翻譯
2020 MCM週末2問題B：持續時間最長的沙堡
無論世界上哪裏有休閒的沙灘，似乎都有兒童（和成人）在海邊製作沙堡。使用工具，玩具和想象力，沙灘行人創造了沙堡，沙堡的範圍從簡單的沙丘到實際城堡的複雜複製品，其牆壁，塔樓，護城河和其他模仿真實城堡的特徵。在所有這些方面，通常會形成一個初始的基礎，該基礎由一個單一的，無形的溼沙丘組成，然後將其切割併成形爲可識別的3維幾何形狀，從而在其上構建更具城堡意義的特徵。
不可避免的是，海浪的流入加上潮汐的上升侵蝕了沙堡。但是，即使建造的沙丘大小和距離同一海灘的水域大致相同，似乎並非所有沙堡對海浪和潮汐的反應方式都相同。因此，人們想知道是否存在用於沙堡基礎的最佳3維幾何形狀。
要求:
1.構造一個數學模型以識別最佳的3維幾何形狀，以用作沙堡基礎，該沙丘基礎將在以下情況下在經歷波浪和潮汐的海濱上持續最長的時間：
•與同一海灘上的水大致保持相同的距離，並且
•使用相同類型的沙子，大致相同的沙子量和相同的水砂比例建造。
2.假設您不使用其他添加劑或材料（例如，塑料或木製支撐物，石頭等），請使用您的模型確定城堡基礎的最佳沙水混合比。
3.根據需要調整模型，以確定需求1中確定的最佳3維沙堡基礎如何受到雨水的影響，以及在下雨時它是否仍保留爲用作城堡基礎的最佳3維幾何形狀。
4.您還可以使用其他哪些策略來延長沙堡的使用壽命？
5.最後，寫一篇內容豐富的一到兩頁的文章，描述您的模型及其結果，以供發表在度假雜誌上：《陽光中的樂趣》，其讀者主要是非技術人員。
您提交的內容應包括：
•一頁摘要表
•目錄
•一到兩頁的文章
•您的解決方案不超過20頁，最多包含摘要，目錄和文章的24頁。
注意：參考列表和任何附錄不計入頁數限制，應在完成解決方案後出現。 您不應使用未經版權法限制使用的未經授權的圖像和材料。 確保您引用了想法的來源和報告中使用的材料。
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
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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. -
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.).
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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?
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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.