Toxic Waste Dump problem
'Can Voronoi diagrams help us select a site that is as far as possible from all existing sites?'
Russia’s Kamchatka peninsula has one of the highest densities of volcanoes in the world. The map, in the 'Resources' section below shows the rough positions of four volcanoes. A new hotel, with observation posts, is planned in the area below. The developers want to build it not further than 80km from each of the 4 volcanoes (so as to have a reasonable view of all of them), but at a point that is as far as possible, given this condition (1 unit = 20km). Which position meets this criteria?
Aims
This activity aims to help students develop their own, conceptual understanding of the "toxic waste dump (aka largest circle) problem". It provides a concrete means (the applets in the 'Resource' section) through which they can experiment, using their current knowledge, get feedback, and gradually move towards an appreciation, and understanding, of the voronoi approach.
Resources - 1: Experiment/Brainstorm point furthest away from ALL volcanoes
In your groups, brainstorm some ideas you may have on how to approach/work towards a solution for the 'hotel' question above using the applet below and your existing mathematical knowledge and reasoning. You can add points, measure lengths/distances, construct circles, perpendicular bisectors, mid-points etc.
If you team/group gets really stuck, click on the "EYE" icon below HINTS underneath this applet.
HINTS
Click on the eye below to see three different hotel positions (A, B and C pictures). In which are the hotel guests safest?
Resources - 2: Where's Safest?
Is the solution shown in applet 2 really "the safest place to be" on the mapped area shown? Use the applet below to experiment, by placing the point 'MoveMe' in different positions, if there are points that are safer than this solution.
Change the position of the "MoveMe" point to confirm that point "I" does satisfy the hotel developers requirements: "not further than 80km from each of the 4 volcanoes (so as to have a good reasonable view of all of them), but at a point that is as far as possible from each, given this condition (1 unit = 20km)".
IB Learner Profile - Thinker/Reflective
Why was it necessary to include, in the question, the condition that the new hotel "not be further than 80km from each of the 4 volcanoes".?
What would the coordinates of the solution be if this condition were not included? Why might this solution not be of interest to the hotel developer?[Not a mathematics question (using commercial/economics knowledge)]
Teacher only boxDescription
- Students experiment with the first applet. Depending on whether this activity/parts of this activity are used prior to learning something about voronoi diagrams will likely influence the methods and approaches applied in the first applet. The aims of the first applet are to provide concrete feedback for students on their existing knowledge and motivate a need for the use of a voronoi diagram (though some may realise this for themselves!).
- The second applet provides an opportunity to test if the solution shown in applet 2 really is "the safest place to be" on the mapped area shown.
It helps students to gain a visual appreciation and understanding of how voronoi diagrams help solve this type of problem. It should, hopefully, also encourage students to use the solution to this one problem to speculate/hypothesise/conjecture as to the general solution for all such problems . . .