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

The toxic waste dump problem Problem set A One question at a time

Try each question on paper first. Open the working a line at a time, then say whether you had it.

Skills

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

    1. (a) Find the exact distance from V1 to S1.

      • \(d=\sqrt{(5-(-6))^2+(-1-(-4))^2}\)
      • \(d=\sqrt{130}\)

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

      • \(d=\sqrt{(-4-2)^2+(6-(-5))^2}\)
      • \(d=\sqrt{157}\) = 12.5

  2. Three buoys, A(−4, 5), B(1, 0) and C(−5, 0), are in a lake. The Voronoi diagram is shown, with the vertex V at (−2, 2). One unit is 100 m.

    −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: \(d^2=(-2-(-4))^2+(2-5)^2=13\)
      • V to B: \(d^2=(-2-1)^2+(2-0)^2=13\)
      • V to C: \(d^2=(-2-(-5))^2+(2-0)^2=13\)
      • All three equal 13, so V is equidistant from A, B and C.

    2. (b) A waste site is placed at V. Find its distance from the nearest buoy, to 3 significant figures.

      • Distance = \(\sqrt{13}\) = 3.61 units

  3. A farm has 5 wells, A, B, C, D and E. 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 well. The vertices inside the region are P(4, 3), Q(4, 5.5), R(4.5, 2.25).

    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, 3): equidistant from A, C, E. Distance to A(1, 5): \(\sqrt{13}\) = 3.61
      • Q(4, 5.5): equidistant from A, D, E. Distance to A(1, 5): \(\sqrt{9.25}\) = 3.04
      • R(4.5, 2.25): equidistant from B, C, E. Distance to B(8, 1): \(\sqrt{13.8125}\) = 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 desert region has 6 villages, A, B, C, D, E and F. The diagram shows their Voronoi diagram; one unit is 10 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(3.5, 3.5), Q(3.75, 4), R(4, 2), S(7.5, 4).

    1234567891234567xyABCDEF

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

      • P(3.5, 3.5): equidistant from B, C, D. Distance to B(1, 1): \(\sqrt{12.5}\) = 3.54
      • Q(3.75, 4): equidistant from C, D, E. Distance to C(7, 3): \(\sqrt{11.5625}\) = 3.40
      • R(4, 2): equidistant from A, B, C. Distance to A(7, 1): \(\sqrt{10}\) = 3.16
      • S(7.5, 4): equidistant from C, E, F. Distance to C(7, 3): \(\sqrt{1.25}\) = 1.12
      • The greatest distance is at P(3.5, 3.5).

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

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

  2. Three schools, A(−5, 6), B(−3, 0) and C(7, 0), are in a valley. One unit is km. 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: \((-4, 3)\)
      • Gradient of AB = \(-3\); perpendicular gradient = \(\dfrac{1}{3}\)
      • Bisector: \(y=\dfrac{1}{3}x+\dfrac{13}{3}\)

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

      • Midpoint of BC: \((2, 0)\)
      • BC is horizontal, so the bisector is vertical: \(x=2\)

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

      • Solve \(y=\dfrac{1}{3}x+\dfrac{13}{3}\) and \(x=2\) together.
      • From the calculator: x = 2, y = 5

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

      • A(−5, 6): \(d=\sqrt{(2-(-5))^2+(5-6)^2}=\sqrt{50}\) = 7.07

  3. A coastal region has 6 ports, A, B, C, D, E and F. The diagram shows their Voronoi diagram; one unit is 10 km. A waste site is to be placed inside the region (a rectangle 9 units by 7 units), as far as possible from every port. The vertices inside the region are P(2.5, 4), Q(4.5, 4.5), R(5, 4), S(5.5, 4). A rule says the waste site must be at least 2.5 units from every port.

    1234567891234567xyABCDEF

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

      • P(2.5, 4): equidistant from C, E, F. Distance to C(2, 6): \(\sqrt{4.25}\) = 2.06
      • Q(4.5, 4.5): equidistant from B, C, F. Distance to B(7, 6): \(\sqrt{8.5}\) = 2.92
      • R(5, 4): equidistant from B, D, F. Distance to B(7, 6): \(\sqrt{8}\) = 2.83
      • S(5.5, 4): equidistant from A, B, D. Distance to A(8, 4): \(\sqrt{6.25}\) = 2.5
      • The greatest is Q(4.5, 4.5).

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

      • The best place is Q, which is 2.92 units from the nearest port.
      • 2.92 is at least 2.5, so the rule can be met at Q.

Stretch

  1. A forest has 5 fire lookouts, 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 fire lookout. The vertices inside the region are P(4.5, 3.5), Q(5, 1.5), R(7.5, 1.5), S(7.5, 3.5). 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 fire lookout.

      • P(4.5, 3.5): equidistant from B, C, E. Distance to B(3, 1): \(\sqrt{8.5}\) = 2.92
      • Q(5, 1.5): equidistant from A, B, C. Distance to A(7, 1): \(\sqrt{4.25}\) = 2.06
      • R(7.5, 1.5): equidistant from A, C, D. Distance to A(7, 1): \(\sqrt{0.5}\) = 0.707
      • S(7.5, 3.5): equidistant from C, D, E. Distance to C(7, 2): \(\sqrt{2.5}\) = 1.58
      • The greatest is 2.92.

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

      • Corner (0, 0): nearest B, d = \(\sqrt{10}\) = 3.16
      • Corner (9, 0): nearest A, d = \(\sqrt{5}\) = 2.24
      • Corner (9, 7): nearest E, d = \(\sqrt{8}\) = 2.83
      • Corner (0, 7): nearest B, d = \(\sqrt{45}\) = 6.71

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

      • The corner (0, 7) is 6.71, greater than any vertex (2.92).
      • On the boundary the farthest point can be a corner of the region.

  2. A city district has 7 hospitals, 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 hospital. The vertices inside the region are P(1.5, 1.5), Q(1.5, 3), R(5, 5), S(5.25, 3), T(5.5, 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(1.5, 1.5): equidistant from B, D, G. Distance to B(1, 2): \(\sqrt{0.5}\) = 0.707
      • Q(1.5, 3): equidistant from A, B, G. Distance to A(2, 4): \(\sqrt{1.25}\) = 1.12
      • R(5, 5): equidistant from A, E, F. Distance to A(2, 4): \(\sqrt{10}\) = 3.16
      • S(5.25, 3): equidistant from A, C, G. Distance to A(2, 4): \(\sqrt{11.5625}\) = 3.40
      • T(5.5, 3.5): equidistant from A, C, F. Distance to A(2, 4): \(\sqrt{12.5}\) = 3.54
      • The greatest is at T(5.5, 3.5).

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

      • Area = π r² = π × 12.5 = 39.3

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