The Ultimate Brownie Pan
(MCM 2013A)
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.
超级布朗尼锅
(美国竞赛2013年A题)
当我们用一个长方形的锅烘烤食物的时候，热量总是集中在四个角上，从而导致四个角上的食物被过度烹饪，而边上的则烹饪程度不够。在一个圆形的锅里，热量均匀分布在整个外部边上，那么在边上的食物就不会被过度烹饪。但是，大多数的烤箱仍然是长方形的，从形状上来讲用圆形的烤盘并不能有效率的运用烤箱的空间。
试建立一个模型来说明在不同形状（如长方形，圆形或者介于两者之间的其他某个形状）的锅里，锅的外部边的热量的分布情况。
假定：
1. 形状为长方形的烤箱的宽度与长度之比为$W/L$.
2. 每个锅的面积都是$A$.
3. 最初，烤箱中有两层烤箱架，并且在空间上是均匀分布的。
建立一个模型，使得我们能用该模型选择满足以下条件的最佳类型（形状）的锅：
1. 希望能放在烤箱中的锅的数目（$N$）最大
2. 希望锅里的平均分布热量（$H$）最大
3. 对条件1与2分别赋予权重 $p$ 与 $1-p$，希望将条件1与2的组合尽量优化，并分析说明在不同的 $W/L$ 及 $p$ 值的变化下，结果会如何变化.
特别的，除了你的MCM的结论外，请为BG杂志准备一到两页的广告页，以凸显你的设计和结果。
各种数模竞赛赛题
参考资料
材料 及 优秀论文
F
优
周伯尧、王森博、傅霞君(M)
袁博、龚政、邵雪婷(M)
范超、陈沛秋、孙聪慧(M)
王闻杰、施亮、熊峰(M)