RIDGE

Airport Problem



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Source(s):

 The inhabitants of three towns want to locate an airport so that the roads joining the towns to the airport will have the shortest possible total length.  Where should the airport be located?

 de Villiers 1999 p. 115

Suppose an airport is planned to service three cities of approximately equal size. The planners decide to locate the airport so that the sum of the distances to the three cities is a minimum. Where should the airport be located?

Stellen Sie sich vor, dass ein Flughafen geplant ist, der von drei Städten aus gut erreichbar sein soll.

3 a) Die Planer entscheiden sich, den Flughafen so zu positionieren, dass der Flughafen von allen drei Städten gleich weit entfernt ist. Wo muss der Flughafen gebaut werden? Begründen Sie Ihre Lösung!

3 b) Noch bevor irgendwelche Maßnahmen für den Bau des Flughafens realisiert worden sind, findet ein politischer Machtwechsel statt. Die neue Regierungspartei ist primär an Sparmaßnahmen orientiert. Sie verlangt, dass möglichst wenig Geld für die erforderlichen neuen Straßen ausgegeben wird. Die Planer entscheiden daher, dass die Summe der Entfernungen zu den drei Städten minimal sein soll. Wo muss der Flughafen positioniert werden? Begründen Sie Ihre Antwort!

[Imagine that an airport is being planned, which should be easily accessible from three cities.

3 a) The planners decide to position the airport in such a way that the airport is equally far from all three cities. Where does the airport have to be built? Justify your solution!

3 b) Before the building of the airport begins, a political change of power takes place. The new government party is primarily oriented to economic measures. It requires that as little money as possible be spent on new
roads. The planners therefore decide that the total of the distances to the three cities should be minimal. Where does the airport have to be positioned? Justify your response!]

Knipping 2005

Other Variants: The airport could be a well or another resource.

Possible solutions


1. Recognise this as a minimal surface problem, and empirically locate a point that makes all the angles between the roads 120 degrees.  Observe that the extended roads pass through vertices of equilateral triangles on the sides. Prove that the angles formed by connecting vertices of equilateral triangles on the sides to the opposite vertices form 120 degree angles, using congruent angles.

2. Rotate sides so that the three roads make a joined path.  When the path is a segment the minimal length is achieved. 

Supported by a research grant from the Social Sciences and Humanities Research Council of Canada

Page last updated July 2008 by David Reid

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