小扣抖
范文一：时针与分针重合问题
前段时间～在徐庄小学听了一节关于时钟计算课～课后细细想想有些问题值得深入探讨:
1、小学数学教材能解决的问题
在3点到4点之间～几点几分,闹钟上的时针与分针第一次重合?
方法:分针每5分钟转动30度～时针每分钟转过的角度是分针的1/12～所以分针每分钟转6度～时针每分钟转动0.5度。这个问题就变成了路程追赶问题:每分钟分针追上时针,6-0.5,度。
3点整:时针领先分针90度～要使时针与分针第一次形成30度角～分针需要追60度。
60/(6-0.5)=10.91(分钟)=10分55秒
所以闹钟上的时针与分针第一次形成30度角的时间是3点10分55秒
当分针转360度时～时针转了30度。设时针和分针从三点整开始转～时针转的角度为X～则分针转的角度为,X+90,～根据比例得:360/30=(X+90)/X解得X=90/11～则分针转了90/(11*6)分～当3:,15+15/11,=3:,16+4/11,分重合
2、初中数学教材能解决的问题
闹钟12时整～时针和分针重合～当时针与分针再次重合是几时几分,第一次构成直角是几时几分,有算式,
方法:把表盘分为60个格～一小时内分针走60格～时针走5格～这样当分针走1格时～时针走5/60=1/12格。
首先可以断定出这次重合是在1点到2点之间～这就像长跑比赛中的最快的选手赶上最慢的选手造成的套圈～在12点到1点范围内肯定分针是比时针快很多的～不可能重合。这样有设1点后重合时分针走了x格～而时针在“领跑”5格基础上又走了x/12格～有x=x/12+5～得到了x=60/11～是1时5分27秒。
而第一次构成直角就在12点到1点范围内了～直角是15格～还是设分针走了x格～有x-x/12=15～为12时16分21秒。 分针每分钟转过6度～时针每分钟转过0.5度,即分针每分钟赶上5.5度
设x分钟后时针与分针完全重合
得:360=5.5x
解得:x=1时,5+4/11,分
即下次重合是1点,5+4/11,分
范文二：数学建模时针与分针重合问题
时 针
数 学 实 践 与 建 模
分 针 重 合 问 题 的
姓名： 学号： 班级： 专业： 分 析 与
时针与分针重合问题的分析
摘要
“时针和分针一昼夜重合多少次？”这是美国著名的微软公司招聘职员的一道面试题。我们经常接触钟表，对钟表的认识可以说是再熟悉不过了，然而，面对这个看似简单的问题，许多应聘的优秀大学毕业生都不假思索的回答：24次。在他们认为，一昼夜是24个小时，每小时时针和分针重合一次，所以一昼夜时针和分针共重合24次。
答案看起来似乎并没有什么问题，可是，这个答案是错误的。只要我们稍加思考，就会感觉有点不对劲。那么时针与分针一昼夜到底重合多少次呢？
面对这种问题，乍一看似乎并无头绪，其实，我们完全可以通过建立数学模型的方法去求取。所谓数学建模，就是从定量的角度分析和研究一个实际问题，在深入调查研究、了解对象信息、作出简化假设、分析内在规律等工作的基础上，用数学的符号和语言作表述，然后用通过计算得到的结果来解释实际问题，并接受实际的检验。
对于时针和分针的相遇问题，我们可以将其类比为时针与分针相互追击的问题，，只不过他们行走的路径是一个圆。其中，分针比时针跑的快。然后通过假设一次相遇，两次相遇，多次相遇的办法，推理出相遇的规律，最后通过数学模型将一昼夜时针与分针相遇的次数求出。
关键词： 时针； 分针； 追击； 重合； 类推；
1、模型假设
①、钟面上分为60个格；
②、分针1分针转过一格，一小时转过60格，时针一小时转过5格。 ③、钟面上一个小格有6o，分针转动一周时，时针转了30o，如下图。
2、符号说明
①、xn：分针的位置； ②、yn：时针的位置；
③、x：分针和时针重合的位置。
3、模型建立
从k（k=1,2,3……，11）点开始，
xn=5k（1+
11
+?+n-1）； 1212111
yn=5k（1++?+n-1+n）；
121212
n→∞
n→∞
x=limxn=limyn。
这就是所建立的数学模型。
4、模型求解
由假设可知分针的速度是时针的12倍。不妨设k=1，即从1点开始，此时时针在分针前5格。当分针走5格（分针走到时针原来的位置）时，时针走了格，此时分针的位置记为x1=5，时针的位置记为y1=5+
5
12
55；当分针再走格1212515
时，时针走了?，此时分针的位置记为x2=5+，时针的位置记为
121212
51511515
y2=5++?，当分针再走?格时，时针走了??，
1212121212121212
515
此时分针的位置为x3=5++?，
121212
515115
时针的位置为y3=5++?+??；……
121212121212
1151++?+n-1）以此类推，当分针的位置是xn=（时，时针的位置为
1212
1155
yn=（51++?+n），分针与时针相差n，因为limn=0，所以分针与时
n→∞12121212
116016051++?+n-1）=lim1-n)=格。 针重合的位置是x=limxn=lim（
n→∞n→∞n→∞1112121211
60
≈5.4545，故此时时间约为1点5分27.27秒。 因为11
在1点5分27.27秒重合之后，由于分针速度是时针的12倍，故到2点为止，分针和时针不会再重合。
类似的，k=2时，即从2点开始，此时时针在分针前10（5×2）格。当分针
10
格，此时分针的位置为12
1010110
x1=10，时针的位置为y1=10+；当分针再走格时，时针走了?，此
121212121010110
时分针的位置为x2=10+，时针的位置为y2=10++?；……以此类推，
12121212
11（1++?+n-1）时，时针的位置为当分针位置是xn=10
1212
111010
yn=10（1++?+n)，分针与时针相差n，因为limn=0，所以分针与时
n→∞12121212
111201120（1++?+n-1）=lim1-n）=针重合的位置是x=limxn=lim10格。
n→∞n→∞n→∞1212111211
120
≈10.9091，故此时时间是2点10分54.55秒。 因为11走10格（分针走到时针原来的位置）时，时针走了
综上所述，k（k=1，2，…，11）点到k+1点之间，分针与时针重合的位置
1+是x=limxn=lim5k（
1160k160
+?+n-1）=lim1-n）=k格。
n→∞n→∞n→∞1212111211
60
