2015年美国大学生数学建模竞赛A题

二、 问题分析
问题一要建立直杆影子长度变化的数学模型， 首先需知道太阳影子长度计算 公式，故引入太阳高度角[1]这个概念。即若已知某时刻太阳高度角的大小和直 杆高度，根据其满足的三角函数关系便可得到此时太阳影子长度。太阳高度角与 观测地地理纬度、地方时角和太阳的赤纬[2]相关。其中太阳赤纬是太阳直射点 所在纬度，与日期有关；时角由当地经度及其所用时区时间决定，故根据影长、 太阳赤纬、时角计算公式可求得直杆影子长度变化模型，并根据模型分析影子长 度关于各参数的变化规律。将附件一中直杆的有关数据直杆影长变化模型中，可 求出该直杆的具体影长变化公式。根据所建立的模型，运用 MATLAB 软件便可得 到影子长度随时间的变化曲线。 问题二需根据某固定直杆在水平地面上的太阳影子顶点坐标数据， 建立数学 模型确定直杆所处的地点。首先由问题一可推测影子长度与时间的关系，故可将 太阳影子长度与对应时间进行拟合，得到影长与时间关系模型。当某个时刻影长 得到极小值时，该时刻为太阳与直杆距离最近，即地方时正午 12 时，结合当地 所使用的标准时间便可得到当地经度。 最后利用太阳高度角与直杆长度以及影长 满足的三角关系式，便可得到影长关于直杆高度、直杆所在地点的纬度的函数关 系式，即得到了有关太阳影子顶点坐标与直杆地点经纬度的模型。将附件一中影 子顶点坐标数据应用于该直杆位置模型，可得到直杆所在位置。用相对误差分析 法分析误差[3](168-169 页)，若所得的相对误差小于 2.5%，认为得到的模型合 理。 问题三可根据光照成影原理和太阳高度角计算公式建立影长与时间变 化模型，根据相关数据，运用 MATLAB 软件拟合可得到直杆所在位置的经纬 度。令年份均为 2015 年，根据太阳赤纬角计算公式，可求解具体的日期。 将附件 2 和附件 3 时间和对应直杆影长数据分别代入模型中，通过拟合计

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地球资源的消耗速度快,越来越多的人关注人类社会的未来.自1960年以来，已经有许多专家研究可持续发展.然而大多数人的研究对象是整个世界，一个国家或一个地区。

几乎没有人选择48个最不发达国家(LDC)在联合国为研究对象列表。

然而,LDC国家集团共享许多相同的点。

他们的发展道路也有法律的内涵。

本文选择这些国家为研究对象针对发现常规的可持续发展道路。

本文组织如下.第二部分介绍研究的背景和本研究的意义。

第三节描述了我们对可持续发展的理解细节和显示我们的评估系统的建立过程和原理，那么我们估计每一个国家的LDC和获得可持续发展的能力和等级。

第四节提供了一个最糟糕的国家毛里塔尼亚计划指数在第三节。

第五节演示了在第四节的合理性和可用性计划。

最后在第六节总结本文的主要结论和讨论的力量和潜在的弱点。

地球上的资源是有限的。

三大能源石油、天然气和煤炭可再生。

如何避免人类的发展了资源枯竭和实现可持续发展目标是现在的一个热门话题.在过去的两个世纪,发达国家已经路上,先污染，再控制和达到高水平的可持续发展。

发展中国家希望发展和丰富。

然而,因为他们的技术力量和低水平的经济基础薄弱,浪费和低效率的发展在这些国家是正常的.所以本文主要关注如何帮助发展中国家特别是48在联合国最不发达国家实现可持续发展是列表可持续发展的理解是解决问题的关键.可持续发展的定义经历了一个长期发展的过程.在这里，布伦特兰可持续发展委员会的简短定义的"能力发展可持续- — - — - -以确保它既满足现代人的需求又不损害未来的能力代来满足自己的需求"［1]无疑是最被广泛接受的一个在各种内吗定义.这个定义方面发挥了重要作用在很多国家的政策制定的过程.然而，为了证明一个国家的现状是否可持续不可持续的,更具体的定义是必要的更具体的概念,我们认为,如果一个国家的发展是可持续的,它应该有一个基本的目前的发展水平，一个平衡的国家结构和一个光明的未来。

