4. (a) Explain what is meant by the term 'path'.
(2)
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{2ae673c0-206a-468b-ae6f-ac55e5970f7b-4_957_1414_408_363}
\end{figure}
Figure 3 shows a network of cycle tracks. The number on each edge represents the length, in miles, of that track. Mary wishes to cycle from \(A\) to \(I\) as part of a cycling holiday. She wishes to minimise the distance she travels.
(b) Use Dijkstra's algorithm to find the shortest path from \(A\) to \(I\). Show all necessary working in the boxes in Diagram 1 in the answer book. State your shortest path and its length.
(6)
(c) Explain how you determined the shortest path from your labelling.
(2)
Mary wants to visit a theme park at \(E\).
(d) Find a path of minimal length that goes from \(A\) to \(I\) via \(E\) and state its length.
(2)
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 4}
\includegraphics[alt={},max width=\textwidth]{2ae673c0-206a-468b-ae6f-ac55e5970f7b-5_688_1469_322_315}
\end{figure}
An engineering project is modelled by the activity network shown in Figure 4. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest time.
(a) Calculate the early time and late time for each event. Write these in boxes in Diagram 1 in the answer book.
(b) State the critical activities.
(c) Find the total float on activities \(D\) and \(F\). You must show your working.
(d) On the grid in the answer book, draw a cascade (Gantt) chart for this project.
The chief engineer visits the project on day 15 and day 25 to check the progress of the work. Given that the project is on schedule,
(e) which activities must be happening on each of these two days?
(3)