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Gavin's Modeling Project 5

the 1991 MCM

Following are two problems, both from the Mathematical Contest in Modeling (in 1991). One is a ``discrete'' problem, which may loosely be interpreted as having to do with the mathematics of integers, while the other is a ``continuous'' problem, similarly characterized as having to do with the mathematics of real numbers. Your group should pick (only) one of the two projects to solve.
  1. The cost for a communication line between two stations is proportional to the length of the line. The cost for conventional minimum spanning trees of a set of stations can often be cut by introduction of ``phantom'' stations and then constructing a new Steiner tree. this device allows costs to be cut by up to 13.4% (=1 - \sqrt(3)/2). Moreover, a network with n stations never requires more than n-2 points to construct the cheapest Steiner tree. Two simple cases are shown in figure 1.
    [figure 1]

    For local networks, it often is necessary to use rectilinear or ``checkerboard'' distances, instead of straight Euclidean lines. distances in this metric are computed as shown in figure 2.

    [figure 2]

    Suppose you wish to design a minimum cost spanning tree for a local network for 9 stations. Their rectangular coordinates are: a(0,15), b(5,20), c(16,24), d(20,20), e(33,25), f(23,11), g(35,7), h(25,0), i(10,3). You are restricted to using rectilinear lines. Moreover, all ``phantom'' stations must be located at lattice points (i.e., the coordinates must be integers). The cost for each line is its length.

    1. Find a minimal cost tree for the network.
    2. Suppose each station has a cost d3/2 w, where d= the degree of the station. If w=1.2, find a minimal cost tree.
    3. Try to generalize this problem.

  2. 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 a 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 the water in the tank reached 31.10 feet. The tank is a vertical circular cylinder, with a height of 40 feet and 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.

    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    

Gavins Modeling Project 5
Last Modified: Mon Nov 22 09:29:07 CST 1999
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