分针与时针重合的时间是k点k分（k=1，2，…，11），即1点5分27.27
11
秒，2点10分54.55秒，3点16分21.82秒，4点21分49.09秒，5点27分16.36秒，6点32分43.64秒，7点38分10.91秒，8点43分38.18秒，9点49分5.45秒，10点54分32.73秒，11点60分（12点）。一天之中，时针和分针共重合22次。
5、模型结果的分析与检验
从上述的模型求解过程中，可以看出一昼夜的时间内，时针与分针共重合了22次，而并非24次，从以上的结果中，也可以看出，时针与分针重合一次所需要的时间为
120
分钟。计算的结果和实际的情况完全符合，故该模型对于求取此11
问题是正确的。
6、对模型的评价与推广
运用数学建模的方法，让我们对于求解某些实际问题有了更加明确的思路和方法。从实际问题中建立数学模型，解决数学问题从而解决实际问题，而此过程的关键是建立数学模型。此模型是近似的将钟表的时针与分针看做是一个特殊的圆形轨道上2人追及或相遇的问题，要把时钟问题当做行程问题来看，分针快，时针慢，所以分针与时针的问题，就是他们之间的追及问题。其实，在实际的求解关于钟表时针、分钟甚至秒针重合问题的时候，都可以采用此种方法进行求解。但是，从总体上看，求解此类题目时所需的计算比较复杂，当然求解此类题目还有更为简单的方法，即拿一个钟表，拨动指针，使其旋转观察，将时针转24周，数其重合的次数即可得出答案。
参考文献： [1]《把数学建模融入高等数学教学中的两个案例》 崔海英、侯文 宇、李林杉 北京联合大学
[2]《关于同一个问题的不同解决方法》
http://www.doc88.com/p-9912711098443.html
范文三：数学建模时针与分针重合问题
数 学 实 践 与 建 模
时 针 与 分 针 重 合 问 题 的 分 析
姓名:
学号:
班级:
专业:
steel welding procedure 6 SH3525-2004 petrochemical JB/ T4708-2000 of welding procedure qualification for steel pressure vessels 7 JB/4709-2000 8 JB4730-2005 pressure vessel welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury
时针与分针重合问题的分析
摘要
“时针和分针一昼夜重合多少次,”这是美国著名的微软公司招聘职员的一道面试题。我们经常接触钟表,对钟表的认识可以说是再熟悉不过了,然而,面对这个看似简单的问题,许多应聘的优秀大学毕业生都不假思索的回答,24次。在他们认为,一昼夜是24个小时,每小时时针和分针重合一次,所以一昼夜时针和分针共重合24次。
答案看起来似乎并没有什么问题,可是,这个答案是错误的。只要我们稍加思考,就会感觉有点不对劲。那么时针与分针一昼夜到底重合多少次呢,
面对这种问题,乍一看似乎并无头绪,其实,我们完全可以通过建立数学模型的方法去求取。所谓数学建模,就是从定量的角度分析和研究一个实际问题,在深入调查研究、了解对象信息、作出简化假设、分析内在规律等工作的基础上,用数学的符号和语言作表述,然后用通过计算得到的结果来解释实际问题,并接受实际的检验。
对于时针和分针的相遇问题,我们可以将其类比为时针与分针相互追击的问题,,只不过他们行走的路径是一个圆。其中,分针比时针跑的快。然后通过假设一次相遇,两次相遇,多次相遇的办法,推理出相遇的规律,最后通过数学模型将一昼夜时针与分针相遇的次数求出。
关键词, 时针, 分针, 追击, 重合, 类推,
1、模型假设
1
?、钟面上分为60个格,
?、分针1分针转过一格,一小时转过60格,时针一小时转过5格。 ?、钟面上一个小格有6o,分针转动一周时,时针转了30o,如下图。
2、符号说明
?、x,分针的位置, n
y?、,时针的位置, n
?、x,分针和时针重合的位置。
3、模型建立
从k,k=1,2,3……,11,点开始,
11x,5k1，，？，(), nn-11212
111y,5k1，，？，，(), nn-1n121212
x,limx,limy。 nn,,,,nn
steel welding procedure 6 SH3525-2004 petrochemical JB/ T4708-2000 of welding procedure qualification for steel pressure vessels 7 JB/4709-2000 8 JB4730-2005 pressure vessel welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury
这就是所建立的数学模型。
4、模型求解
由假设可知分针的速度是时针的12倍。不妨设k=1,即从1点开始,此时
5时针在分针前5格。当分针走5格,分针走到时针原来的位置,时,时针走了12
55,，y5格,此时分针的位置记为x=5,时针的位置记为,当分针再走格111212
515x5,，时,时针走了,此时分针的位置记为,时针的位置记为，2121212
51515115y5,，，，，，,当分针再走格时,时针走了, ，21212121212121212
515x5,，，，此时分针的位置为, 3121212
515115y5,，，，，，，时针的位置为,…… 3121212121212
11x,51，，？，()以此类推,当分针的位置是时,时针的位置为nn-11212
1155y,51，，？，lim,0(),分针与时针相差,因为,所以分针与时nnnn,,n12121212
1160160xlimxlim51lim1),,(，，？，),(,,针重合的位置是格。 nn-1n,,,,,,nnn1212111211
60,5.4545因为,故此时时间约为1点5分27.27秒。 11
在1点5分27.27秒重合之后,由于分针速度是时针的12倍,故到2点为止,分针和时针不会再重合。
类似的,k=2时,即从2点开始,此时时针在分针前10,5×2,格。当分针
3
10走10格,分针走到时针原来的位置,时,时针走了格,此时分针的位置为
12
1011010,时针的位置为y10,当分针再走格时,时针走了,此,，x,10，1112121212
1011010y10时分针的位置为x10,时针的位置为,……以此类推,,，,，，，2212121212