For office use only T1 ________________ T2 ________________ T3 ________________ T4 ________________
Team Control Number
For office use only F1 ________________ F2 ________________
32150
Problem Chosen
A
F3 ________________ F4 ________________
2015 Mathematical Contest iow to Eradicate Ebola? The breakout of Ebola in 2014 triggered global panic. How to control and eradicate Ebola has become a universal concern ever since. Firstly, we build up an epidemic model SEIHCR (CT) which takes the special features of Ebola into consideration. These are treatment from hospital, infectious corpses and intensified contact tracing. This model is developed from the traditional SEIR model. The model’s results (Fig.4,5,6), whose parameters are decided using computer simulation, match perfectly with the data reported by WHO, suggesting the validity of our improved model. Secondly, pharmaceutical intervention is studied thoroughly. The total quantity of the medicine needed is based on the cumulative number of individuals CUM (Fig.7). Results calculated from the WHO statistics and from the SEIHCR (CT) model show only minor discrepancy, further indicating the feasibility of our model. In designing the delivery system, we apply the weighted Fuzzy c- Means Clustering Algorithm and select 6 locations (Fig.10, Table.2) that should serve as the delivery centers for other cities. We optimize the delivery locations by each city’s location and needed medicine. The percentage each location shares is also figured out to facilitate future allocation (Table.3,4). The average speed of manufacturing should be no less than 106.2 unit dose per day and an increase in the manufacturing speed and the efficacy of medicine will reinforce the intervention effect. Thirdly, other critical factors besides those discussed early in the model, safer treatment of corpses, and earlier identification/isolation also prove to be relevant. By varying the value of parameters, we can project the future CUM . Results (Fig.12,13) show that these interventions will help reduce CUM to a lower plateau at a faster speed. We then analyze the factors for controlling and the time of eradication of Ebola. For example, when the rate of the infectious being isolated is 33% - 40%, the disease can be successfully controlled (Table.5). When the introduction time for treatment decreases from 210 to 145 days, the eradication of Ebola arrives over 200 days earlier. Finally, we select three parameters: the transmission rate, the incubation period and the fatality rate for sensitivity analysis. Key words: Ebola, epidemic model, cumulative cases, Clustering Algorithm

比赛时间：美国东部时间：2015年2月5日（星期四）下午8点-2月9日下午8点（共4天）北京时间：2015年2月6日（星期五）上午9点-2月10日上午9点农历：十二月十八～十二月廿二重要说明：●COMAP是所有的规则和政策的最后仲裁者，对不遵循竞赛规则和程序的任何队伍，拥有唯一的自由裁量权，取消参赛资格或拒绝登记。

●评委、竞赛组织者、以及UMAP杂志的编辑拥有最终裁定权。

●如果参赛队伍违反竞赛规则，其指导老师一年内将不能指导其他团队，其所在参赛单位将被处以一年的察看处理。

●如果同一机构第二次被抓到违反规则的队伍，该学校将至少不被允许参加下一年度的赛事。

●以下所有时间都是美国东部时间EST（北京时间比美国东部时间早13个小时）●递交参赛论文后，意味参赛者同意以下条款：⏹论文提交后，出版权归COMAP, Inc所有；⏹COMAP可以使用，编辑，引用和出版论文，用于宣传或任何其他目的，包括在线展示，出版电子版，在UMAP杂志刊登或其他方式，并且没有任何形式的补偿；⏹COMAP可以在没有进一步的通知，许可，或补偿的情形下，使用这次比赛相关材料，团队成员、指导老师的名字，以及和他们的背景资料。

●递交参赛论文后，意味参赛者作出以下承诺：⏹论文中出现的所有的图像，数据，照片，图表，图画，如果未注明，都是由参赛者创建；如果引用其它资源，都在参考文献中列出，并在引用的具体位置标注来源。

