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The table below shows the distances, in kilometres, between a number of towns - HSC - SSCE Mathematics Standard - Question 20 - 2022 - Paper 1
Question 20
The table below shows the distances, in kilometres, between a number of towns. Towns Snowtown (S) Clairville (C) Yuma (Y) Bosten (B) Morrella (M) (S) ... show full transcript
Worked Solution & Example Answer:The table below shows the distances, in kilometres, between a number of towns - HSC - SSCE Mathematics Standard - Question 20 - 2022 - Paper 1
Step 1
Draw a weighted network diagram
Answer
To represent the distances between the towns (S, C, Y, B, M), we draw a weighted graph as follows:
- Connect S to Y with a weight of 280 km.
- Connect S to B with a weight of 275 km.
- Connect C to Y with a weight of 60 km.
- Connect C to B with a weight of 150 km.
- Connect Y to M with a weight of 530 km.
- Connect B to M with a weight of 790 km.
Thus, the resultant diagram visually depicts all the towns and their respective distances.
S
/|\
280/ | \275
/ | \
Y---C---B
|\ | /
530\ 150/
\ /
M
Step 2
Draw the minimum spanning tree and determine its length
Answer
To find the minimum spanning tree (MST), we assess the distances and select edges that connect all towns with the least total distance:
- Start with the smallest weights: S to Y (280 km), S to B (275 km), and C to Y (60 km).
- Include the connection from C to B (150 km) to finalize connections.
- Avoid any cycles while maintaining connectivity among all vertices.
This results in:
S
/ \
280/ \275
/
Y---C
|
60 |
M
Calculating total length of the MST: Length of minimum spanning tree = 275 + 280 + 60 + 150 + 530 = 1015 km.