【正文】 an A in a class in which all students obtain an A, then this student is only average in this class. On the other hand, if a student obtain the only A in a class, then that student is clearly above average. Combining information from several classes might allow students to be placed in deciles (top 10%,next 10%,ect.)across the college. Problem Assuming that the grades given out are(A+,A,B+,B,...)can the dean39。s idea be made to work? Can any other schemes produce a desired ranking? A concern is that the grade in a single class could change many student39。s exits in case of an emergency. Similarly, elevators and other facilities often have maximum capacities posted. Develop a mathematical model for deciding what number to post on such a sign as being the lawful capacity. As part of your solution discuss criteria, other than public safety in the case of a fire or other emergency, that might govern the number of people considered unlawful to occupy the room (or space). Also, for the model that you construct, consider the differences between a room with movable furniture such as a cafeteria (with tables and chairs), a gymnasium, a public swimming pool, and a lecture hall with a pattern of rows and aisles. You may wish to pare and contrast what might be done for a variety of different environments: elevator, lecture hall, swimming pool, cafeteria, or gymnasium. Gatherings such as rock concerts and soccer tournaments may present special conditions. Apply your model to one or more public facilities at your institution (or neighboring town). Compare your results with the stated capacity, if one is posted. If used, your model is likely to be challenged by parties with interests in increasing the capacity. Write an article for the local newspaper defending your analysis. 2022 Mathematical Contest in Modeling Problem A Air traffic Control Dedicated to the memory of Dr. Robert Machol, former chief scientist of the Federal Aviation Agency To improve safety and reduce air traffic controller workload, the Federal Aviation Agency (FAA) is considering adding software to the air traffic control system that would automatically detect potential aircraft flight path conflicts and alert the controller. To that end, an analyst at the FAA has posed the following problems. Requirement A: Given two airplanes flying in space, when should the air traffic controller consider the objects to be too close and to require intervention? Requirement B: An airspace sector is the section of threedimensional airspace that one air traffic controller controls. Given any airspace sector, how do we measure how plex it is from an air traffic workload perspective? To what extent is plexity determined by the number of aircraft simultaneously passing through that sector (1) at any one instant? (2) during any given interval of time?(3) during a particular time of day? How does the number of potential conflicts arising during those periods affect plexity? Does the presence of additional software tools to automatically predict conflicts and alert the controller reduce or add to this plexity? In addition to the guidelines for your report, write a summary (no more than two pages) that the FAA analyst can present to Jane Garvey, the FAA Administrator, to defend your conclusions. Problem B Radio Channel Assignments We seek to model the assignment of radio channels to a symmetric work of transmitter locations over a large planar area, so as to avoid interference. One basic approach is to partition the region into regular hexagons in a grid (honeybstyle), as shown in Figure 1, where a transmitter is located at the center of each hexagon. Figure 1 An interval of the frequency spectrum is to be allotted for transmitter frequencies. The interval will be divided into regularly spaced channels, which we represent by integers 1, 2, 3, ... . Each transmitter will be assigned one positive integer channel. The same channel can be used at many locations, provided that interference from nearby transmitters is avoided. Our goal is to minimize the width of the interval in the frequency spectrum that is needed to assign channels subject to some constraints. This is achieved with the concept of a span. The span is the minimum, over all assignments satisfying the constraints, of the largest channel used at any location. It is not required that every channel smaller than the span be used in an assignment that attains the span. Let s be the length of a side of one of the hexagons. We concentrate on the case that there are two levels of interference. Requirement A: There are several constraints on frequency assignments. First, no two transmitters within distance of each other can be given the same channel. Second, due to spectral spreading, transmitters within distance 2s of each other must not be given the same or adjacent channels: Their channels must differ by at least 2. Under these constraints, what can we say about the span in, Requirement B: Repeat Requirement A, assuming the grid in the example spreads arbitrarily far in all directions. Requirement C: Repeat Requirements A and B, except assume now more generally that channels for transmitters within distance differ by at least some given integer k, while those at distance at most must still differ by at least one. What can we say about the span and about efficient strategies for designing assignments, as a function of k? Requirement D: Consider generalizations of the problem, such as several levels of interference or irregular transmitter placements. What other factors may be important to consider? Requirement E: Write an article (no more than 2 pages) for the local newspaper explaining your findings.