Maximize spanning tree weight

7 A construction company has built eight wind turbines on a moorland site. The network below shows nodes which represent the site entrance, \(E\), and the wind turbine positions, \(S , T , \ldots , Z\) \includegraphics[max width=\textwidth, alt={}, center]{372edcfa-c3cd-4c83-89e9-2bb5fd9825f1-12_924_1294_479_356} Each arc represents an access track with its length given in metres.
These 17 tracks were created in order to build the wind turbines. Eight of the tracks are to be retained so that each turbine can be accessed for maintenance, directly or indirectly, from the site entrance. The other nine tracks will be removed. 7

1. 1. To save money the construction company wants to maximise the total length of the eight tracks to be retained. Determine which tracks the construction company should retain.
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      1. (ii) Find the total length of the eight tracks that are to be retained. 7
   2. The total length of the 17 tracks is 14.6 km
      The cost of removing all 17 tracks would be \(\pounds 87,600\) Using your answer to part (a)(ii), calculate an estimate for the cost of removing the nine tracks that will not be retained.
      [0pt] [2 marks]
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   3. Comment on why the modelling used in part (b) may not give an accurate estimate for the cost of removing the nine tracks.

3 A mining company wants to open a new mine in an area where the ground contains a precious metal. The mining company has carried out a survey of the area. The network below shows nodes which represent the entrance to the new mine, \(X\), and the 8 ventilation shafts, \(A , B , \ldots , H\), which have been installed to prevent the build up of dangerous gases underground. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-04_846_1228_623_404} Each arc represents a possible underground tunnel which could be mined.
The weight on each arc represents the estimated amount of precious metal in that possible underground tunnel in tonnes. Due to geological reasons, the mining company can only create 8 underground tunnels. All 8 ventilation shafts must be accessible from the entrance of the mine. 3

1. 1. The mining company wants to maximise the amount of precious metal it can extract from the new mine. Determine the tunnels the mining company should use.
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      1. (ii) Estimate the maximum amount of precious metal the mining company can extract from the new mine. 3
   2. Comment on why the maximum amount of precious metal the mining company can extract from the new mine may be different from your answer to part (a)(ii).
      [0pt] [2 marks]
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   3. Before the mining company begins work on the new mine, a government survey prevents the mining company drilling the tunnel represented by \(C F\). Determine the effect, if any, the government survey has on your answers to part (a)(i) and part (a)(ii).