美赛2013A题论文(终极布朗尼盘)

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A

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2013Mathematical Contest in Modeling(MCM/ICM)Summary Sheet

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There are plenty of people who are fond of Brownies.Industrial manufacturing of them requires to bake the biggest amount of Brownies in the same batch and also to avoid overcooking of the batter next to the corners of the pan,in order to avoid losses. Overcooking can affect the volumes of batter that are in direct contact with more than one hot surface of the pan,for instance the bottom of it and one of its sides.The corners of the pan are the hottest regions,as three surfaces deliver heat simultaneously(two sides plus the bottom).That’s why overcooking is frequent in the corners.

Round pans distribute heat in a more even way,due to the fact that there are never three surfaces delivering heat to a volume of batter(two at most),.The problem with them is that they cause a big loss of oven space as compared to square ones,even to hexagonal ones.As round pans minimize overcooking and rectangular pans minimize oven-space loss,our model posits a compromise between these two shapes,namely modified rectangular pans having a 90°centred-angle arc instead of being standard rectangles.

In order to clarify which factor will be advantaged by variations of this type of shape(either space saving or heat evenness),we put weights p and(1-p)on the sum of the line lengths and on the sum of the arc lengths.As we consider that the area of the pan is fixed and its width-to-length ratio W/L is given,we propose3equations expressing the effect of having longer or shorter arcs on the pans performances(the longer the arcs,the evener the heat distribution;the shorter the arcs,the better the space in the oven is filled).

We use matlab to solve these equations.Once we give p and W/L two values between0to1, we can get a shape of pan.By changing the weights p by small steps while the ratio W/L is fixed,we can plot the figure in a three-dimentional space layer by layer,so that we can see clearly how the figure changes with weights changing for a given W/L ratio.From these graphs,we can conclude that the space the pan takes increases with increasing weights on the arc factor(we provide some charts to make clear).

In general,if the producer wants to increase the quality of the brownies by avoiding overcooking of volumes placed in critical locations,he needs to decrease p.If he wants to increase the production by filling the oven as much as possible,he needs to increase p. Large p values amount to short arcs,small p values mean long arcs.