2012 年美国大学生数学建模竞赛 MCM

2023年12月10日发(作者：2017中考数学试卷宁德)

2012 年美国大学生数学建模竞赛

MCM、ICM 试题

2012 MCM A: The Leaves of a Tree

“How much do the leaves on a tree weigh?” How might one estimate the actual weight of

the leaves (or for that matter any other parts of the tree)? How might one classify leaves?

Build a mathematical model to describe and classify leaves. Consider and answer the

following:

Why do leaves have the various shapes that they have?

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Do the shapes “minimize” overlapping individual shadows that are cast, so as to

maximize exposure? Does the distribution of leaves within the “volume” of the tree

and its branches effect the shape?



Speaking of profiles, is leaf shape (general characteristics) related to tree

profile/branching structure?



How would you estimate the leaf mass of a tree? Is there a correlation between the

leaf mass and the size characteristics of the tree (height, mass, volume defined by the

profile)?

In addition to your one page summary sheet prepare a one page letter to an editor of a

scientific journal outlining your key findings.



2012 MCM B: Camping along the Big Long River

Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water

rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip

that requires several days of camping. River trips all start at First Launch and exit the river

at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts,

which travel on average 4 mph or motorized boats, which travel on average 8 mph. The

trips range from 6 to 18 nights of camping on the river, start to finish.. The government

agency responsible for managing this river wants every trip to enjoy a wilderness

experience, with minimal contact with other groups of boats on the river. Currently, X trips

travel down the Big Long River each year during a six month period (the rest of the year it

is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly

uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park

managers have been asked to allow more trips to travel down the river. They want to

determine how they might schedule an optimal mix of trips, of varying duration (measured

in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the

best way possible. In other words, how many more boat trips could be added to the Big

Long River’s rafting season? The river managers have hired you to advise them on ways

in which to develop the best schedule and on ways in which to determine the carrying

capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo

to the managers of the river describing your key findings.