2001年美国大学生数学建模竞赛试题 2001 Mathematical Contest in Modeling (mcm) 问题A选择自行车车轮 有不同类型的车轮可以让自行车手们用在自己的自行车上。两种基本的车轮类型是分别用金属辐条和 实体圆盘组装而成(见图1)。辐条车轮较轻,但实体车轮更符合空气动力学原理。对于一场公路竞赛 实体车轮从来不会用作自行车的前轮但可以用作后轮。 职业自行车手们审视竞赛路线,并且请一位识文断字的人推断应该使用哪种车轮。选择决定是根据沿 途山丘的数量和陡度,天气,风速,竞赛本身以及其他考虑作出的。你所喜爱的参赛队的教练希望准备妥 当一个较好的系统,并且对于给定的竞赛路线己经向你的参赛队索取有助于确定宜用哪种车轮的信息 这位教练需要明确的信息来帮助作出决定,而且已经要求你的参赛队完成下面列出的各项任务。对于 每项任务都假定,同样的辐条车轮将总是装在前面,而装在后面的车轮是可以选择的。 任务1:提供一个给出风速的表格,在这种速度下实体后轮所需要的体能少于辐条后轮。这个表格应 当包括相应于从百分之零到百分之十增量为百分之一的不同公路陡度的风速。(公路陡度定义为一座山丘 的总升高除以公路长度。如果把山丘看作一个三角形,它的陡度是指山脚处倾角的正弦。)一位骑手以初 始速度45kph从山脚出发,他的减速度与公路陡度成正比。对于百分之五的陡度,骑上100米车速要下降 8kph左右。 任务2:提供一个例证,说明这个表格怎样用于一条时间试验路线。 任务3:请判明这个表格是不是一件决定车轮配置的适当工具,并且关于如何作出这个决定提出其他 建议。 王强译 图1:左图表示一个实体车轮,而辐条车轮画在右边 Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right. Problem A: Choosing a Bicycle wheel Cyclists have d ifferent types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk(see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear Task 1. Provide a table giving the wind speed at which the power required for a solid rear wheel less than for a spoked rear wheel. The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments.( Road grade is defined to be the ratio of the total rise of a hill divided by the length of the road. If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill. ) a rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road grade. a rider will lose about 8 kph for a five percent grade over 100 meters Task 2 Provide an example of how the table could be used for a specific time trial course Task 3. Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision
2001 年美国大学生数学建模竞赛试题 2001 Mathematical Contest in Modeling（MCM） 问题 A 选择自行车车轮 有不同类型的车轮可以让自行车手们用在自己的自行车上。两种基本的车轮类型是分别用金属辐条和 实体圆盘组装而成（见图 1）。辐条车轮较轻，但实体车轮更符合空气动力学原理。对于一场公路竞赛， 实体车轮从来不会用作自行车的前轮但可以用作后轮。 职业自行车手们审视竞赛路线，并且请一位识文断字的人推断应该使用哪种车轮。选择决定是根据沿 途山丘的数量和陡度，天气，风速，竞赛本身以及其他考虑作出的。你所喜爱的参赛队的教练希望准备妥 当一个较好的系统，并且对于给定的竞赛路线已经向你的参赛队索取有助于确定宜用哪种车轮的信息。 这位教练需要明确的信息来帮助作出决定，而且已经要求你的参赛队完成下面列出的各项任务。对于 每项任务都假定，同样的辐条车轮将总是装在前面，而装在后面的车轮是可以选择的。 任务 1：提供一个给出风速的表格，在这种速度下实体后轮所需要的体能少于辐条后轮。这个表格应 当包括相应于从百分之零到百分之十增量为百分之一的不同公路陡度的风速。（公路陡度定义为一座山丘 的总升高除以公路长度。如果把山丘看作一个三角形，它的陡度是指山脚处倾角的正弦。）一位骑手以初 始速度 45kph 从山脚出发，他的减速度与公路陡度成正比。对于百分之五的陡度，骑上 100 米车速要下降 8kph 左右。 任务 2：提供一个例证，说明这个表格怎样用于一条时间试验路线。 任务 3：请判明这个表格是不是一件决定车轮配置的适当工具，并且关于如何作出这个决定提出其他 建议。 王强译 图 1:左图表示一个实体车轮，而辐条车轮画在右边。 Figure 1: A solid wheel is shown on the left and a spoked wheel is shown on the right. Problem A: Choosing a Bicycle Wheel Cyclists have different types of wheels they can use on their bicycles. The two basic types of wheels are those constructed using wire spokes and those constructed of a solid disk (see Figure 1) The spoked wheels are lighter, but the solid wheels are more aerodynamic. A solid wheel is never used on the front for a road race but can be used on the rear of the bike. Professional cyclists look at a racecourse and make an educated guess as to what kind of wheels should be used. The decision is based on the number and steepness of the hills, the weather, wind speed, the competition, and other considerations. The director sportif of your favorite team would like to have a better system in place and has asked your team for information to help determine what kind of wheel