5 Answer this question on the insert provided. The network represents a system of irrigation channels along which water can flow. The weights on the arcs represent the maximum flow in litres per second.
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1. Calculate the capacity of the cut that separates \(\{ S , B , C , E \}\) from \(\{ A , D , F , G , H , T \}\).
2. Explain why neither arc \(S C\) nor arc \(B C\) can be full to capacity. Explain why the arcs \(E F\) and \(E H\) cannot both be full to capacity. Hence find the maximum flow along arc \(H T\). When arc \(H T\) carries its maximum flow, what is the flow along arc \(H G\) ?
3. Show a flow of 58 litres per second on the diagram in the insert, and find a cut of capacity 58. The direction of flow in \(H G\) is reversed.
4. Use the diagram in the insert to show the excess capacities and potential backflows for your flow from part (iii) in this case.
5. Without augmenting the labels from part (iv), write down flow augmenting routes to enable an additional 2 litres per second to flow from \(S\) to \(T\).
6. Show your augmented flow on the diagram in the insert. Explain how you know that this flow is maximal.