Multivariable Calculus with Project Report | MATH 234, Study Guides, Projects, Research of Advanced Calculus

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Rinky Dinky Railroad Co. Erewhon, NV February 4, 2008

Multivariable Calculus, Inc. Loyola Marymount University Los Angeles, CA 90045 Dear MCI, I am writing to offer you a lucrative contract. The Rinky Dinky Railroad Company is planning a new railroad in a valley in Nevada, running between the growing commercial centers along the north side of the valley, and the residential communities along the south side of the valley, 10 miles away. Currently, there is no good mass transit between these communities, and we see a great business opportunity. Our original plan was to lay about 16 miles of straight track along each side of the valley, connecting the communities on each side of the valley, and then connect the ends with semicircles, as shown below: Fortunately, before we finalized our plans, we heard the sad tale of the Lollapalooza Railroad Company in Utah, which had laid out a similar track last year. They discovered that if their trains ran faster than 30 mph, the transition from the straight track to the circular track was jolting and uncomfortable for the passengers. They are now in bankruptcy after settling $50 million worth of personal injury lawsuits. We want to avoid their error. Clearly the problem is the change in curvature between the two track segments. We’d like you to design a transition curve between the straight track and the circular track along which the curvature increases linearly until it matches the curvature of the circle – we think that will provide the smoothest ride. In order to bend the rails properly, we will need an explicit parameterization for the curve. If you find it impossible to produce an explicit formula for the exact curve, an approximation is acceptable – but the curvatures where two pieces of track come together cannot differ by more than 1%. Also, small adjustments can be made to the exact distance between the two parallel stretches of straight track – but they must stay within the right-of-way we have purchased, so the distance between the tracks cannot vary from 10 miles by more than 100 feet. Finally, we would like to know the total length of the final track, so we can place our order for rails. A second problem we have recently become aware of is the necessity of banking the track along the curved portions of the track (i.e. elevating the outer rail). If this is not done, the passengers will be pushed uncomfortably towards the outer side of the rail car; the centripetal forces also cause considerable wear on the rails and railcar wheels. We’d like you to tell us at each point along the track, the angle at which to bank it so as to eliminate the forces acting tangentially to the surface of the track. We expect our trains to be traveling at an average speed of 80 mph on the curved sections of the track.