Modeling flow from a cylinder

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needsmathhelp101

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This problem comes from a 1991 problem given at a math competition, since it is available online I just copied and pasted it from http://www.grafitto.com/private/comap/MCMICM/First10/1991/1991B.pdf , but what is says is given the data in the table, estimate the flow from the tank at any time, even when the pump is on, what I've thought of/my problem follows the text:

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.
The table gives the time, in seconds, 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.
Table 1.
Water-tank levels over a single day for a small town. Time is in seconds and level is in 0.01 ft.
Time Level Time Level Time Level
0 3175 35932 pump on 68535 2842
3316 3110 39332 pump on 71854 2767
6635 3054 39435 3550 75021 2697
10619 2994 43318 3445 79254 pump on
13937 2947 46636 3350 82649 pump on
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

At first I was thinking I'd take a related rates approach, but I kept getting stuck and I'm not sure if that would be the same as a modeling approach.
I just don't know how to use the raw data to get anything that I need, the graph is so weird because of the 'pump on' values that I don't know what to do with it. Any guidance would be greatly appreciated, as I just don't know where to start.

Thanks a lot
 
needsmathhelp101 said:
This problem comes from a 1991 problem given at a math competition, since it is available online I just copied and pasted it from http://www.grafitto.com/private/comap/M ... /1991B.pdf , but what is says is given the data in the table, estimate the flow from the tank at any time, even when the pump is on, what I've thought of/my problem follows the text:

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.
The table gives the time, in seconds, 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.
Table 1.
Water-tank levels over a single day for a small town. Time is in seconds and level is in 0.01 ft.
Code: 
Time  Level    Time     Level       Time     Level
0      3175     35932   pump on   68535    2842
3316   3110    39332    pump on   71854    2767
6635   3054    39435    3550        75021    2697
10619  2994   43318    3445        79254    pump on
13937  2947   46636    3350        82649    pump on
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
At first I was thinking I'd take a related rates approach, but I kept getting stuck and I'm not sure if that would be the same as a modeling approach.
I just don't know how to use the raw data to get anything that I need, the graph is so weird because of the 'pump on' values that I don't know what to do with it. Any guidance would be greatly appreciated, as I just don't know where to start.

Thanks a lot
Click to expand...
 
estimate the flow from the tank at any time
Click to expand...

Here is one approach:

First calculate the rate the tank level is dropping during the “pump on” period by some simple averaging. Use the two data points right before the pump comes on to find a drop rate at that time. Do the same with the two data points right after the pump turns off. Average these two drop rates.

Use the average drop rate to figure out how much the level would have dropped if the pumps had not been turned on.

Now comes the data adjustment: calculate and subtract the water level drop (during the “pump on” period) from time 32284 to time 39435, thus creating a “new level for time 39435.

From this “new” level, adjust all subsequent “level” data points.

Repeat this procedure for the next time the pump comes on at time 79254.

This should “smooth” all your data, making it flow more continuously when plotted. It will eliminate the “hiccups” due to the water added by the pump.

Since this is a “real world” problem, we can proceed by making some assumptions. First, I’d assume that this usage pattern would be repeated daily, except that weekend usage would be different than work-week usage. There is nothing we can do about that, though, since we only have data for about one day.

I might consider throwing out the data after time 86400, as that is the number of seconds in one day, and I’m expecting the pattern to repeat itself every 86400 seconds. That would just eliminate the last two data points.

Next convert all the water level data into volume data.

Create a scatter plot of Volume versus Time.

Since this is a repeating pattern, my first guess would be that it could be modeled with a sinusoidal function of the form V(t) = Asin[B(t-C)]+D. Check your sinusoid by plotting it on top of the scatter lot.

After determining the sinusoid function, find the derivative, dV/dT, of this function. That is the answer to this problem since it gives the flow rate at any time.
 
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