2013 Contest Problems MCM PROBLEMS
PROBLEM A: The Ultimate Brownie Pan
When baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges.
However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.
Develop a model to show the distribution of heat across the outer edge 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 in 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 (shape) 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 and (1- p) are assigned to illustrate how the results vary with different values of W/L and p.
In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results.
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, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage and movement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility and costs, and why it is the “best water strategy choice.”
Countries: United States, China, Russia, Egypt, or Saudi Arabia
2013年赛题 MCM问题
问题A：终极布朗尼潘
当在一个矩形的锅热烘烤时的4个角落中浓缩，并在拐角处（以及在较小程度上在边缘处）：产品会过头。在一个圆形盘的热量被均匀地分布在整个外缘和在边缘处的产品不过头。然而，由于大多数烤炉是矩形的形状使用圆形平底锅是效率不高，相对于使用的空间的烘箱中。 开发一个模型来显示不同形状的平底锅锅的外边缘之间的热分布-矩形到圆形和在两者之间的其它形状。 假设 1。的宽度长度比W / L为烘箱是矩形的形状。 2。每盘必须有一个区域的一个。 3。最初两个机架的烘箱中，均匀地间隔开。 开发模型，可用于选择的最佳类型的盘（形状）在下列条件下： 1。最大化数量的锅，可以适合在烤箱中（N） 2。最大化均匀分布的热（H）为锅 3。优化的组合的条件（1）和（2）式中的权重p和（1 - p）被分配的结果来说明如何与W / L 和 p的 不同的值的变化。 在除了到您的MCM的格式化的溶液中，制备一个到新的布朗尼美食杂志，突出自己的设计和结果两页的广告片。
问题B：水，水，无处不在
新鲜的白 开水是在世界大部分地区的发展限制约束。建立一个数学模
型，为确定有效的，可行的和具有成本效益的水资源战略于2013年，以满足预计的用水需求，从下面的列表]中选择一个国家，到2025年，确定最佳的水战略。特别是，您的数学模型必须解决存储和运动，去盐碱化和保护。如果可能的话，用你的模型，探讨经济，物理和环境的影响，你的战略。提供一个非技术性的立场文件，政府领导介绍你的方法，其可行性和成本，以及为什么它是“最好的水战略的选择。” 国家：美国，中国，俄罗斯，埃及，沙特阿拉伯
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