11x,101，，？，当分针位置是时,时针的位置为()nn-11212
111010lim,0y,10(1，，？，),分针与时针相差,因为,所以分针与时nnnn,,n12121212
111201120xlimxlim101lim1针重合的位置是格。 ,,(，，？，),(,),nn-1n,,,,,,nnn1212111211
120,10.9091因为,故此时时间是2点10分54.55秒。 11
综上所述,k,k=1,2,…,11,点到k+1点之间,分针与时针重合的位置
1160k160是x,limx,lim5k(1，，？，),lim(1,),k格。 nn-1n,,,,,,nnn1212111211
60分针与时针重合的时间是k点分,k=1,2,…,11,,即1点5分27.27k11
秒,2点10分54.55秒,3点16分21.82秒,4点21分49.09秒,5点27分16.36秒,6点32分43.64秒,7点38分10.91秒,8点43分38.18秒,9点49分5.45秒,10点54分32.73秒,11点60分,12点,。一天之中,时针和分针共重合22次。
5、模型结果的分析与检验
从上述的模型求解过程中,可以看出一昼夜的时间内,时针与分针共重合了22次,而并非24次,从以上的结果中,也可以看出,时针与分针重合一次所需
120要的时间为分钟。计算的结果和实际的情况完全符合,故该模型对于求取此11
问题是正确的。
steel welding procedure 6 SH3525-2004 petrochemical JB/ T4708-2000 of welding procedure qualification for steel pressure vessels 7 JB/4709-2000 8 JB4730-2005 pressure vessel welding procedures of steel pressure vessel NDT 9 JB/T4744-2000 steel pressure vessel products mechanical properties test of welded plate II, mechanical equipment installation engineering 1 GB150-98 2 GB50128-2005 vertical cylindrical steel pressure vessel steel welded specification for construction and acceptance of oil tank 3 JB/ T4735-1997 steel welded atmospheric pressure vessel 4 GB50231-2009 mechanical equipment installation engineering construction and acceptance specification for 5 GB50275-98 compressor, fan and pump installation engineering construction and acceptance specification for lifting 6 HG20201-2000 construction installation engineering construction standard 7, HG ... Pressure gauges, using an installed spark arrestor for acetylene cylinders, illegal carrying, using gas cylinders, each operator fined 20 Yuan. Gas bottle without the hot sun exposure measures, responsibilities of team a fine of 50 Yuan. (7) at height and the opening and provisional protective measures have been taken, the responsible unit fined 200 Yuan, construction person in charge a fine of 50 Yuan. (8) no permit to work and the job in accordance with the provisions, on the job unit fined 300 yuan, the unit will charge a fine of 100 Yuan. 7.5.3 accident penalties (1) injuries fatalities directly punished 1~2 million. Direct responsibility for the accident and have the corresponding responsibility of leadership, such as concerning administrative sanctions, should be brought to the company or the relevant Department. (2) personal injury
6、对模型的评价与推广
运用数学建模的方法,让我们对于求解某些实际问题有了更加明确的思路和方法。从实际问题中建立数学模型,解决数学问题从而解决实际问题,而此过程的关键是建立数学模型。此模型是近似的将钟表的时针与分针看做是一个特殊的圆形轨道上2人追及或相遇的问题,要把时钟问题当做行程问题来看,分针快,时针慢,所以分针与时针的问题,就是他们之间的追及问题。其实,在实际的求解关于钟表时针、分钟甚至秒针重合问题的时候,都可以采用此种方法进行求解。但是,从总体上看,求解此类题目时所需的计算比较复杂,当然求解此类题目还有更为简单的方法,即拿一个钟表,拨动指针,使其旋转观察,将时针转24周,数其重合的次数即可得出答案。
参考文献, [1]《把数学建模融入高等数学教学中的两个案例》 崔海英、侯文
宇、李林杉 北京联合大学
[2]《关于同一个问题的不同解决方法》
