6\\
The network shows a system of pipes, where $S$ is the source and $T$ is the sink.\\
The lower and upper capacities, in litres per second, of each pipe are shown on each arc.\\
\includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-09_649_1399_580_424}

6
\begin{enumerate}[label=(\alph*)]
\item There is a feasible flow from $S$ to $T$.

6 (a) (i) Explain why arc $A D$ must be at its lower capacity.\\[0pt]
[1 mark]

6 (a) (ii) Explain why arc $B E$ must be at its upper capacity.\\[0pt]
[1 mark]

6
\item Explain why a flow of 11 litres per second through the network is impossible.\\[0pt]
[1 mark]

6
\item The network in Figure 2 shows a second system of pipes, where $S$ is the source and $T$ is the sink.

The lower and upper capacities, in litres per second, of each pipe are shown on each edge.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
  \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_760_1372_680_470}
\end{center}
\end{figure}

Figure 3 shows a feasible flow of 17 litres per second through the system of pipes.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
  \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_750_1371_1811_466}
\end{center}
\end{figure}

6 (c) (i) Using Figures 2 and 3, indicate on Figure 4 potential increases and decreases in the flow along each arc.\\[0pt]
[2 marks]

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 4}
  \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-11_749_1384_457_426}
\end{center}
\end{figure}

6 (c) (ii) Use flow augmentation on Figure 4 to find the maximum flow from $S$ to $T$.\\
You should indicate any flow augmenting paths clearly in the table below and modify the potential increases and decreases of the flow on Figure 4.\\[0pt]
[3 marks]

\begin{center}
\begin{tabular}{ | l | l | }
\hline
Augmenting Path & Flow \\
\hline
 &  \\
\hline
 &  \\
\hline
 &  \\
\hline
\end{tabular}
\end{center}

6 (c) (iii) Prove the flow found in part (d) (ii) is maximum.\\

6 (c) (iv) Due to maintenance work, the flow through node $E$ is restricted to 9 litres per second.\\[0pt]
Interpret the impact of this restriction on the maximum flow through the system of pipes. [2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Further Paper 3 Discrete  Q6 [11]}}