Route inspection with time constraint

4. \begin{figure}[h] \end{figure} Figure 4 represents the network of sensor wires used in a medical scanner. The number on each arc represents the length, in cm, of that section of wire. After production, each scanner is tested.
A machine will be programmed to inspect each section of wire.
It will travel along each arc of the network at least once, starting and finishing at A. Its route must be of minimum length.

1. Use the route inspection algorithm to find the length of a shortest inspection route. You must make your method and working clear. The machine will inspect 15 cm of wire per second.
2. Calculate the total time taken, in seconds, to test 120 scanners. It is now possible for the machine to start at one vertex and finish at a different vertex. An inspection route of minimum length is still required.
3. Explain why the machine should be programmed to start at a vertex with odd degree. Due to constraints at the factory, only B or D can be chosen as the starting point and there will also be a 2 second pause between tests.
4. Determine the new minimum total time now taken to test 120 scanners. You must state which vertex you are starting from and make your calculations clear.
   (4)

6 A council wants to monitor how long cars are being parked for in short-stay parking bays in a town centre. They employ a traffic warden to walk along the streets in the town centre and issue fines to drivers who park for longer than the stated time. The network below shows streets in the town centre which have short-stay parking bays. Each node represents a street corner and the weight of each arc represents the length, in metres, of the street. The short-stay parking bays are positioned along only one side of each street. \includegraphics[max width=\textwidth, alt={}, center]{22f11ce2-8d07-4f51-9326-b578d1e454f9-08_483_1108_733_466} The council assumes that the traffic warden will walk at an average speed of 4.8 kilometres per hour when not issuing fines. 6

1. To monitor all of the parking bays, the traffic warden needs to walk along every street in the town centre at least once, starting and finishing at the same street corner. Find the shortest possible time, to the nearest minute, it can take the traffic warden to monitor all of the parking bays. Fully justify your answer.
   6
2. Explain why the actual time for the traffic warden to walk along every street in the town centre at least once may be different to the value found in part (a).

6 A council wants to grit all of the roads on a housing estate. The network shows the roads on a housing estate. Each node represents a junction between two or more roads and the weight of each arc represents the length, in metres, of the road. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-08_1145_1458_539_292} The total length of all of the roads on the housing estate is 9175 metres.
In order to grit all of the roads, the council requires a gritter truck to travel along each road at least once. The gritter truck starts and finishes at the same junction. 6

1. The gritter truck starts gritting the roads at 7:00 pm and moves with an average speed of 5 metres per second during its journey. Find the earliest time for the gritter truck to have gritted each road at least once and arrived back at the junction it started from, giving your answer to the nearest minute. Fully justify your answer.
   [0pt] [6 marks]

   6	Explain how a refinement to the council's requirement, that the gritter truck must start and finish at the same junction, could reduce the time taken to grit all of the roads at least once. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   The planning involves producing an activity network for the project, which is shown in Figure 1 below. The duration of each activity is given in weeks. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-10_965_1600_559_221}
   \end{figure}

A company delivers parcels to houses in a village, using a van. The network below shows the roads in the village. Each node represents a road junction and the weight of each arc represents the length, in miles, of the road between the junctions. \includegraphics{figure_6} The total length of all of the roads in the village is 31.4 miles. On one particular day, the driver is due to make deliveries to at least one house on each road, so the van must travel along each road at least once. However, the driver has forgotten to add fuel to the van and it only has 4.5 litres of fuel to use to make its deliveries. The van uses, on average, 1 litre of fuel to travel 7.8 miles along the roads of this village. Whilst making each delivery, the driver turns off the van's engine so it does not use any fuel. Determine whether the van has enough fuel for the driver to make all of the deliveries to houses on each road of the village, starting and finishing at the same junction. Fully justify your answer. [6 marks]