http://www.doc88.com/p-9912711098443.html
5
范文四：数学建模时针与分针重合问题
数 学 实 践 与 建 模
时 针 与 分 针 重 合 问 题 的 分 析
姓名,
学号,
班级,
专业,
while the lawn is still in a great stage of life activities, compared with the previous two periods, lawn water requirements increase at this stage, if not timely irrigation, not only affect the growth of grass, also causes yellow. At this stage, according to 4-5 of irrigation. In short, grass types, water requirements are different, you mustAccording to the climate conditions and plant species to determine irrigation of the lawn. About a day, when irrigation is better depends on irrigation. In theory, the same period as long as the amount of irrigation water is less than the infiltration capacity of the soil, irrigation all the time in a day. Then depends on the irrigation way. If intermittent intermittent spray or spray irrigation (high degree of atomization), irrigation is best with the Sun. Not only can add moisture, but also significantly improve the microclimate and photosynthesis and transpiration and gas exchange, facilitated the coordination of water, air, fertilizer and soil thermal and vegetative organs of root and root expansion, and could lead the entire transit storage of the plant material, breathing, lawns and lush. If using water, irrigation, and so on, should be watching season, early spring, late autumn around the heel as well and the rest take twilight as much. (2) irrigation irrigation quantity should be based on the soil, growing, grass and other factors to determine. General lawn growing season drought period, takes about a week to pay 20-40mm; vigorously growing lawn in the heat and severe drought situations, need to pay 50-60mm per week or more. In General, regardless of the method, should be irrigated more than once, each time less water, maximum surface runoff that just occurred. 2, fertilizing to maintain lawn dark green leaves, grows, portions must be fertilized. Lawn plants for leaf growth and blossom-free requirements, so nitrogen is more important, response to nitrogen fertilizer is the most obvious. Should be basic application
while the lawn is still in a great stage of life activities, compared with the previous two periods, lawn water requirements increase at this stage, if not timely irrigation, not only affect the growth of grass, also causes yellow. At this stage, according to 4-5 of irrigation. In short, grass types, water requirements are different, you mustAccording to the climate conditions and plant species to determine irrigation of the lawn. About a day, when irrigation is better depends on irrigation. In theory, the same period as long as the amount of irrigation water is less than the infiltration capacity of the