeping inaccessibly underground, and the location of their source, on the
basis of very few measurements taken only around, but not necessarily
directly in, the suspected polluted region.
Example
The data set (an Excel file at http:// www.comap.com/mcm/procdata.xls,
downloadable into most spreadsheets) shows measurements of pollutants in
underground water from 10 monitoring wells (MW) from 1990 to 1997. The
units are micrograms per liter (μg/1). The location and elevation for
eight wells are known and given in Table 1. The first two numbers are the
coordinates of the location of the well on a Cartesian grid on a map. The
third number is the altitude in feet above Mean Sea Level of the water
level in the well.
Table 1.
Well Number x-Coordinate(ft) y-Coordinate(ft) Elevation(ft)
MW-1 4187.5 6375.0 1482.23
MW-3 9062.5 4375.0 1387.92
MW-7 7625.0 5812.5 1400.19
MW-9 9125.0 4000.0 1384.53
MW-11 9062.5 5187.5 1394.26
MW-12 9062.5 4562.5 1388.94
MW-13 9062.5 5000.0 1394.25
MW-14 4750.0 2562.5 1412.00
The locations and elevations of the other two wells in the data set (MW-27
and MW-33) are not known. In the data set, you will also see the letter T,
M, or B after the well number, indicating that the measurements were taken
at the Top,. Middle, or Bottom of the aquifer in the well. Thus, MW-7B and
MW-7M are from the same well, but from the bottom and from the middle.
Also, other measurements indicate that water tends to flow toward well
MW-9in this area.
Problem One
Build a mathematical model to determine whether any new pollution has
begun during this time period in the area represented by the data set. If
so, identify the new pollutants and estimate the location and time of
their source.
Problem Two
Before the collection of any data, the question arises whether the
intended type of data and model can yield the desired assessment of the
location and amount of pollutants. Liquid chemicals may have leaked from
one of the storage tanks among many similar tanks in a storage facility
built over a homogeneous soil. Because probing under the many large tanks
would be prohibitively expensive and dangerous, measuring only near the
periphery of the storage facility or on the surface of the terrain seems
preferable. Determine what type and number of measurements, taken only
outside the boundary or on the surface of the entire storage facility, can
be used in a mathematical model to determine whether a leak has occurred,
when it occurred, where (from which tank) it occurred, and how much liquid
has leaked.作者: lovecan 时间: 2009-2-1 10:40
谢谢分享！
procdata.rar
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