2013美国大学生数学建模竞赛题目和翻译和A题图解.doc

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?panthe?heat?is?distributed?evenly?over?the?entire?outer?edge?andthe 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?theouter?edge?of?a?pan?for?pans?of?different?shapes?-?rectangular?to?circular?and?other?shapes?in?between.Assume? A?width?to?length?ratio?of?W/L?for?the?oven?which?isrectangular?in?shape.? 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? ? 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。 电烤箱构造 电烤箱由箱体、箱门、电热元件、控温与定时装置组成。 ①箱体多用薄钢板制成，一