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A network of running tracks connects the points A, B, C, D, E, F, G, H, as shown - HSC - SSCE Mathematics Standard - Question 19 - 2023 - Paper 1
Question 19
A network of running tracks connects the points A, B, C, D, E, F, G, H, as shown. The number on each edge represents the time, in minutes, that a typical runner shou... show full transcript
Worked Solution & Example Answer:A network of running tracks connects the points A, B, C, D, E, F, G, H, as shown - HSC - SSCE Mathematics Standard - Question 19 - 2023 - Paper 1
Step 1
Which path could a typical runner take to run from point A to point D in the shortest time?
Answer
To determine the shortest path from point A to point D, we can analyze the given network.
Evaluating the paths:
- A to B (6) then B to C (11) and C to D (9) totals to 26 minutes.
- A to F (5) then F to G (5) and G to D (9) totals to 19 minutes.
- Alternatively, A to F (5) then F to E (4) and E to D (15) totals to 24 minutes.
- Finally, A to B (6) then B to F (5) and F to D (10) totals to 21 minutes.
The quickest route is A -> F -> G -> D, taking a total of 19 minutes. Therefore, a correct path is: AFGD.
Step 2
Is it a minimum spanning tree? Give a reason for your answer.
Answer
It is not a minimum spanning tree. The path BC is not the shortest path to connect B to C in the tree, as there exists a shorter alternative route available. Minimum spanning trees should connect all vertices with the least total edge weight; hence BC can be improved to ensure optimal connectivity.