should be used for a given course. The director sportif needs specific information to help make a decision and has asked your team to accomplish the tasks listed below. For each of the tasks assume that the same spoked wheel will always be used on the front but there is a choice of wheels for the rear. Task 1.Provide a table giving the wind speed at which the power required for a solid rear wheel is less than for a spoked rear wheel. The table should include the wind speeds for different road grades starting from zero percent to ten percent in one percent increments. (Road grade is defined to be the ratio of the total rise of a hill divided by the length of the road.If the hill is viewed as a triangle, the grade is the sine of the angle at the bottom of the hill.) A rider starts at the bottom of the hill at a speed of 45 kph, and the deceleration of the rider is proportional to the road grade. A rider will lose about 8 kph for a five percent grade over 100 meters. Task 2.Provide an example of how the table could be used for a specific time trial course. Task 3.Determine if the table is an adequate means for deciding on the wheel configuration and offer other suggestions as to how to make this decision
问题B逃避飓风怒吼(一场恶风……) 1999年,在 Floyd飓风预报登陆之前,撤离南卡罗来纳州沿海地区的行动导致一场永垂青史的交通拥 塞。车水马龙停滞在州际公路I-26上,那是内陆上从 Charleston通往该州中心 Columbia相对安全处所的 主要干线。正常时轻松的两个小时驱车路要用上18个小时才能开到头。许多车竟然沿途把汽油消耗净尽 幸运的是, Floyd飓风掉头长驱北上,这次放过了南卡罗来纳州,但是,公众的喧嚷正在迫使该州官员们 寻找各种办法,以求避免这场交通恶梦再度出现 倾力解决这个问题的主要提议是Ⅰ-26公路上的车辆转向疏散,因此,包括通往海岸的多条次级公路在 内,从两个侧面疏导车流在内陆从 Charleston开往 Columbia 把提议付诸实施的计划已经由 South Carolina Emergency Preparedness Division准备好(而且贴在互联 网上)。从 Myrtle Beach和 Hilton Head通往内地的主干道上车辆转向疏散的方案也在规划中。 这里有一张南卡罗来纳州的简化地图。 Charleston有近500000, Myrtle Beach有200,000人左右 而另一个250,000人分散在沿岸其余地区。(如果查找,更精确的数据随处可用。)州与州之间有两条车 辆往来的次级公路,自然大都市地区除外,那里有三条。 Columbia,又一个50000.左右的大都市地区, 没有充足的旅店空间为撤退者提供食宿(包括沿其他路线来自大北边的一些人),所以,若干车辆继续撤 离,沿着I-26公路开往 Spartanburg市;沿着-77公路北上 Charlotte市;而且沿着L-20公路东进 Atlanta 市。在1999年,从 Columbia开往西北方向的车辆行进得非常慢。对这个问题建立一个模型,调查研究哪 种策略可以降低在1999年观察到的拥挤。这里有一些问题需要加以考虑: 在什么条件下,把I-26的两条开往海岸的次级公路变成开往 Columbia的两条次级公路,特别是把整 个1-26变成单行道会使撤离交通状况得到重大改善? 在1999年,南卡罗来纳州的整个沿海地区奉命同时撤离。如果采取另一种策略,逐个郡按某个时间 段错开撤离,同时与飓风对沿岸影响的模式相协调,撤离交通状况会改善吗? 在1-26公路旁边有若干较小的高速公路从海岸延伸到内陆。在什么条件下,把车辆流转向这些道路会 改善撤离交通? 在 Columbia建立更多临时收容所来减少离开 Columbia的车辆,这会对撒离交通状况有什么影响? 在1999年,离开海岸的许多家庭一路上携带他们的船只,露营设备和汽车住宅。许多家庭驾驶他们 的所有汽车。在什么条件下,应当对携带的车辆类型或车辆数目加以限制以求保证适时撒离? 在1999年,人们还会记得,若干 Georgia州 and Florida州的沿岸居民逃避较早预报的 Floyd飓风南部 登陆,沿着I-95公路北上而加重了南卡罗来纳州交通问题。他们对于撤离交通的冲击会有多大? 要清楚地指明,为了比较各种策略,使用什么方法对实施状况予以评测。 要求:预备一篇简短的报刊文章,不超过两页,向公众解释你的研究成果和结论 王强译 Problem B: Escaping a Hurricane's Wrath(An Ill wind.) Evacuating the coast of South Carolina ahead of the predicted land fall of hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, includ ing the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads lead ing inland from Myrtle Beach and Hilton Head is also planned A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of
