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Date	November Example question	Marks available	1	Reference code	EXN.2.SL.TZ0.5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	State	Question number	5	Adapted from	N/A

Question

The living accommodation on a university campus is in the shape of a rectangle with sides of length $200 m$ $200 m$ and $300 m$ $300 m$ .

There are three offices for the management of the accommodation set at the points $A$ $A$ , $B$ $B$ and $C$ $C$ . These offices are responsible for all the students in the areas closest to the office. These areas are shown on the Voronoi diagram below. On this coordinate system the positions of $A$ $A$ , $B$ $B$ and $C$ $C$ are $(100, 160)$ $(100, 160)$ , $(100, 40)$ $(100, 40)$ and $(250, 100)$ $(250, 100)$ respectively.

The equation of the perpendicular bisector of $[AC]$ $[AC]$ is $5 x - 2 y = 615$ $5 x - 2 y = 615$ .

The manager of office $C$ $C$ believes that he has more than one third of the area of the campus to manage.

Find the area of campus managed by office $C$ $C$ .

[3]

Hence or otherwise find the areas managed by offices $A$ $A$ and $B$ $B$ .

[3]

State a further assumption that must be made in order to use area covered as a measure of whether or not the manager of office $C$ $C$ is responsible for more students than the managers of offices $A$ $A$ and $B$ $B$ .

[1]

A new office is to be built within the triangle formed by $A$ $A$ , $B$ $B$ and $C$ $C$ , at a point as far as possible from the other three offices.

Find the distance of this office from each of the other offices.

[2]

Markscheme

Divides area into two appropriate shapes

For example,

Area of triangle $(\frac{1}{2} \times 200 \times 40 =) 4000 m^{2}$ $(\frac{1}{2} \times 200 \times 40 =) 4000 m^{2}$ (A1)

Area of rectangle $(200 \times 97) = 19400 m^{2}$ $(200 \times 97) = 19400 m^{2}$ (A1)

$23400 (m^{2})$ $23400 (m^{2})$ A1

Note: The area can be found using different divisions. Award A1 for any two correct areas found and A1 for the final answer.

[3 marks]

EITHER

$\frac{200 \times 300 - 23400}{2} = 18300 m^{2}$ $\frac{200 \times 300 - 23400}{2} = 18300 m^{2}$ (M1)A1

$\frac{1}{2} \times 100 \times (203 + 163) = 18300 m^{2}$ $\frac{1}{2} \times 100 \times (203 + 163) = 18300 m^{2}$ (M1)A1

THEN

Area managed by both offices $A$ $A$ and $B$ $B$ is $18300 m^{2}$ $18300 m^{2}$ A1

[3 marks]

Density of accommodation/students is uniform R1

[1 mark]

$250 - 163 = 87 (m)$ $250 - 163 = 87 (m)$ (M1)A1

Note: M1 is for an attempt to find the distance from the intersection point to one of the offices.

[2 marks]

Examiners report

[N/A]

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.6—Voronoi diagrams

Show 27 related questions

22M.3.AHL.TZ1.2c.i:
Find the equation of the perpendicular bisector of $[IF]$ $[IF]$ .
22M.3.AHL.TZ1.2b:
Sketch a Voronoi diagram showing the regions within the metropolitan area that are closest to each town.
22M.3.AHL.TZ1.2c.iii:
Sketch this new Voronoi diagram showing the regions within the metropolitan area which are closest to each town.
EXM.2.SL.TZ0.4a:
Sketch a Voronoi diagram to represent this information.
22M.3.AHL.TZ1.2e.ii:
Make one possible criticism of the council’s choice of location.
22M.3.AHL.TZ1.2c.ii:
Given that the coordinates of one vertex of the new Voronoi diagram are $(20, 37.5)$ $(20, 37.5)$ , find the coordinates of the other two vertices within the metropolitan area.
22M.3.AHL.TZ1.2e.i:
Find the location of the toxic waste dump, given that this location is not on the edge of the metropolitan area.
21M.1.SL.TZ1.5a:
Explain why Tim will receive the strongest signal from tower $T4$ $T4$ .
21N.1.SL.TZ0.7b.i:
Write down the equation of the perpendicular bisector of $[PC]$ $[PC]$ .
EXN.2.SL.TZ0.5e:
A new office is to be built within the triangle formed by $A$ $A$ , $B$ $B$ and $C$ $C$ , at a point as far as possible from the other three offices.

Find the distance of this office from each of the other offices.
22M.2.SL.TZ2.3a:
Find the midpoint of $[BD]$ $[BD]$ .
22M.3.AHL.TZ1.2a:
The model assumes that each town is positioned at a single point. Describe possible circumstances in which this modelling assumption is reasonable.
EXM.2.SL.TZ0.4b:
Sketch a new Voronoi diagram to represent this new situation.
EXM.2.SL.TZ0.4c:
State what the shape of the land, owned by the youngest child, is.
SPM.1.SL.TZ0.7c:
In the context of the question, explain the significance of the Voronoi cell containing site E.
SPM.1.SL.TZ0.7a:
Calculate the gradient of the line segment AE.
EXM.2.SL.TZ0.4e:
Find how much land an older child has lost.
EXM.1.SL.TZ0.4:
The diagram below is part of a Voronoi diagram.

Diagram not to scale

A and B are sites with B having the co-ordinates of (4, 6). L is an edge; the equation of this perpendicular bisector of the line segment from A to B is $y = - 2 x + 9$ $y = - 2x + 9$

Find the co-ordinates of the point A.
EXM.2.SL.TZ0.4f:
State, with a reason, if all five children now own an equal amount of land.
21M.1.SL.TZ1.5b:
Write down the coordinates of tower $T4$ $T4$ .
21M.1.SL.TZ1.5c:
Tower $T1$ $T1$ has coordinates $(- 13, 3)$ $(- 13, 3)$ .

Find the gradient of the edge of the Voronoi diagram between towers $T1$ $T1$ and $T2$ $T2$ .
SPM.1.SL.TZ0.7b:
Find the equation of the line which would complete the Voronoi cell containing site E.

Give your answer in the form $a x + b y + d = 0$ $ax + by + d = 0$ where $a$ $a$ , $b$ $b$ , $d \in \mathbb{Z}$ .
EXM.2.SL.TZ0.4d:
Find the area of the youngest child’s land.
EXN.2.SL.TZ0.5b:
Find the area of campus managed by office $C$ .
EXN.2.SL.TZ0.5c:
Hence or otherwise find the areas managed by offices $A$ and $B$ .
21N.1.SL.TZ0.7a:
Show that the new station should be built at $P$ .
21N.1.SL.TZ0.7b.ii:
Hence draw the missing boundaries of the region around $P$ on the following diagram.

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Topic 3—Geometry and trigonometry

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