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IB Mathematics: Applications and Interpretation HL · 12.8: Voronoi Diagrams · 12.8.6

The toxic waste dump problem Problem set B Worked solutions

Skills

  1. 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.

      • \(d=\sqrt{(1-6)^2+(1-(-4))^2}\)
      • \(d=\sqrt{50}\)

    2. (b) Find the distance from V2 to S2, to 3 significant figures.

      • \(d=\sqrt{(-5-5)^2+(-2-2)^2}\)
      • \(d=\sqrt{116}\) = 10.8

  2. 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.

    −8−6−4−22468−8−6−4−22468xyABC

    1. (a) Show that V is the same distance from A, B and C.

      • V to A: \(d^2=(-4-(-7))^2+(2-(-5))^2=58\)
      • V to B: \(d^2=(-4-(-1))^2+(2-(-5))^2=58\)
      • V to C: \(d^2=(-4-3)^2+(2-(-1))^2=58\)
      • All three equal 58, so V is equidistant 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.

      • Distance = \(\sqrt{58}\) = 7.62 units

  3. 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).

    1234567891234567xyABCDE

    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.

      • P(4, 4): equidistant from B, C, E. Distance to B(2, 1): \(\sqrt{13}\) = 3.61
      • Q(4.5, 1.5): equidistant from A, B, C. Distance to A(7, 1): \(\sqrt{6.5}\) = 2.55
      • R(4.5, 4.75): equidistant from C, D, E. Distance to C(7, 2): \(\sqrt{13.8125}\) = 3.72

    2. (b) Which vertex is farthest from its nearest sites?

      • The greatest distance is at R.

    3. (c) What is the greatest distance?

      • R is \(\sqrt{13.8125}\) = 3.72 units from its nearest sites.

In context

  1. 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).

    1234567891234567xyABCDEF

    1. (a) Find the coordinates of the best place for the waste site.

      • P(2.5, 2.5): equidistant from A, B, C. Distance to A(2, 4): \(\sqrt{2.5}\) = 1.58
      • Q(5, 2.5): equidistant from A, C, D. Distance to A(2, 4): \(\sqrt{11.25}\) = 3.35
      • R(5, 5): equidistant from A, E, F. Distance to A(2, 4): \(\sqrt{10}\) = 3.16
      • S(5.5, 3.5): equidistant from A, D, F. Distance to A(2, 4): \(\sqrt{12.5}\) = 3.54
      • The greatest distance is at S(5.5, 3.5).

    2. (b) How far is it from the nearest house?

      • Distance = \(\sqrt{12.5}\) = 3.54 units

  2. 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.

    −8−6−4−22468−8−6−4−22468xyABC

    1. (a) Find the equation of the perpendicular bisector of A and B.

      • Midpoint of AB: \((2, 1)\)
      • Gradient of AB = \(\dfrac{4}{5}\); perpendicular gradient = \(-\dfrac{5}{4}\)
      • Bisector: \(y=-\dfrac{5}{4}x+\dfrac{7}{2}\)

    2. (b) Find the equation of the perpendicular bisector of B and C.

      • Midpoint of BC: \((2, -3)\)
      • BC is horizontal, so the bisector is vertical: \(x=2\)

    3. (c) Solve the two equations to find the coordinates of the vertex.

      • Solve \(y=-\dfrac{5}{4}x+\dfrac{7}{2}\) and \(x=2\) together.
      • From the calculator: x = 2, y = 1

    4. (d) Find the distance from the vertex to each gate, to 3 significant figures.

      • A(7, 5): \(d=\sqrt{(2-7)^2+(1-5)^2}=\sqrt{41}\) = 6.40

  3. 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.

    1234567891234567xyABCDEF

    1. (a) Find the best place for the waste site and its distance from the nearest hut, to 3 significant figures.

      • P(2, 3): equidistant from A, B, D. Distance to A(1, 5): \(\sqrt{5}\) = 2.24
      • Q(2.5, 3.25): equidistant from A, B, E. Distance to A(1, 5): \(\sqrt{5.3125}\) = 2.30
      • R(5.5, 2.5): equidistant from B, C, E. Distance to B(3, 1): \(\sqrt{8.5}\) = 2.92
      • S(6, 3): equidistant from C, E, F. Distance to C(8, 1): \(\sqrt{8}\) = 2.83
      • The greatest is R(5.5, 2.5).

    2. (b) Can the rule be met? Give a reason.

      • No point in the region can be farther than 2.92 units from the nearest hut.
      • 2.92 < 3.5, so the answer is no.

Stretch

  1. 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.

    1234567891234567xyABCDE

    1. (a) Find the greatest distance from a vertex inside the region to its nearest farm.

      • P(3, 3): equidistant from B, C, E. Distance to B(5, 4): \(\sqrt{5}\) = 2.24
      • Q(3, 5.5): equidistant from C, D, E. Distance to C(4, 5): \(\sqrt{1.25}\) = 1.12
      • R(5, 5): equidistant from A, B, C. Distance to A(6, 5): \(\sqrt{1}\) = 1
      • The greatest is 2.24.

    2. (b) Find the distance from each corner of the region to its nearest farm. Which corner is farthest?

      • Corner (0, 0): nearest E, d = \(\sqrt{29}\) = 5.39
      • Corner (9, 0): nearest B, d = \(\sqrt{32}\) = 5.66
      • Corner (9, 7): nearest A, d = \(\sqrt{13}\) = 3.61
      • Corner (0, 7): nearest D, d = \(\sqrt{5}\) = 2.24

    3. (c) Where should the waste site go, and what is its distance from the nearest farm?

      • The corner (9, 0) is 5.66, greater than any vertex (2.24).
      • On the boundary the farthest point can be a corner of the region.

  2. 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).

    1234567891234567xyABCDEFG

    1. (a) Find the best place for the waste site and the radius of the largest empty circle, to 3 significant figures.

      • P(3, 3): equidistant from B, E, G. Distance to B(2, 1): \(\sqrt{5}\) = 2.24
      • Q(4, 3): equidistant from B, C, G. Distance to B(2, 1): \(\sqrt{8}\) = 2.83
      • R(4.5, 2.5): equidistant from A, B, C. Distance to A(7, 1): \(\sqrt{8.5}\) = 2.92
      • S(7.5, 3.25): equidistant from A, C, D. Distance to A(7, 1): \(\sqrt{5.3125}\) = 2.30
      • T(8, 3.5): equidistant from C, D, F. Distance to C(6, 5): \(\sqrt{6.25}\) = 2.5
      • The greatest is at R(4.5, 2.5).

    2. (b) The circle centred there is a no-village zone. Find its area to 3 significant figures, in square units.

      • Area = π r² = π × 8.5 = 26.7

jmaths.xyz · AI HL · 12.8.6 The toxic waste dump problem