The toxic waste dump problem Problem set B

A Voronoi vertex V1 is at (1, 1), and a site S1 is at (6, −4). In a second diagram a vertex V2 is at (−5, −2) and a site S2 is at (5, 2).

1. (a) Find the exact distance from V1 to S1.
2. (b) Find the distance from V2 to S2, to 3 significant figures.

Three airports, A(−7, −5), B(−1, −5) and C(3, −1), are in a county. The Voronoi diagram is shown, with the vertex V at (−4, 2). One unit is 10 km.

1. (a) Show that V is the same distance from A, B and C.
2. (b) A waste site is placed at V. Find its distance from the nearest airport, to 3 significant figures.

A training area has 5 bases, A, B, C, D and E. The diagram shows their Voronoi diagram; one unit is km. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every base. The vertices inside the region are P(4, 4), Q(4.5, 1.5), R(4.5, 4.75).

1. (a) For each vertex, name two of the sites next to it and find its distance from one of them. Write down the distances to 3 significant figures.
2. (b) Which vertex is farthest from its nearest sites?
3. (c) What is the greatest distance?

A wind-farm site has 6 houses, A, B, C, D, E and F. The diagram shows their Voronoi diagram; one unit is 100 m. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every house. The vertices inside the region are P(2.5, 2.5), Q(5, 2.5), R(5, 5), S(5.5, 3.5).

1. (a) Find the coordinates of the best place for the waste site.
2. (b) How far is it from the nearest house?

Three gates, A(7, 5), B(−3, −3) and C(7, −3), are in a country estate. One unit is 100 m. A waste site is to be as far as possible from all three. Assume it goes at the vertex of the Voronoi diagram.

1. (a) Find the equation of the perpendicular bisector of A and B.
2. (b) Find the equation of the perpendicular bisector of B and C.
3. (c) Solve the two equations to find the coordinates of the vertex.
4. (d) Find the distance from the vertex to each gate, to 3 significant figures.

A moor has 6 huts, A, B, C, D, E and F. The diagram shows their Voronoi diagram; one unit is km. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every hut. The vertices inside the region are P(2, 3), Q(2.5, 3.25), R(5.5, 2.5), S(6, 3). A rule says the waste site must be at least 3.5 units from every hut.

1. (a) Find the best place for the waste site and its distance from the nearest hut, to 3 significant figures.
2. (b) Can the rule be met? Give a reason.

A marsh has 5 farms, A, B, C, D and E. The diagram shows their Voronoi diagram; one unit is km. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every farm. The vertices inside the region are P(3, 3), Q(3, 5.5), R(5, 5). The waste site may be anywhere in the region, including its boundary.

1. (a) Find the greatest distance from a vertex inside the region to its nearest farm.
2. (b) Find the distance from each corner of the region to its nearest farm. Which corner is farthest?
3. (c) Where should the waste site go, and what is its distance from the nearest farm?

A reservoir area has 7 villages, A, B, C, D, E, F and G. The diagram shows their Voronoi diagram; one unit is km. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every village. The vertices inside the region are P(3, 3), Q(4, 3), R(4.5, 2.5), S(7.5, 3.25), T(8, 3.5).

1. (a) Find the best place for the waste site and the radius of the largest empty circle, to 3 significant figures.
2. (b) The circle centred there is a no-village zone. Find its area to 3 significant figures, in square units.