高中数学补习班2013美国大学生数学建模竞赛题目和翻译

2020年11月19日发(作者：梁俊一)
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；