2020 MCM问题B真题

问题标题：最持久的沙堡

年份：2020
学生等级：本科生
来源：MCM

问题

世界上任何有休闲沙滩的地方，似乎都有孩子（和成人）在海边堆沙堡。沙滩游客利用工具、玩具和想象力，堆起的沙堡从简单的沙堆到复杂的真实城堡复制品，带有墙壁、塔楼、护城河和其他模仿真实城堡的特征。在所有这些沙堡中，人们通常先形成一个由单一、不起眼的湿沙堆组成的初始地基，然后继续切割和塑造这个底座，使其成为可识别的三维几何形状，在此基础上建造更具城堡特色的特征。

不可避免的是，海浪的涌入和潮汐的上涨会侵蚀沙堡。然而，看起来并不是所有的沙堡对海浪和潮汐的反应都一样，即使它们建在同一片海滩上，大小大致相同，距离水面的距离也大致相同。因此，人们不禁想知道，是否存在一个最适合沙堡地基的三维几何形状。

要求

1. 建立一个数学模型，确定最适合用作沙堡地基的三维几何形状，该形状将在以下条件下在经历海浪和潮汐的海岸上持续最长时间：

建在同一片海滩上，距离水面的距离大致相同，
使用相同类型的沙子、大致相同的沙子数量以及相同的水与沙子的比例建造。

2. 使用您的模型，确定城堡地基的最佳沙水混合比例，假设您不使用其他添加剂或材料（例如塑料或木质支架、石头等）。

3. 根据需要调整模型，以确定您在要求 1 中确定的最佳 3 维沙堡地基如何受到雨水的影响，以及它在下雨时是否仍然是用作城堡地基的最佳 3 维几何形状。

4. 如果有的话，您还可以使用哪些其他策略来让您的沙堡保存更长时间？

5. 最后，撰写一篇一到两页的内容丰富的文章，描述您的模型及其结果，并在度假杂志《阳光下的乐趣》上发表，该杂志的读者主要是非技术人员。

您的提交内容应包括：

单页摘要表
目录
一至两页的文章
您的解决方案不超过 20 页，最多 24 页，包括摘要、目录和文章。

注意：参考文献列表和任何附录不计入页数限制，应在完成解决方案后显示。您不应使用未经授权的图像和材料，这些图像和材料的使用受版权法限制。确保您引用了您的想法和报告中使用的材料的来源。

可到文末下载完整版中英文真题

以下是英文版真题

Problem

Wherever there are recreational sandy ocean beaches in the world, there seem to be children (and adults) creating sandcastles on the seashore. Using tools, toys, and imagination, beach goers create sandcastles that range from simple mounds of sand to complicated replicas of actual castles with walls, towers, moats, and other features that mimic real castles. In all these, one typically forms an initial foundation consisting of a single, nondescript mound of wetted sand, and then proceeds to cut and shape this base into a recognizable 3-dimensional geometric shape upon which to build the more castle-defining features.

Inevitably, the inflow of ocean waves coupled with rising tides erodes sandcastles. It appears, however, that not all sandcastles react the same way to waves and tides, even if built roughly the same size and at roughly the same distance from the water on the same beach. Consequently, one wonders if there exists a best 3-dimensional geometric shape to use for a sandcastle foundation.

Requirements

1. Construct a mathematical model to identify the best 3-dimensional geometric shape to use as a sandcastle foundation that will last the longest period of time on a seashore that experiences waves and tides under the following conditions:

built at roughly the same distance from the water on the same beach, and
built using the same type of sand, roughly the same amount of sand, and the same water-to-sand proportion.

2. Using your model, determine an optimal sand-to-water mixture proportion for the castle foundation, assuming you use no other additives or materials (e.g. plastic or wooden supports, stones, etc.).

3. Adjust your model as needed to determine how the best 3-dimensional sandcastle foundation you identified in requirement 1 is affected by rain, and whether it remains the best 3-dimensional geometric shape to be used as a castle foundation when it is raining.

4. What other strategies, if any, might you use to make your sandcastle last longer?

5. Finally, write an informative, one- to two-page article describing your model and its results for publication in the vacation magazine: Fun in the Sun, whose readers are mainly non-technical.

Your submission should consist of:

One-page Summary Sheet
Table of Contents
One- to Two-page Article
Your solution of no more than 20 pages, for a maximum of 24 pages with your summary, table of contents, and article.

Note: Reference List and any appendices do not count toward the page limit and should appear after your completed solution. You should not make use of unauthorized images and materials whose use is restricted by copyright laws. Ensure you cite the sources for your ideas and the materials used in your report.

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