9 Herman is packing some hampers. Each day, he packs three types of hamper: basic, standard and luxury. Each basic hamper has 6 tins, 9 packets and 6 bottles.
Each standard hamper has 9 tins, 6 packets and 12 bottles.
Each luxury hamper has 9 tins, 9 packets and 18 bottles.
Each day, Herman has 600 tins and 600 packets available, and he must use at least 480 bottles. Each day, Herman packs \(x\) basic hampers, \(y\) standard hampers and \(z\) luxury hampers.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find three inequalities in \(x , y\) and \(z\) that model the above constraints, simplifying each inequality.
2. On a particular day, Herman packs the same number of standard hampers as luxury hampers.
   1. Show that your answers in part (a) become $$\begin{aligned} x + 3 y & \leqslant 100
      3 x + 5 y & \leqslant 200
      x + 5 y & \geqslant 80 \end{aligned}$$
   2. On the grid opposite, draw a suitable diagram to represent Herman's situation, indicating the feasible region.
   3. Use your diagram to find the maximum total number of hampers that Herman can pack on that day.
   4. Find the number of each type of hamper that Herman packs that corresponds to your answer to part (b)(iii).
      (1 mark)