3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{905f2578-e4b2-4d4d-8455-298170fd824b-4_781_1159_365_551}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 models the flow of fluid through a system of pipes from a source, S , to a sink, T . The weights on the arcs show the capacities of the corresponding pipes in litres per minute. Two cuts \(C _ { 1 }\) and \(C _ { 2 }\) are shown.
- Find the capacity of
- cut \(C _ { 1 }\)
- cut \(C _ { 2 }\)
- Using only the capacities of cuts \(C _ { 1 }\) and \(C _ { 2 }\) state what can be deduced about the maximum possible flow through the system.
- On Diagram 1 in the answer book, show how a flow of 120 litres per minute from S to T can be achieved. You do not need to apply the labelling procedure to find this flow.
- Prove that 120 litres per minute is the maximum possible flow through the system.
A new pipe is planned from S to A . Let the capacity of this pipe be \(x\) litres per minute.
- Find, in terms of \(x\) where necessary, the maximum possible flow through the new system.