Home / IB Mathematics SL 3.6 sites, vertices, edges, cells AI SL Paper 1- Exam Style Questions

IB Mathematics SL 3.6 sites, vertices, edges, cells AI SL Paper 1- Exam Style Questions- New Syllabus

IB DP Maths AI SL Paper 1 - All Topics
Question 
The Voronoi diagram shows three identical cellular phone towers, T1, T2, and T3. A fourth tower, T4, is located in the shaded region. The dashed lines represent the edges of the Voronoi diagram.
Horizontal scale: 1 unit represents 1 km.
Vertical scale: 1 unit represents 1 km.
(a) Alex stands in the shaded region. Explain why Alex receives the strongest signal from tower T4. [2]
(b) Tower T2 is at coordinates (-9, 5), and the edge connecting vertices A and B has equation \( y = 3 \). Determine the coordinates of tower T4. [2]
(c) Tower T1 is at coordinates (-13, 3). Determine the gradient of the edge of the Voronoi diagram between towers T1 and T2. [2]
▶️ Answer/Explanation
Markscheme
(a)
In a Voronoi diagram, each region contains points closest to its respective tower. A1
Since Alex is in the shaded region (T4’s Voronoi cell), Alex is closer to T4 than to T1, T2, or T3. Signal strength is stronger from the nearest tower. A1
[2 marks]
(b)
T2 coordinates: (-9, 5). Edge A-B: \( y = 3 \). A1
T4 is in the shaded region below \( y = 3 \), vertically aligned with T2 at \( x = -9 \), and marked at \( y = 1 \) in the diagram.
Coordinates of T4: (-9, 1). A1
[2 marks]
(c)
T1: (-13, 3), T2: (-9, 5). M1
Gradient of T1-T2: \( \dfrac{5 – 3}{-9 – (-13)} = \dfrac{2}{4} = \dfrac{1}{2} \).
Voronoi edge is the perpendicular bisector, so the gradient is the negative reciprocal: \( -\dfrac{1}{\dfrac{1}{2}} = -2 \). A1
[2 marks]
Total Marks: 6
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