By Murray S. Klamkin

Designed for school room use, this ebook includes brief, self-contained mathematical versions of difficulties within the actual, mathematical, and organic sciences first released within the school room Notes part of the SIAM assessment from 1975-1985. the issues supply a good way to make advanced subject material extra available to the coed by using concrete functions. every one part has huge supplementary references supplied by means of the editor from his years of expertise with mathematical modelling.

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KLAMKINt In this note we extend the following problem proposed by M. F. Gardner [1]: "A swimmer can swim with speed v in still water. He is required to swim for a given length of time T in a stream whose speed is w < v. If he is also required to start and finish at the same point, what is the longest path (total arc length) that he can complete? " It is physically intuitive that the longest and shortest paths must be perpendicular and parallel, respectively, to the velocity of the stream. We generalize and prove these results by considering an aeroplane flying with a speed v with respect to ground in a bounded irrotational wind field given by \V-V

A. Bender. Copyright © 1978 by John Wiley & Sons, Inc. t Mathematics Department, University of California, San Diego, La Jolla, California 92093. 15 16 BENDER FIG. 1 We wish to maximize / so we set its derivative equal to zero and use (2): Our basic equations are (2)-(4). To make further progress we must describe how 0 varies throughout the turn. Baylis assumes 0 = K —

Engineers find the tensile strength more accessible to experimental measurement. The tensile strength (cr) for fresh water ice is 150 psi, and for sea water ice is 90 psi (see, for example, [4]). The relationship between bending moment and tensile strength is derived in most texts on strengths of materials, for example [3]. An explicit derivation for a wedge is given in [2a] and is Therefore, equating m max to (rh"/6 we find This equation relates the variables of interest: r, p. the radius of load or size of air cushion; the air pressure or cushion pressure; h, the thickness of the ice field.