⏹不论是直接，还是转述方式的文字引用，都在参考文献中列出，并在引用的具体位置标注来源；直接的文字引用使用引号标注。

比赛之前注册报名1.报名截至时间：2015年2月5日下午2：00 EST。

截止日期后，注册系统将自动关闭，不再接受任何新的注册，没有例外。

2.每支参赛队伍都必须有一位来自参赛机构（institute）的教师担任导师（faculty advisor），不允许学生担任导师。

由指导老师负责为其指导队伍注册报名，每位指导老师可注册的队伍数目没有限制。

PROBLEM A: Eradicating EbolaThe world medical association has announced that their new medication could stop Ebola and cure patients whose diseases not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.A消除埃博拉病毒世界医学协会已经宣布他们的新疗法可以阻止埃博拉疫情和治愈非晚期患者。

构建一个现实的、合理的和有用的模型,不仅要考虑疾病的传播,所需药物的数量,可能且可行的给药系统,给药地点,生产疫苗或药物的速度,而且还要考虑其他关键因素（你的团队认为有必要要考虑的）作为模型的一部分以优化消除埃博拉病毒，或至少是现行毒株。

2015 Contest ProblemsMCM PROBLEMSPROBLEM A: Eradicating EbolaThe world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other critical factors your team considers necessary as part of the model to optimize the eradication of Ebola, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.PROBLEM B: Searching for a lost planeRecall the lost Malaysian flight MH370. Build a generic mathematical model that could assist "searchers" in planning a useful search for a lost plane feared to have crashed in open water such as the Atlantic, Pacific, Indian, Southern, or Arctic Ocean while flying from Point A to Point B. Assume that there are no signals from the downed plane. Your model should recognize that there are many different types of planes for which we might be searching and that there are many different types of search planes, often using different electronics or sensors. Additionally, prepare a 1-2 page non-technical paper for the airlines to use in their press conferences concerning their plan for future searches.ICM PROBLEMSPROBLEM C: Managing Human Capital in OrganizationsClick the title below to download a PDF of the 2015 ICM Problem C.Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.Managing Human Capital in OrganizationsPROBLEM D: Is it sustainable?Click the title below to download a PDF of the 2015 ICM Problem D.Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.Is it sustainable?。

Problem1What is the value ofProblem2Two of the three sides of a triangle are20and15.Which of the following numbers is not a possible perimeter of the triangle?Problem3Mr.Patrick teaches math to15students.He was grading tests and found that when he graded everyone's test except Payton's,the average grade for the class was80.after he graded Payton's test,the class average became81.What was Payton's score on the test?Problem4The sum of two positive numbers is5times their difference.What is the ratio of the larger number to the smaller?Problem5Amelia needs to estimate the quantity,where and are large positiveintegers.She rounds each of the integers so that the calculation will be easier to do mentally.In which of these situations will her answer necessarily be greater than the exact value of?Problem6Two years ago Pete was three times as old as his cousin Claire.Two years before that, Pete was four times as old as Claire.In how many years will the ratio of their ages be ?Problem7Two right circular cylinders have the same volume.The radius of the second cylinderis more than the radius of the first.What is the relationship between the heights of the two cylinders?Problem8The ratio of the length to the width of a rectangle is:.If the rectangle hasdiagonal of length,then the area may be expressed as for some constant.What is?Problem9A box contains2red marbles,2green marbles,and2yellow marbles.Carol takes2 marbles from the box at random;then Claudia takes2of the remaining marbles at random;and then Cheryl takes the last2marbles.What is the probability that Cheryl gets2marbles of the same color?Problem10Integers and with satisfy.What is?Problem11On a sheet of paper,Isabella draws a circle of radius,a circle of radius,and all possible lines simultaneously tangent to both circles.Isabella notices that she has drawn exactly lines.How many different values of are possible?Problem12The parabolas and intersect the coordinate axes in exactly four points,and these four points are the vertices of a kite of area.What is?Problem13A league with12teams holds a round-robin tournament,with each team playing every other team exactly once.Games either end with one team victorious or else end in a draw.A team scores2points for every game it wins and1point for every game it draws.Which of the following is NOT a true statement about the list of12scores?Problem14What is the value of for which?Problem15What is the minimum number of digits to the right of the decimal point needed toexpress the fraction as a decimal?Problem16Tetrahedron has and .What is the volume of the tetrahedron?Problem17Eight people are sitting around a circular table,each holding a fair coin.All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem18The zeros of the function are integers.What is the sum of the possible values of?Problem19For some positive integers,there is a quadrilateral with positive integerside lengths,perimeter,right angles at and,,and.Howmany different values of are possible?Problem20Isosceles triangles and are not congruent but have the same area and the sameperimeter.The sides of have lengths of and,while those of have lengthsof and.Which of the following numbers is closest to?Problem21A circle of radius passes through both foci of,and exactly four points on,the ellipse with equation.The set of all possible values of is an interval. What is?Problem22For each positive integer n,let be the number of sequences of length n consisting solely of the letters and,with no more than three s in a row and nomore than three s in a row.What is the remainder when is divided by12?Problem23Let be a square of side length1.Two points are chosen independently at random onthe sides of.The probability that the straight-line distance between the points is atleast is,where and are positive integers and.Whatis?Problem24Rational numbers and are chosen at random among all rational numbers in the interval that can be written as fractions where and are integers with .What is the probability that is a real number?Problem25A collection of circles in the upper half-plane,all tangent to the-axis,is constructedin layers as yer consists of two circles of radii and that areexternally tangent.For,the circles in are ordered according to their points of tangency with the-axis.For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair.Layer consists of the circles constructed in this way.Let,andfor every circle denote by its radius.What isDIAGRAM NEEDED。

