2012美国数学建模比赛赛题(中文)

IMPORTANT CHANGE TO CONTEST RULES FOR MCM/ICM 2012:

2012年MCM/ICM的竞赛规则的重大变化：

Teams (Student or Advisor) are now required to submit an electronic copy (summary sheet and solution) of their solution paper by email to solutions@. Your email MUST be received at COMAP by the submission deadline of 8:00 PM EST, February 13, 2012. Teams are free to choose between MCM Problem A, MCM Problem B or ICM Problem C.

团队（学生或指导老师）必须将解决方案文件的电子副本（汇总表及解决方案）以电子邮件的形式发送到solutions@。你们的电子邮件必须在美国东部时间2012年2月13日之前发送到COMAP。团队可以自由选择MCM中的A、B题或ICM中的C题。

COMAP Mirror Site: For more in:

/undergraduate/contests/mcm/

COMAP是镜像网站：欲了解更多：

/undergraduate/contests/mcm/

MCM: The Mathematical Contest in Modeling

MCM：数学建模竞赛

ICM: The Interdisciplinary Contest in Modeling

ICM的：交叉学科建模竞赛

2012 Contest Problems

MCM PROBLEMS

PROBLEM A: The Leaves of a Tree

一个树的叶子

"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:

“一颗树上的叶子有多重？”如何估计这些叶子（或者一颗树的任何其他部分）的实际重量？又如何分类这些叶子？建立一个数学模型来描述和分类叶子。考虑并回答下列问题：

• Why do leaves have the various shapes that they have?

为什么叶子有各种形状？

• 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?

这些叶子形状是以最小化来重叠自身阴影或者说是最大化来曝光吗？树叶“量”上的分布和树的分支影响了树叶的形状吗？

• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?

就树型而言，叶子的形状（一般特征）和树型/分支结构有关吗？

• 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)?

你将如何估计一棵树的叶质量？在树叶质量和树的大小特性（高度、重量或者体积）之间有一个关系吗？

In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.