6 The diagram shows a nature reserve which has its entrance at \(A\), eight information signs at \(B , C , \ldots , I\), and fifteen grass paths. The length of each grass path is given in metres.
The total length of the grass paths is 1465 metres.
\includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-10_812_1192_584_424} To cut the grass, Ashley starts at the entrance and drives a mower along every grass path in the nature reserve. The mower moves at 7 kilometres per hour. 6

1. Find the least possible time that it takes for Ashley to cut the grass on all fifteen paths in the nature reserve and return to the entrance. Fully justify your answer.
   6
2. Brook visits every information sign in the nature reserve to update them, starting and finishing at the entrance. For the eight information signs, the minimum connecting distance of the grass paths is 510 metres. 6
3. 1. Determine a lower bound for the distance Brook walks to visit every information sign.
      Fully justify your answer.
      [0pt] [2 marks]
      6
4. (ii) Using the nearest neighbour algorithm starting from the entrance, determine an upper bound for the distance Brook walks to visit every information sign.
   [0pt] [2 marks]
   6
5. Brook takes one minute to update the information at one information sign. Brook walks on the grass paths at an average speed of 5 kilometres per hour. Ashley and Brook start from the entrance at the same time. 6
6. 1. Use your answers from parts (a) and (b) to show that Ashley and Brook will return to the entrance at approximately the same time. Fully justify your answer.
      6
7. (ii) State an assumption that you have used in part (c)(i).
   \includegraphics[max width=\textwidth, alt={}, center]{dcf97b92-d067-41d4-89a6-ea5bab9ea4ff-13_2488_1716_219_153}
   \(7 \quad\) Ali and Bex play a zero-sum game. The game is represented by the following pay-off matrix for Ali.

   \multirow{2}{*}{}	Bex
   	Strategy	\(\mathbf { B } _ { \mathbf { 1 } }\)	\(\mathbf { B } _ { \mathbf { 2 } }\)	\(\mathbf { B } _ { \mathbf { 3 } }\)
   \multirow{4}{*}{Ali}	\(\mathbf { A } _ { \mathbf { 1 } }\)	2	-1	3
   	\(\mathbf { A } _ { \mathbf { 2 } }\)	-4	-2	2
   	\(\mathbf { A } _ { \mathbf { 3 } }\)	0	1	1
   	\(\mathrm { A } _ { 4 }\)	-3	2	-2