What is the value ofProblem2A box contains a collection of triangular and square tiles.There are tiles in the box, containing edges total.How many square tiles are there in the box?Problem3Ann made a3-step staircase using18toothpicks.How many toothpicks does she need to add to complete a5-step staircase?Problem4Pablo,Sofia,and Mia got some candy eggs at a party.Pablo had three times as many eggs as Sofia,and Sofia had twice as many eggs as Mia.Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs.What fraction of his eggs should Pablo give to Sofia?Problem5Mr.Patrick teaches math to students.He was grading tests and found that when hegraded everyone's test except Payton's,the average grade for the class was.After he graded Payton's test,the test average became.What was Payton's score on thetest?The sum of two positive numbers is times their difference.What is the ratio of thelarger number to the smaller number?Problem7How many terms are there in the arithmetic sequence,,,...,,?Problem8Two years ago Pete was three times as old as his cousin Claire.Two years before that, Pete was four times as old as Claire.In how many years will the ratio of their ages be :?Problem9Two right circular cylinders have the same volume.The radius of the second cylinderis more than the radius of the first.What is the relationship between the heights of the two cylinders?Problem10How many rearrangements of are there in which no two adjacent letters are alsoadjacent letters in the alphabet?For example,no such rearrangements could include either or.The ratio of the length to the width of a rectangle is:.If the rectangle hasdiagonal of length,then the area may be expressed as for some constant.What is?Problem12Points and are distinct points on the graph of.What is?Problem13Claudia has12coins,each of which is a5-cent coin or a10-cent coin.There are exactly17different values that can be obtained as combinations of one or more of her coins.How many10-cent coins does Claudia have?Problem14Problem15Consider the set of all fractions where and are relatively prime positive integers. How many of these fractions have the property that if both numerator anddenominator are increased by,the value of the fraction is increased by?Problem16If,and,what is the value of?Problem17A line that passes through the origin intersects both the line and the line.The three lines create an equilateral triangle.What is the perimeter of the triangle?Problem18Hexadecimal(base-16)numbers are written using numeric digits through as well as the letters through to represent through.Among the firstpositive integers,there are whose hexadecimal representation contains onlynumeric digits.What is the sum of the digits of?Problem19The isosceles right triangle has right angle at and area.The raystrisecting intersect at and.What is the area of?Problem20A rectangle has area and perimeter,where and are positive integers.Which of the following numbers cannot equal?Problem21Tetrahedron has,,,,,and.What is the volume of the tetrahedron?Problem22Eight people are sitting around a circular table,each holding a fair coin.All eight people flip their coins and those who flip heads stand while those who flip tails remain seated.What is the probability that no two adjacent people will stand?Problem23The zeros of the function are integers.What is the sum of the possible values of?Problem24For some positive integers,there is a quadrilateral with positive integerside lengths,perimeter,right angles at and,,and.Howmany different values of are possible?Problem25Let be a square of side length.Two points are chosen independently at random onthe sides of.The probability that the straight-line distance between the points is atleast is,where,,and are positive integers with.Whatis?。

