Some state water-right agencies require from communities data on the rate of water use, in gallons per hour, and the total amount of water used each day. Many communities do not have equipment to measure the flow of water in or out of the municipal tank. Instead, they can measure only the level of water in the tank, within 0.5% accuracy, every hour. More importantly, whenever the level in the tank drops below some minimum level L, a pump fills the tank up to the maximum level, H; however, there is no measurement of the pump flow either. Thus, one cannot readily relate the level in the tank to the amount of water used while the pump is working, which occurs once or twice per day, for a couple of hours each time. Estimate the flow out of the tank f(t) at all times, even when the pump is working, and estimate the total amount of water used during the day. Table 1 gives real data, from an actual small town, for one day[ table omitted]. The table gives the time, in, since the first measurement, and the level of water in the tank, in hundredths of a foot. For example, after 3316 seconds, the depth of water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and a diameter of 57 feet. Usually, the pump starts filling the tank when the level drops to about 27.00 feet, and the pump stops when the level rises back to about 35.50 feet. 

 

表91A-1 某小镇某天的水塔水位

时间(秒) 水位

(0.01英尺) 时间(秒) 水位

(0.01英尺) 时间(秒) 水位

(0.01英尺)  

0 3175 35932 水泵工作 68535 2842  

3316 3110 39332 水泵工作 71854 2767  

6635 3054 39435 3550 75021 2697  

10619 2994 43318 3445 79154 水泵工作  

13937 2947 46636 3350 82649 水泵工作  

17921 2892 49953 3260 85968 3475  

21240 2850 53936 3167 89953 3397  

25223 2797 57254 3087 93270 3340  

28543 2752 60574 3012      

32284 2697 64554 2927      

 

水箱流量(美国竞赛1991年A题) 

 

美国某州的各用水管理机构要求各社区提供以每小时多少加仑计的用水率以及每天所用的总水量, 但许多社区并没有测量流人或流出当地水塔的水量的设备，他们只能代之以每小时测量水塔中的水位，其精度在0.5％以内。更为重要的是，无论什么时候，只要水塔中的水位下降到某一最低水位L时，水泵就启动向水塔重新充水直至某一最高水位只，但也无法得到水泵的供水量的测量数据。因此，在水泵正在工作时，人们不容易建立水塔中的水位与水泵工作时的用水量之间的关系。水泵每天向水塔充水一次或两次，每次约二小时。 

 

试估计在任何时刻，甚至包括水泵正在工作的时间内，水从水塔流出的流量 f(t)，并估计一天的总用水量。表9lA-1给出了某个真实小镇某一天的真实数据。