2020美赛B题翻译

翻译

2020 MCM周末2问题B：持续时间最长的沙堡

无论世界上哪里有休闲的沙滩，似乎都有儿童（和成人）在海边制作沙堡。使用工具，玩具和想象力，沙滩行人创造了沙堡，沙堡的范围从简单的沙丘到实际城堡的复杂复制品，其墙壁，塔楼，护城河和其他模仿真实城堡的特征。在所有这些方面，通常会形成一个初始的基础，该基础由一个单一的，无形的湿沙丘组成，然后将其切割并成形为可识别的3维几何形状，从而在其上构建更具城堡意义的特征。
不可避免的是，海浪的流入加上潮汐的上升侵蚀了沙堡。但是，即使建造的沙丘大小和距离同一海滩的水域大致相同，似乎并非所有沙堡对海浪和潮汐的反应方式都相同。因此，人们想知道是否存在用于沙堡基础的最佳3维几何形状。

要求:
1.构造一个数学模型以识别最佳的3维几何形状，以用作沙堡基础，该沙丘基础将在以下情况下在经历波浪和潮汐的海滨上持续最长的时间：
•与同一海滩上的水大致保持相同的距离，并且
•使用相同类型的沙子，大致相同的沙子量和相同的水砂比例建造。
2.假设您不使用其他添加剂或材料（例如，塑料或木制支撑物，石头等），请使用您的模型确定城堡基础的最佳沙水混合比。
3.根据需要调整模型，以确定需求1中确定的最佳3维沙堡基础如何受到雨水的影响，以及在下雨时它是否仍保留为用作城堡基础的最佳3维几何形状。
4.您还可以使用其他哪些策略来延长沙堡的使用寿命？
5.最后，写一篇内容丰富的一到两页的文章，描述您的模型及其结果，以供发表在度假杂志上：《阳光中的乐趣》，其读者主要是非技术人员。

您提交的内容应包括：
•一页摘要表
•目录
•一到两页的文章
•您的解决方案不超过20页，最多包含摘要，目录和文章的24页。
注意：参考列表和任何附录不计入页数限制，应在完成解决方案后出现。 您不应使用未经版权法限制使用的未经授权的图像和材料。 确保您引用了想法的来源和报告中使用的材料。

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原文

Problem B: The Longest Lasting Sandcastle(s)

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 waterto-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?

4. 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.