合理的假设1假设论文的抽样网评是完全随机的具有代表性2假设评委打分数据是客观真实有效的出现偏差是评委水平导致3假设当时评委的精神处于最佳状态评委分数可信度不受客观因素影响4假设论文的网评和集中评审的评阅信息是相互独立的各评委打分之间没有相互交流影响个评委对第j份试卷打分份论文的第j份试卷的平均分数份论文的第j份试卷平均分数矩阵数据的预处理51初步分析数据根据数据文件给出的数据分别算出某一个评委所评阅的所有论文的平均分为x均方差为并利用附录给出的公式算出该评委的第k份论文的标准分具体数据见下表1
2.2问题二的分析
第二个问题根据对竞赛评委有不同的基本素质要求，给出合理的度量评委基本素质的指标体系。我们根据题目附件给出的数据，去发掘测评评委基本素质要求的一些指标体系。测评基本素质指标体系主要三个方面构成：指标一是评委打分的准确度，指标二是评委打分的稳定度，指标三是评委打分的偏差度。为了使指标准确可靠，需要把不同的论文的结果分为两大类，一个是得奖论文，另一个是未得奖论文。为简化问题的复杂度，我们从得奖论文入手，分别找到这三个指标的评价标准：
序号
阅卷号
评委
打分
标准分
1
A1
评委A04
35
46.25937
2
A2
评委A11
53
55.66406
3
A3
评委A06
46
60.54732
……
……
……
……
……
353
A9020
评委A03
62
61.27679
354
A9021
评委A12
28
46.8965
355
A9022
评委A11
30
36.32556

针对问题三，题中虽然没有给出采样日期，但其整体思路与问题二是一致的。 在选取假设采样点后，将经度、纬度和日期作为变量，使用问题一中的模型求出 该假设采样点的影子长度。最后使用最小二乘法将这些假设采样点数据与原始影 子长度数据进行拟合，在 MATLAB 中编程计算，得到的结果为：附件 2 中的采样 点在东经 79°，北纬 43°，采样日期为 6 月 12 日；附件 3 中的采样点在东经 107°， 北纬 28°，采样日期为 11 月 25 日。
针对问题二，首先，我们通过影子的顶点坐标得到各个时刻的影子长度。之 后进行数据标准化，消除直杆长度对影子长度的影响。任意选取某一经纬度为假 设采样点，将经度、纬度作为变量，使用问题一中的模型求出该假设采样点的影 子长度。最后使用最小二乘法将这些假设采样点数据与原始影子长度数据进行拟 合，在 MATLAB 中编程计算，得到的最小目标函数值������ = 1.2981 × 10−7 ，该假设 采样点为东经 109°，北纬 17°（见正文图 11），其周边海南三亚市、越南沿海地 区都可以认为是采样点的可能位置。
太阳影子定位技术的研究
摘要
本文针对太阳影子定位问题，通过运用天球模型和最小二乘法，研究了直杆 太阳影子长度与直杆长度、太阳高度角、采样点经纬度、采样日期和采样时间等 参数的关系，实现了利用物体的太阳影子变化来确定视频拍摄地点和日期。
针对问题一，在已知直杆长度的情况下，太阳影子长度和太阳高度角满足一 个确定的函数关系。因此，我们可以将研究对象从太阳影子长度转换为太阳高度 角。引入天球模型后，使用天球坐标系统中的赤道坐标系和地平坐标系来描述太 阳的运动和位置，得到了太阳高度角与采样地点经度、纬度、日期和当天具体时 间的函数关系，进而得到了影子长度与各参数的关系。之后，使用控制变量法分 别得到了影子长度关于直杆长度、经度、纬度、日期和时间这 5 个参数的变化规 律（见正文图 5、6、7、8、9）。最后，运用该模型画出了天安门广场上 3 米高的 直杆的太阳影子长度的变化曲线（见正文图 10）。