soil, irrigation all the time in a day. Then depends on the irrigation way. If intermittent intermittent spray or spray irrigation (high degree of atomization), irrigation is best with the Sun. Not only can add moisture, but also significantly improve the microclimate and photosynthesis and transpiration and gas exchange, facilitated the coordination of water, air, fertilizer and soil thermal and vegetative organs of root and root expansion, and could lead the entire transit storage of the plant material, breathing, lawns and lush. If using water, irrigation, and so on, should be watching season, early spring, late autumn around the heel as well and the rest take twilight as much. (2) irrigation irrigation quantity should be based on the soil, growing, grass and other factors to determine. General lawn growing season drought period, takes about a week to pay 20-40mm; vigorously growing lawn in the heat and severe drought situations, need to pay 50-60mm per week or more. In General, regardless of the method, should be irrigated more than once, each time less water, maximum surface runoff that just occurred. 2, fertilizing to maintain lawn dark green leaves, grows, portions must be fertilized. Lawn plants for leaf growth and blossom-free requirements, so nitrogen is more important, response to nitrogen fertilizer is the most obvious. Should be basic application
时针与分针重合问题的分析
摘要
“时针和分针一昼夜重合多少次,”这是美国著名的微软公司招聘职员的一道面试题。我们经常接触钟表,对钟表的认识可以说是再熟悉不过了,然而,面对这个看似简单的问题,许多应聘的优秀大学毕业生都不假思索的回答,24次。在他们认为,一昼夜是24个小时,每小时时针和分针重合一次,所以一昼夜时针和分针共重合24次。
答案看起来似乎并没有什么问题,可是,这个答案是错误的。只要我们稍加思考,就会感觉有点不对劲。那么时针与分针一昼夜到底重合多少次呢,
面对这种问题,乍一看似乎并无头绪,其实,我们完全可以通过建立数学模型的方法去求取。所谓数学建模,就是从定量的角度分析和研究一个实际问题,在深入调查研究、了解对象信息、作出简化假设、分析内在规律等工作的基础上,用数学的符号和语言作表述,然后用通过计算得到的结果来解释实际问题,并接受实际的检验。
对于时针和分针的相遇问题,我们可以将其类比为时针与分针相互追击的问题,,只不过他们行走的路径是一个圆。其中,分针比时针跑的快。然后通过假设一次相遇,两次相遇,多次相遇的办法,推理出相遇的规律,最后通过数学模型将一昼夜时针与分针相遇的次数求出。
关键词, 时针, 分针, 追击, 重合, 类推,
1、模型假设
while the lawn is still in a great stage of life activities, compared with the previous two periods, lawn water requirements increase at this stage, if not timely irrigation, not only affect the growth of grass, also causes yellow. At this stage, according to 4-5 of irrigation. In short, grass types, water requirements are different, you mustAccording to the climate conditions and plant species to determine irrigation of the lawn. About a day, when irrigation is better depends on irrigation. In theory, the same period as long as the amount of irrigation water is less than the infiltration capacity of the soil, irrigation all the time in a day. Then depends on the irrigation way. If intermittent intermittent spray or spray irrigation (high degree of atomization), irrigation is best with the Sun. Not only can