The toxic waste dump problem asks where to place an undesirable facility so that it is as far as possible from the nearest existing site, such as a town or water source. A Voronoi diagram solves it by identifying the centre of the largest circle that contains no sites.
The Reasoning
A Voronoi diagram divides a region into cells. Every point within a cell is closer to that cell’s site than to any other site.
Suppose the proposed dump is at point , and the sites are . The distance to the nearest site is
The objective is to choose so that is as large as possible. This is a maximin problem: maximize the minimum distance.
A Voronoi vertex is the intersection of three Voronoi edges and is equidistant from three sites. It is therefore the centre of an empty circle passing through those sites, with no other site inside it.
| Step | Method |
|---|---|
| 1 | Identify every Voronoi vertex in the permitted region. |
| 2 | For each vertex, calculate its distance to any one of the three surrounding sites. |
| 3 | Compare these distances. |
| 4 | Select the vertex with the greatest distance: the centre of the largest empty circle. |
This chosen point maximizes the distance from the dump to its nearest site. In the general mathematical problem, boundaries may also need checking, but IB Mathematics: Applications and Interpretation questions specify that the solution is at an intersection of three Voronoi edges.
Exam Technique
This is SL 3.6: Voronoi diagrams. Show the candidate vertices and their distances, then justify the selected vertex by stating that it has the greatest minimum distance to a site. A common misconception is to choose the visually largest cell; the answer is a vertex determined by distance comparison, not the largest cell.