问题 B 逃避飓风怒吼（一场恶风……） 1999 年，在 Floyd 飓风预报登陆之前，撤离南卡罗来纳州沿海地区的行动导致一场永垂青史的交通拥 塞。车水马龙停滞在州际公路 I-26 上，那是内陆上从 Charleston 通往该州中心 Columbia 相对安全处所的 主要干线。正常时轻松的两个小时驱车路要用上 18 个小时才能开到头。许多车竟然沿途把汽油消耗净尽。 幸运的是，Floyd 飓风掉头长驱北上，这次放过了南卡罗来纳州，但是，公众的喧嚷正在迫使该州官员们 寻找各种办法，以求避免这场交通恶梦再度出现。 倾力解决这个问题的主要提议是 I-26 公路上的车辆转向疏散，因此，包括通往海岸的多条次级公路在 内，从两个侧面疏导车流在内陆从 Charleston 开往 Columbia。 把提议付诸实施的计划已经由 South Carolina Emergency Preparedness Division 准备好（而且贴在互联 网上）。从 Myrtle Beach 和 Hilton Head 通往内地的主干道上车辆转向疏散的方案也在规划中。 这里有一张南卡罗来纳州的简化地图。Charleston 有近 500,000 人，Myrtle Beach 有 200,000 人左右， 而另一个 250,000 人分散在沿岸其余地区。（如果查找，更精确的数据随处可用。）州与州之间有两条车 辆往来的次级公路，自然大都市地区除外，那里有三条。Columbia，又一个 500,000 人左右的大都市地区， 没有充足的旅店空间为撤退者提供食宿（包括沿其他路线来自大北边的一些人），所以，若干车辆继续撤 离，沿着 I-26 公路开往 Spartanburg 市；沿着 I-77 公路北上 Charlotte 市；而且沿着 I-20 公路东进 Atlanta 市。在 1999 年，从 Columbia 开往西北方向的车辆行进得非常慢。对这个问题建立一个模型，调查研究哪 种策略可以降低在 1999 年观察到的拥挤。这里有一些问题需要加以考虑： 在什么条件下，把 I-26 的两条开往海岸的次级公路变成开往 Columbia 的两条次级公路，特别是把整 个 I-26 变成单行道会使撤离交通状况得到重大改善？ 在 1999 年，南卡罗来纳州的整个沿海地区奉命同时撤离。如果采取另一种策略，逐个郡按某个时间 段错开撤离，同时与飓风对沿岸影响的模式相协调，撤离交通状况会改善吗？ 在 I-26 公路旁边有若干较小的高速公路从海岸延伸到内陆。在什么条件下，把车辆流转向这些道路会 改善撤离交通？ 在 Columbia 建立更多临时收容所来减少离开 Columbia 的车辆，这会对撤离交通状况有什么影响？ 在 1999 年，离开海岸的许多家庭一路上携带他们的船只，露营设备和汽车住宅。许多家庭驾驶他们 的所有汽车。在什么条件下，应当对携带的车辆类型或车辆数目加以限制以求保证适时撤离？ 在 1999 年，人们还会记得，若干 Georgia 州 and Florida 州的沿岸居民逃避较早预报的 Floyd 飓风南部 登陆，沿着 I-95 公路北上而加重了南卡罗来纳州交通问题。他们对于撤离交通的冲击会有多大？ 要清楚地指明，为了比较各种策略，使用什么方法对实施状况予以评测。 要求：预备一篇简短的报刊文章，不超过两页，向公众解释你的研究成果和结论。 王强译 Problem B: Escaping a Hurricane's Wrath (An Ill Wind...) Evacuating the coast of South Carolina ahead of the predicted landfall of Hurricane Floyd in 1999 led to a monumental traffic jam. Traffic slowed to a standstill on Interstate I-26, which is the principal route going inland from Charleston to the relatively safe haven of Columbia in the center of the state. What is normally an easy two-hour drive took up to 18 hours to complete. Many cars simply ran out of gas along the way. Fortunately, Floyd turned north and spared the state this time, but the public outcry is forcing state officials to find ways to avoid a repeat of this traffic nightmare. The principal proposal put forth to deal with this problem is the reversal of traffic on I-26, so that both sides, including the coastal-bound lanes, have traffic headed inland from Charleston to Columbia. Plans to carry this out have been prepared (and posted on the Web) by the South Carolina Emergency Preparedness Division. Traffic reversal on principal roads leading inland from Myrtle Beach and Hilton Head is also planned. A simplified map of South Carolina is shown. Charleston has approximately 500,000 people, Myrtle Beach has about 200,000 people, and another 250,000 people are spread out along the rest of
the coastal strip ( More accurate data, if sought, are widely available The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees(including some coming from farther north by other routes), so some traffic continues outbound on 1-26 towards Spartanburg; on I-77 north to Charlotte: and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed 1. Under what cond itions does the plan for turning the two coastal-bound lanes of I-26 into two lanes of Columbia-bound traffic, essentially turning the entire I-26 into one-way traffic significantly improve evacuation traffic flow? <2.In 1999, the simultaneous evacuation of the state's entire coastal region was ordered. Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation perhaps county-by-county over some time period consistent with the pattern of how hurricanes affect the coast? 