36837
Problem Chosen
A
F4 ________________
2015
Mathematical Contest in Modeling (MCM/ICM) Summary Sheet
Summary
Ebola viral disease(EVD) has become a problem threatening not only western Africa, but the world. The world medical association has announced their new medication. This paper aims at tackling the following problems: Describing and predicting the spread of EVD based on current statistics; determing the quantity of medicine needed; deciding the speed of manufacturing the vaccine or drug under the principle of meeting demands and lowering costs; locating the delivery center and establish an efficient delivery system. When analyzing demand and supply, an additional factor, inventory level is taken into account. First an improved SIR epidemic model is built. The population of epidemic area is divided into 5 parts instead of 3, to be more comprehensive. By solving a set of differential equations and computing several parameters, we obtained the fitting function of cumulative cases of EVD. Compared with current statistics, the imitative effect is satisfying. Second, by predicting through the fitting function obtained from Model one, we get the approximate demand curve of medicine. Considering efficiency and inventory costs, an approximate speed function of manufacturing is obtained. In the updated model, the proper inventory level is discussed based on EOQ model of short-supply not allowed and time-needed supplying. Third, a three-level delivery system is built based on Model two where supply is guaranteed to be sufficient. Centroid method and analytic hierarchy process (AHP) are combined to select optimal location of distribution center, and the severeness of the epidemic in different places, the distances and transportation costs are considered. In addition, an emergency dispatch scheme is built to response to sudden outbreak. By randomly simulation, we test the sensitivity of Model three, and conclude that it is steady and effective.Team Nhomakorabea#36837

嫦娥三号软着陆轨道设计与控制策略摘要在整个“嫦娥三号”软着陆过程中发动机的燃耗问题是整个着陆过程的关键问题之一，其利用率直接影响到整个着陆过程的成果与否，本文主要利用数学建模的方法对整个软着陆过程进行分析，使得整个软着陆过程发动机能耗最优。

针对问题一，首先需建立一个三维立体坐标系，根据牛顿第二定律，结合科氏定律整理得到嫦娥三号在月固定坐标系中的运动方程，再以卫星运行轨道切面为基面建立二维平面坐标系，将嫦娥三号软着陆问题简化为平面几何问题，求解出主减速阶段嫦娥三号水平位移的距离。

通过坐标变换求得位置。

最后根据天体运动规律得到近日点与远日点速度分别为s6226.1。

km.1、skm7006针对问题二，通过寻找一个制导律u,来调整推力的大小和方向,使嫦娥三号在月面实现燃耗最优着陆轨道，应用极大值原理设计这个最优制导律。

在障碍规避过程中，将动力学模型进行进一步简化，忽略了月球的自转角速度等相关因素。

再利用双线性插值的方法求取规则的采样点处的高程值，这样有利于方便的建立障碍检测算法并对着陆区表面的障碍进行提取，最后利用基于平面拟合的障碍检测算法取得着陆区域内某局部区域内的地形平面，我们将利用这个地表平面来对障碍物进行识别，达到安全着陆的目的。

针对问题三，影响制导精度的误差源主要有偏离标准飞行轨迹的初始条件误差和导航与控制传感器误差。

初始条件误差由主制动段以前的任务决定，传感器误差则由导航系统和传感器本身决定，通过建立误差模型，可以很好地对初始状态偏差、传感器测量偏差等不同因素造成的误差进行分析。

关键词：月球着陆轨道能耗最优打靶法最优制导律控制策略一问题重述嫦娥三号于2013年12月2日1时30分成功发射，12月6日抵达月球轨道。

嫦娥三号在着陆准备轨道上的运行质量为2.4t，其安装在下部的主减速发动机能够产生1500N到7500N的可调节推力，其比冲（即单位质量的推进剂产生的推力）为2940m/s，可以满足调整速度的控制要求。

