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2005 年美国大学生数学建模竞赛 MCM试题
2005 MCM A: Flood Planning 洪水计划
Lake Murray in central South Carolina is formed by a large earthen dam, which was completed in 1930 for power production Model the flooding downstream in the event there is a catastrophic earthquake that breaches the dam
Two particular questions:
Rawls Creek is a year-round stream that flows into the Saluda River a short distance downriver from the dam How much flooding will occur in Rawls Creek from a dam failure, and how far back will it extend?
Could the flood be so maive downstream that water would reach up to the SC State Capitol Building, which is on a hill overlooking the Congaree River? South Carolina中部的Murray河是由北部的一个巨大水坝形成的，这是在1930年为了发电而修建的。模拟一起洪水淹没下游的事件，这起事件是由于一次灾难性的地震损坏了水坝造成的。两个问题：
1、Rawls Creek是水坝下游流入Saluda河的一条终年流动的河流，在当水坝损坏后，在Rawls Creek将会出现多大的洪流，洪水的波及面将有多大？
2、S.C.国会大厦大楼在一座小山上，在S.C.国会大厦大楼能俯视Congaree河。洪水能如此巨大顺流以至于水将扩展到S.C.国会大厦大楼吗？
2005 MCM B: Tollbooths收费亭
Heavily-traveled toll roads such as the Garden State Parkway, Interstate 95, and so forth, are multi-lane divided highways that are interrupted at intervals by toll plazas Because collecting tolls is usually unpopular, it is desirable to minimize motorist annoyance by limiting the amount of traffic disruption caused by the toll plazas Commonly, a much larger number of tollbooths is provided than the number of travel lanes entering the toll plaza Upon entering the toll plaza, the flow of vehicles fans out to the larger number of tollbooths, and when leaving the toll plaza, the flow of vehicles is required to squeeze back down to a number of travel lanes equal to the number of travel lanes before the toll plaza Consequently, when traffic is heavy, congestion increases upon departure from the toll plaza When traffic is very heavy, congestion also builds at the entry to the toll plaza because of the time required for each vehicle to pay the toll
Make a model to help you determine the optimal number of tollbooths to deploy in a barrier-toll plaza Explicitly consider the scenario where there is exactly one tollbooth per incoming travel lane Under what conditions is this more or le effective than the current practice? Note that the definition of “optimal” is up to you to determine
像 Garden State Parkway, Interstate 95这类的长途收费公路，通常是多行道德，被分成几条高速公路，在这些高速公路上每隔一段会设立一个通行税收费广场。因为征收通行税通常不受欢迎，所以应该尽量减少通过通行税收费广场引起的交通混乱给汽车司机带来的烦恼。通常，收费亭的数量要多于进入收费广场的道路的数量。进入通行税收费广场的时候，流到大量收费亭的车辆呈扇形展开，当离开通行税广场的时候，车流只能按照收费广场前行车道路的数量排队，按次通过！从而，当交通拥挤的时候，拥挤在违背通行税广场增加。当交通非常拥挤的时候，因为每辆车付通行费的时间要求，阻塞也会出现在通行税收费广场的入口处。建立一个模型来确定一个容易造成阻塞的通行税收费广场中应该部署的最优的收费亭数量。需要保证每一个进入收费广场的交通路线都仅有一个收费亭。与当今的实践相比较，在什么条件下，这或多或少有效。注意：最佳的定义由你自己决定。
