2002高教社杯全国大学生数学建模竞赛题目

A题、B题)

 

 

l        A题、B题任选一题。

l        答卷用白色A4纸,第一页为空白页(用于赛区或全国组委会对论文进行编号)。

l        论文题目和摘要写在第二页上,从第三页开始是论文正文。

l        论文中不能有任何可能显示答题人身份的标志。

l        提请大家注意:从去年起,提高了摘要在整篇论文评阅中所占的权重。

l        全部题目可以从以下网址之一下载:

http://mcm.edu.cn   http://www.hep.edu.cn   http://mcm.diy.163.com

 

 

 

 

 

 


A  车灯线光源的优化设计

 

安装在汽车头部的车灯的形状为一旋转抛物面,车灯的对称轴水平地指向正前方, 其开口半径36毫米,深度21.6毫米。经过车灯的焦点,在与对称轴相垂直的水平方向,对称地放置一定长度的均匀分布的线光源。要求在某一设计规范标准下确定线光源的长度。

    该设计规范在简化后可描述如下。在焦点F正前方25米处的A点放置一测试屏,屏与FA垂直,用以测试车灯的反射光。在屏上过A点引出一条与地面相平行的直线,在该直线A点的同侧取B点和C点,使AC=2AB=2.6米。要求C点的光强度不小于某一额定值(可取为1个单位),B点的光强度不小于该额定值的两倍(只须考虑一次反射)

    请解决下列问题:

1)在满足该设计规范的条件下,计算线光源长度,使线光源的功率最小。

2)对得到的线光源长度,在有标尺的坐标系中画出测试屏上反射光的亮区。

3)讨论该设计规范的合理性。

 

B  彩票中的数学

 

近年来“彩票飓风”席卷中华大地,巨额诱惑使越来越多的人加入到“彩民”的行列,目前流行的彩票主要有“传统型”和“乐透型”两种类型。

“传统型”采用“106+1”方案:先从60~9号球中摇出6个基本号码,每组摇出一个,然后从0~4号球中摇出一个特别号码,构成中奖号码。投注者从0~9十个号码中任选6个基本号码(可重复),从0~4中选一个特别号码,构成一注,根据单注号码与中奖号码相符的个数多少及顺序确定中奖等级。以中奖号码“abcdef+g”为例说明中奖等级,如表一(X表示未选中的号码)

                                表一

 

 

10    6+16+1/10

                             特别号码

 

一等奖

abcdef                                                  g

7中(6+1

二等奖

abcdef 

7中(6

三等奖

abcdeX    Xbcdef        

7中(5

四等奖

abcdXX   XbcdeX    XXcdef

7中(4

五等奖

abcXXX  XbcdXX    XXcdeX   XXXdef    

7中(3

六等奖

abXXXX  XbcXXX   XXcdXX  XXXdeX   XXXXef 

7中(2

 

“乐透型”有多种不同的形式,比如“337”的方案:先从01~33个号码球中一个一个地摇出7个基本号,再从剩余的26个号码球中摇出一个特别号码。投注者从01~33个号码中任选7个组成一注(不可重复),根据单注号码与中奖号码相符的个数多少确定相应的中奖等级,不考虑号码顺序。又如“366+1”的方案,先从01~36个号码球中一个一个地摇出6个基本号,再从剩下的30个号码球中摇出一个特别号码。从01~36个号码中任选7个组成一注(不可重复),根据单注号码与中奖号码相符的个数多少确定相应的中奖等级,不考虑号码顺序。这两种方案的中奖等级如表二。

                               表二

 

 

33    77/33

36    6+16+1/36

  特别号码

 

  特别号码

 

一等奖

●●●●●●●     

7中(7

●●●●●●    

7中(6+1

二等奖

●●●●●●○  

7中(6+1

●●●●●● 

7中(6

三等奖

●●●●●●○

7中(6

●●●●●○    

7中(5+1

四等奖

●●●●●○○   

7中(5+1

●●●●●○   

7中(5

五等奖

●●●●●○○

7中(5

●●●●○○    

7中(4+1

六等奖

●●●●○○○   

7中(4+1

●●●●○○   

7中(4

七等奖

●●●●○○○

7中(4

●●●○○○    

7中(3+1

注:●为选中的基本号码;★ 为选中的特别号码;○ 为未选中的号码。

 

     以上两种类型的总奖金比例一般为销售总额的50%,投注者单注金额为2元,单注若已得到高级别的奖就不再兼得低级别的奖。现在常见的销售规则及相应的奖金设置方案如表三,其中一、二、三等奖为高项奖,后面的为低项奖。低项奖数额固定,高项奖按比例分配,但一等奖单注保底金额60万元,封顶金额500万元,各高项奖额的计算方法为:

[(当期销售总额 ×总奖金比例) -低项奖总额 ]×单项奖比例

1)根据这些方案的具体情况,综合分析各种奖项出现的可能性、奖项和奖金额的设置以及对彩民的吸引力等因素评价各方案的合理性。

2)设计一种“更好”的方案及相应的算法,并据此给彩票管理部门提出建议。

3)给报纸写一篇短文,供彩民参考。

 

                                             表三

序号

      奖项

方案

一等奖

 

二等奖

 

三等奖

 

四等奖

 

五等奖

 

六等奖

 

七等奖

 

 

1

6+1/10

50%

20%

30%

50

 

 

 

按序

2

6+1/10

60%

20%

20%

300

20

5

 

按序

3

6+1/10

65%

15%

20%

300

20

5

 

按序

4

6+1/10

70%

15%

15%

300

20

5

 

按序

5

7/29

60%

20%

20%

300

30

5

 

 

6

6+1/29

60%

25%

15%

200

20

5

 

 

7

7/30

65%

15%

20%

500

50

15

5

 

8

7/30

70%

10%

20%

200

50

10

5

 

9

7/30

75%

10%

15%

200

30

10

5

 

10

7/31

60%

15%

25%

500

50

20

10

 

11

7/31

75%

10%

15%

320

30

5

 

 

12

7/32

65%

15%

20%

500

50

10

 

 

13

7/32

70%

10%

20%

500

50

10

 

 

14

7/32

75%

10%

15%

500

50

10

 

 

15

7/33

70%

10%

20%

600

60

6

 

 

16

7/33

75%

10%

15%

500

50

10

5

 

17

7/34

65%

15%

20%

500

30

6

 

 

18

7/34

68%

12%

20%

500

50

10

2

 

19

7/35

70%

15%

15%

300

50

5

 

 

20

7/35

70%

10%

20%

500

100

30

5

 

21

7/35

75%

10%

15%

1000

100

50

5

 

22

7/35

80%

10%

10%

200

50

20

5

 

23

7/35

100%

2000

20

4

2

 

 

无特别号

24

6+1/36

75%

10%

15%

500

100

10

5

 

25

6+1/36

80%

10%

10%

500

100

10

 

 

26

7/36

70%

10%

20%

500

50

10

5

 

27

7/37

70%

15%

15%

1500

100

50

 

 

28

6/40

82%

10%

8%

200

10

1

 

 

29

5/60

60%

20%

2lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll