5 The diagram represents a map of seven locations (A to G) and the direct road distances (km) between some of them.
\includegraphics[max width=\textwidth, alt={}, center]{9fe422a0-c498-4ad5-bdfc-f70482960d39-4_382_771_356_644} A delivery driver needs to start from his depot at D , make deliveries at each of \(\mathrm { A } , \mathrm { B } , \mathrm { F }\) and G , and finish at D .

1. Write down a route from A to G of length 70 km . The table shows the length of the shortest path between some pairs of places.

   	D	A	B	F	G
   D	-
   A		-			70
   B			-	84
   F			84	-
   G		70			-

2. (a) Complete the table.
   (b) Use the nearest neighbour method on the table, starting at D , to find the length of a cycle through \(\mathrm { D } , \mathrm { A } , \mathrm { B } , \mathrm { F }\) and G , ignoring possibly repeating E and C .
3. By first considering the table with the row and column for D removed, find a lower bound for the distance that the driver must travel.
4. What can you conclude from your previous answers about the distance that the driver must travel? A new road is constructed between D and F . Using this road the driver starts from D , makes the deliveries and returns to D having travelled just 172 km .
5. Find the length of the new road if
   (a) the driver does not return to D until all the deliveries have been made,
   (b) the driver uses the new road twice in making the deliveries.