add moisture, but also significantly improve the microclimate and photosynthesis and transpiration and gas exchange, facilitated the coordination of water, air, fertilizer and soil thermal and vegetative organs of root and root expansion, and could lead the entire transit storage of the plant material, breathing, lawns and lush. If using water, irrigation, and so on, should be watching season, early spring, late autumn around the heel as well and the rest take twilight as much. (2) irrigation irrigation quantity should be based on the soil, growing, grass and other factors to determine. General lawn growing season drought period, takes about a week to pay 20-40mm; vigorously growing lawn in the heat and severe drought situations, need to pay 50-60mm per week or more. In General, regardless of the method, should be irrigated more than once, each time less water, maximum surface runoff that just occurred. 2, fertilizing to maintain lawn dark green leaves, grows, portions must be fertilized. Lawn plants for leaf growth and blossom-free requirements, so nitrogen is more important, response to nitrogen fertilizer is the most obvious. Should be basic application
?、钟面上分为60个格,
?、分针1分针转过一格,一小时转过60格,时针一小时转过5格。
?、钟面上一个小格有6o,分针转动一周时,时针转了30o,如下图。
2、符号说明
?、x,分针的位置, n
y?、,时针的位置, n
?、x,分针和时针重合的位置。
3、模型建立
从k,k=1,2,3……,11,点开始,
11x,5k1，，？，(), nn-11212
111y,5k1，，？，，(), nn-1n121212
x,limx,limy。 nn,,,,nn
3
这就是所建立的数学模型。
4、模型求解
由假设可知分针的速度是时针的12倍。不妨设k=1,即从1点开始,此时
5时针在分针前5格。当分针走5格,分针走到时针原来的位置,时,时针走了12
55,，y5格,此时分针的位置记为x=5,时针的位置记为,当分针再走格111212
515x5,，时,时针走了,此时分针的位置记为,时针的位置记为，2121212
51515115y5,，，，，，,当分针再走格时,时针走了, ，21212121212121212
515x5,，，，此时分针的位置为, 3121212
515115y5,，，，，，，时针的位置为,…… 3121212121212
11x,51，，？，()以此类推,当分针的位置是时,时针的位置为nn-11212
1155y,51，，？，lim,0(),分针与时针相差,因为,所以分针与时nnnn,,n12121212
1160160xlimxlim51lim1),,(，，？，),(,,针重合的位置是格。 nn-1n,,,,,,nnn1212111211
60,5.4545因为,故此时时间约为1点5分27.27秒。 11
在1点5分27.27秒重合之后,由于分针速度是时针的12倍,故到2点为止,分针和时针不会再重合。
类似的,k=2时,即从2点开始,此时时针在分针前10,5×2,格。当分针while the lawn is still in a great stage of life activities, compared with the previous two periods, lawn water requirements increase at this stage, if not timely irrigation, not only affect the growth of grass, also causes yellow. At this stage, according to 4-5 of irrigation. In short, grass types, water requirements are different, you mustAccording to the climate conditions and plant species to determine irrigation of the lawn. About a day, when irrigation is better depends on irrigation. In theory, the same period as long as the amount of irrigation water is less than the infiltration capacity of the soil, irrigation all the time in a day. Then depends on the irrigation way. If intermittent intermittent spray or spray irrigation (high