Question 1
Paper 1Hard14 marksAn environmental agency models four towns as sites in a Voronoi diagram. Coordinates are in kilometres. Each point in a cell is assigned the annual rainfall measured at that cell’s site (nearest-neighbour interpolation).
| Site | Coordinates | Annual rainfall (mm) |
|---|---|---|
| A | (0, 0) | 820 |
| B | (10, 0) | 960 |
| C | (0, 8) | 1240 |
| D | (10, 10) | 1080 |
A new monitoring station, E, is later opened at (5, 8). Its annual rainfall is 1500 mm. E is also treated as a town site when screening possible toxic-waste facility locations.
A point P has coordinates (2, 3). Determine the site whose cell contains P and the rainfall assigned to P.
A point Q has coordinates (4, 6). Determine the change in its rainfall estimate when E is added to the Voronoi diagram.
After E is added, two relevant Voronoi edge equations are y = 4 and 10x + 16y = 89. Find their point of intersection, W, and use the coordinates of all five sites to determine whether W is a vertex of the updated Voronoi diagram.
Before E was added, V₀ = (5.8, 5) was a vertex where the cells of B, C and D met. Determine whether V₀ remains a Voronoi vertex after E is added.
For the agency’s toxic-waste screening model, the search is restricted to the following three triple-edge vertices in its approved search zone. The agency selects the point whose distance from its nearest town is greatest.
| Candidate | Coordinates |
|---|---|
| U | (5, 39/16) |
| W | (5/2, 4) |
| X | (91/10, 5) |
Determine which candidate the agency should select. Give its distance from the nearest town.