②6 月 22 日，太阳直射北回归线，北回归线及其以北各地的正午太阳高度 达到全年最大，其日影也达到全年最短。
③6 月 22 日—12 月 22 日，在太阳直射点向南移动过程中，北回归线及其 以北各地的正午太阳高度逐渐减小，那么其日影逐渐增长；
④12 月 22 日，太阳直射南回归线，北回归线及其以北各地的正午太阳高度 达到全年最小，其日影也达到全年最长。
一年中，各地的日影长度会随季节变化而变化，这种变化主要体现在正午的 日影长短上。它与当地的正午太阳高度有直接关系：正午太阳高度越大，日影越 短；正午太阳高度越小，日影越长。例如：
①12 月 22 日—6 月 22 日，在太阳直射点向北移动过程中，北回归线及其以 北各地的正午太阳高度逐渐增大，那么其日影逐渐缩短；
图 4 天安门广场 15 年 10 月 22 日影子长度随时间(9 点到 15 点)变化图
在该问题中，影子长度的变化曲线根据计算出是一个关于真太阳时 12 点对 称的二次函数拟合曲线，所以我们利用题中所给的时间数据运用 MATLAB（附 录二）求解该附件的拟合曲线的表达式为
l(t) = 0.3179 t2 - 7.7982t + 51.4250
对于地球上的某个地点，太阳高度角是指太阳光的入射方向和地平面之间的 夹角，专业上讲太阳高度角是指某地太阳光线与通过该地与地心相连的地表切线 的夹角。太阳高度角简称高度角。当太阳高度角为 90°时，此时太阳辐射强度 最大；当太阳斜射地面时，太阳辐射强度就小。
图 1 太阳高度角示意图
太阳方位角即太阳所在的方位，指太阳光线在地平面上的投影与当地经线的 夹角，可近似地看作是竖立在地面上的直线在阳光下的阴影与正南方的夹角。方 位角以目标物正北方向为零，顺时针方向逐渐变大，其取值范围是 0—360°。 因此太阳方位角一般是以目标物的北方向为起始方向，以太阳光的入射方向 为 终止方向，按顺时针方向所测量的角度。

PROBLEM A: Eradicating EbolaThe world medical association has announced that their new medication could stop Ebola and cure patients whose disease is not advanced. Build a realistic, sensible, and useful model that considers not only the spread of the disease, the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug, but also any other cr itical factors your team considers necessary as part of the model to optimize the eradication of Ebo la, or at least its current strain. In addition to your modeling approach for the contest, prepare a 1-2 page non-technical letter for the world medical association to use in their announcement.消除埃博拉病毒世界医学协会已宣布其新的药物治疗可以阻止埃博拉病毒和治愈的病人的疾病不晚期。

建立一项现实、明智的和有用的模型认为不仅传播这种疾病的药量需要，可能可行的运载系统、交货地点、制造疫苗或药物，速度，还任何其他关键的因素，你的团队认为必要组成部分的模型优化的埃博拉病毒，消除或至少其当前的应变。

A题飞越北极2000年6月，扬子晚报发布消息：“中美航线下月可飞越北极，北京至底特律可节省4小时”，摘要如下：7月1日起，加拿大和俄罗斯将允许民航班机飞越北极，此改变可大幅度缩短北美与亚洲间的飞行时间，旅客可直接从休斯敦，丹佛及明尼阿波利斯直飞北京等地。

据加拿大空中交通管制局估计，如飞越北极，底特律至北京的飞行时间可节省4个小时。

由于不需中途降落加油，实际节省的时间不止此数。

假设：飞机飞行高度约为10公里，飞行速度约为每小时980公里；从北京至底特律原来的航线飞经以下10处：A1 (北纬31度，东经122度)； A2 (北纬36度，东经140度)；A3 (北纬 53度，西经165度)； A4 (北纬62度，西经150度);A5 (北纬 59度，西经140度)； A6 (北纬 55度，西经135度)；A7 (北纬 50度，西经130度)； A8 (北纬 47度，西经125度)；A8 (北纬 47度，西经122度)； A10 (北纬 42度，西经87度)。

请对“北京至底特律的飞行时间可节省4小时“从数学上作出一个合理的解释，分两种情况讨论：（1）设地球是半径为6371千米的球体；（2）设地球是一旋转椭球体，赤道半径为6378千米，子午线短半轴为6357千米。

B题：DNA限制性图谱的绘制绘制DNA限制性图谱是遗传生物学中的重要问题。

由于DNA分子很长，目前的实验技术无法对其进行直接测量，所以生物学家们需要把DNA分子切开，一段一段的来测量。

在切开的过程中，DNA片段在原先DNA分子上的排列顺序丢失了，如何找回这些片段的排列顺序是一个关键问题。

为了构造一张限制性图谱，生物学家用不同的生化技术获得关于图谱的间接的信息，然后采用组合方法用这些数据重构图谱。

一种方法是用限制性酶来消化DNA分子。

这些酶在限制性位点把DNA链切开，每种酶对应的限制性位点不一样。

对于每一种酶，每个DNA分子可能有多个限制性位点，此时可以按照需要来选择切开某几个位点（不一定连续）。