2013美国大学生数学建模竞赛题目和翻译

2013 Contest Problems MCM PROBLEMS

PROBLEM A: The Ultimate Brownie Pan

When baking in a rectangular pan heat is concentrated in the 4 corn ers and the product gets overcooked at the corners (and to a lesser e xtent at the edges). In a round pan the heat is distributed evenly ov er the entire outer edge and the product is not overcooked at the edg es. However, since most ovens are rectangular in shape using round pa ns is not efficient with respect to using the space in an oven.

Develop a model to show the distribution of heat across the outer edg e of a pan for pans of different shapes - rectangular to circular and other shapes in between. Assume

1. A width to length ratio of W/L for the oven which is rectangular i n shape.

2. Each pan must have an area of A.

3. Initially two racks in the oven, evenly spaced.

Develop a model that can be used to select the best type of pan (shap

e) under the following conditions:

1. Maximize number of pans that can fit in the oven (N)

2. Maximize even distribution of heat (H) for the pan

3. Optimize a combination of conditions (1) and (2) where weights p a nd (1- p) are assigned to illustrate how the results vary with differ ent values of W/L and p.

In addition to your MCM formatted solution, prepare a one to two pa ge advertising sheet for the new Brownie Gourmet Magazine highlightin g your

当用方形的烤盘烤饼时，热量会集中在四角，食物就在四角（四条边的热量略小于四角）烤焦了。而用一个圆形的烤盘热量会均匀分布在整个外缘，食物就不会被边缘烤焦。但是，因为大多数烤箱是矩形的，使用圆形的烤盘不那么有效地使用空间。建立一个模型来表现热量在不同形状的烤盘的外缘的分布——包括从矩形到圆形以及介于矩形与圆形的过渡形状。

试构建一个模型来显示通过不同烤盘的外沿热量的分布情况：方形到圆形极其两者之间的其他形状。

假定：

1. 方形烤箱宽长比为W/L；

2. 所有参烤盘的面积必须为A；

3. 给定原始条件为两个烤盘支架在烤箱中均距摆放。

构建一个模型用于在如下情境下筛选最佳烤盘形状： 1. 使能放进烤箱中的烤盘数（N）最大；

2. 最大化均匀热度分布(H)的烤盘形状；

3. 综合考虑1和2，给上述两个指标分配权重p和（1-p）。随着W/L与p的变化，展示出结果的变化。

除了提供标准的MCM格式解答之外，为布朗尼美食杂志提供一份1-2页的广告宣传，你需要突出你的设计和结果。

直接热辐射带热风风扇烤箱的空气对流

微波炉基本原理

通电后，电能通过磁控管的工作转化为微波，通过炉内空气传递到食物，使食物内部每个分子都进行热运动，相邻分子之间产生了的摩擦力，并产生大量的摩擦热，从而加热变熟。这个过程是由内而外。

PROBLEM B: Water, Water, Everywhere

Fresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feas ible, and cost-efficient water strategy for 2013 to meet the projecte d water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical mo del must address storage and movement; de-salinization; and conservat ion. If possible, use your model to discuss the economic, physical, a nd environmental implications of your strategy. Provide a non-technic al position paper to governmental leadership outlining your approach, its feasibility and costs, and why it is the “best water strategy c hoice.”

Countries: United States, China, Russia, Egypt, or Saudi Arabia

淡水是世界上许多地方发展的限制因素。建立一个数学模型，来确定一个有效的，可行的低成本的2013年用水计划，来满足某国（从下方的列表中选择一个国家）未来（2025年）的用水需求，并确定最优的水的计划。特别的，你的数学模型必须满足储存、运输、淡化、和节水。如果可能的话，用你的模型来讨论你的计划对经济，自然和环境的影响。提供一个非技术性的意见书给政府领导概述你的方法，以及方法的可行性和成本，以及它为什么是“最好的用水计划的选择”。

国家：美国，中国，俄罗斯，埃及，或者沙特阿拉伯。

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