3. Several smaller highways besides I-26 extend inland from the coast. Under what conditi would it improve evacuation flow to turn around traffic on these? 4. What effect would it have on evacuation flow to establish more temporary shelters in Columbia, to reduce the traffic leaving Columbia? 5. In 1999, many families leaving the coast brought along their boats, campers, and motor homes Many drove all of their cars. Under what cond itions should there be restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation? 6. It has been suggested that in 1999 some of the coastal residents of Georgia and Florida, who were fleeing the earlier pred icted landfalls of Hurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation traffic flow? Clearly identify what measures of performance are used to compare strategies Required: Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public
the coastal strip. (More accurate data, if sought, are widely available.) The interstates have two lanes of traffic in each direction except in the metropolitan areas where they have three. Columbia, another metro area of around 500,000 people, does not have sufficient hotel space to accommodate the evacuees (including some coming from farther north by other routes), so some traffic continues outbound on I-26 towards Spartanburg; on I-77 north to Charlotte; and on I-20 east to Atlanta. In 1999, traffic leaving Columbia going northwest was moving only very slowly. Construct a model for the problem to investigate what strategies may reduce the congestion observed in 1999. Here are the questions that need to be addressed: 1. Under what conditions does the plan for turning the two coastal-bound lanes of I-26 into two lanes of Columbia-bound traffic, essentially turning the entire I-26 into one-way traffic, significantly improve evacuation traffic flow? 2. In 1999, the simultaneous evacuation of the state's entire coastal region was ordered. Would the evacuation traffic flow improve under an alternative strategy that staggers the evacuation, perhaps county-by-county over some time period consistent with the pattern of how hurricanes affect the coast? 3. Several smaller highways besides I-26 extend inland from the coast. Under what conditions would it improve evacuation flow to turn around traffic on these? 4. What effect would it have on evacuation flow to establish more temporary shelters in Columbia, to reduce the traffic leaving Columbia? 5. In 1999, many families leaving the coast brought along their boats, campers, and motor homes. Many drove all of their cars. Under what conditions should there be restrictions on vehicle types or numbers of vehicles brought in order to guarantee timely evacuation? 6. It has been suggested that in 1999 some of the coastal residents of Georgia and Florida, who were fleeing the earlier predicted landfalls of Hurricane Floyd to the south, came up I-95 and compounded the traffic problems. How big an impact can they have on the evacuation traffic flow? Clearly identify what measures of performance are used to compare strategies. Required:Prepare a short newspaper article, not to exceed two pages, explaining the results and conclusions of your study to the public