degree of atomization), irrigation is best with the Sun. Not only can add moisture, but also significantly improve the microclimate and photosynthesis and transpiration and gas exchange, facilitated the coordination of water, air, fertilizer and soil thermal and vegetative organs of root and root expansion, and could lead the entire transit storage of the plant material, breathing, lawns and lush. If using water, irrigation, and so on, should be watching season, early spring, late autumn around the heel as well and the rest take twilight as much. (2) irrigation irrigation quantity should be based on the soil, growing, grass and other factors to determine. General lawn growing season drought period, takes about a week to pay 20-40mm; vigorously growing lawn in the heat and severe drought situations, need to pay 50-60mm per week or more. In General, regardless of the method, should be irrigated more than once, each time less water, maximum surface runoff that just occurred. 2, fertilizing to maintain lawn dark green leaves, grows, portions must be fertilized. Lawn plants for leaf growth and blossom-free requirements, so nitrogen is more important, response to nitrogen fertilizer is the most obvious. Should be basic application
10走10格,分针走到时针原来的位置,时,时针走了格,此时分针的位置为
12
1011010,时针的位置为y10,当分针再走格时,时针走了,此,，x,10，1112121212
1011010y10时分针的位置为x10,时针的位置为,……以此类推,,，,，，，2212121212
11x,101，，？，当分针位置是时,时针的位置为()nn-11212
111010lim,0y,10(1，，？，),分针与时针相差,因为,所以分针与时nnnn,,n12121212
111201120xlimxlim101lim1针重合的位置是格。 ,,(，，？，),(,),nn-1n,,,,,,nnn1212111211
120,10.9091因为,故此时时间是2点10分54.55秒。 11
综上所述,k,k=1,2,…,11,点到k+1点之间,分针与时针重合的位置
1160k160是x,limx,lim5k(1，，？，),lim(1,),k格。 nn-1n,,,,,,nnn1212111211
60分针与时针重合的时间是k点分,k=1,2,…,11,,即1点5分27.27k11
秒,2点10分54.55秒,3点16分21.82秒,4点21分49.09秒,5点27分16.36秒,6点32分43.64秒,7点38分10.91秒,8点43分38.18秒,9点49分5.45秒,10点54分32.73秒,11点60分,12点,。一天之中,时针和分针共重合22次。
5、模型结果的分析与检验
从上述的模型求解过程中,可以看出一昼夜的时间内,时针与分针共重合了22次,而并非24次,从以上的结果中,也可以看出,时针与分针重合一次所需
120要的时间为分钟。计算的结果和实际的情况完全符合,故该模型对于求取此11
问题是正确的。
5
6、对模型的评价与推广
运用数学建模的方法,让我们对于求解某些实际问题有了更加明确的思路和方法。从实际问题中建立数学模型,解决数学问题从而解决实际问题,而此过程的关键是建立数学模型。此模型是近似的将钟表的时针与分针看做是一个特殊的圆形轨道上2人追及或相遇的问题,要把时钟问题当做行程问题来看,分针快,时针慢,所以分针与时针的问题,就是他们之间的追及问题。其实,在实际的求解关于钟表时针、分钟甚至秒针重合问题的时候,都可以采用此种方法进行求解。但是,从总体上看,求解此类题目时所需的计算比较复杂,当然求解此类题目还有更为简单的方法,即拿一个钟表,拨动指针,使其旋转观察,将时针转24周,数其重合的次数即可得出答案。
参考文献, [1]《把数学建模融入高等数学教学中的两个案例》 崔海英、侯文
宇、李林杉 北京联合大学
[2]《关于同一个问题的不同解决方法》
http://www.doc88.com/p-9912711098443.html
while the lawn is still in a great stage of life activities, compared with the previous two periods, lawn water requirements increase at this stage, if not timely irrigation, not only affect the growth of grass, also causes yellow. At this stage, according to 4-5 of irrigation. In short, grass types, water requirements are different, you mustAccording to the climate conditions and plant species to determine irrigation of the lawn. About a day, when irrigation is better depends on irrigation. In theory, the same period as long as the amount of irrigation water is less than the infiltration capacity of the soil, irrigation all the time in a day. Then depends on the irrigation way. If intermittent intermittent spray or spray irrigation (high degree of atomization), irrigation is best with the Sun. Not only can add moisture, but also significantly improve the microclimate and photosynthesis and transpiration and gas exchange, facilitated the coordination of water, air, fertilizer and soil thermal and vegetative organs of root and root expansion, and could lead the entire transit storage of the plant material, breathing, lawns and lush. If using water, irrigation, and so on, should be watching season, early spring, late autumn around the heel as well and the rest take twilight as much. (2) irrigation irrigation quantity should be based on the soil, growing, grass and other factors to determine. General lawn growing season drought period, takes about a week to pay 20-40mm; vigorously growing lawn in the heat and severe drought situations, need to pay 50-60mm per week or more. In General, regardless of the method, should be irrigated more than once, each time less water, maximum surface runoff that just occurred. 2, fertilizing to maintain lawn dark green leaves, grows, portions must be fertilized. Lawn plants for leaf growth and blossom-free requirements, so nitrogen is more important, response to nitrogen fertilizer is the most obvious. Should be basic application
范文五：分针和时针重合
时钟上的秘?密
作业终于做?完了，我长长地伸?了个懒腰:“妈妈，几点了,”
“现在呀，时针走在八?点多，分针和时针?重合。”我嘟囔了一?句:“你说八点四?十多一点点?，不就得了。”
妈妈笑着说?:“那你知道这?一点点是多?少吗,”
“这还不简单?～”我提起笔就?计算起来。
?钟面上每相?邻两个数字?间的夹角为?30 o，时针每旋转?30 o为一小时?，即60分钟?。
?时针每分钟?旋转0.5 o。
同样，分针每分钟?旋转6 o。
假设:以8:00为起始?点，经过X分钟?，分针与时针?重合。那么起始夹?角为240? o。
6X = 240 + 0.5X
480(分) 求得:X = 11
480我耸耸鼻子?，高兴地大喊?:“妈妈，我知道了，是8:分。” 11
妈妈又笑了?:“算得不错。那你知道时?针走一圈，和分针重合?多少次吗,”
我满不在乎?地回答:“时针旋转一?圈，分针就旋转?12圈，分针每旋转?一圈与时针?重合一次，那当然有1?2次重合了?。”妈妈拍拍我?的头，笑眯眯地说?:“我看你还是?不要这么草?率。这样吧，我给你一天?时间，明天再回答?我。”
哼，妈妈小瞧我?～明天我计算?出这十二个?重合点，让妈妈瞧瞧?。
第二天，我早早做完?作业，开始计算起?来。
同样，以整点为起?始点，经过X分钟?，时针与分针?重合。
第一次的重?合:0点整。
60第二次的重?合:1: 6X = 30 + 0.5X 11
60 X = (分) 11
120第三次的重?合:2: 6X = 60 + 0.5X 11
120 X = (分) 11
180第四次的重?合:3: 6X = 90 + 0.5X 11
180 X = (分) 11
哎，慢着慢着，我发现了一?个规律:后一次的重?合时间总是?比前一次的?
60重合时间增?加1小时分?。由此可以推?理出: 11
240第五次的重?合:4: 11
300第六次的重?合:5: 11
??
540第十次的重?合:9: 11
600第十一次的?重合:10: 11
660第十二次的?重合:11: = 11:60 = 12:00 11
第十二次的?重合与第一?次的重合是?在一个点上?，它又回到了?原点。原来在
11?时与12时?之间是没有?重合点的，时针旋转一?周只能与分?针重合11?次～ 还是妈妈说?得对，任何事情都?不能想当然?，而要经过认?真计算、思考。
我们往往认?为当然正确?的事情，却与真相相?差甚远。
