6 Each day, a factory makes three types of widget: basic, standard and luxury. The widgets produced need three different components: type \(A\), type \(B\) and type \(C\). Basic widgets need 6 components of type \(A , 6\) components of type \(B\) and 12 components of type \(C\).
Standard widgets need 4 components of type \(A , 3\) components of type \(B\) and 18 components of type \(C\).
Luxury widgets need 2 components of type \(A , 9\) components of type \(B\) and 6 components of type \(C\).
Each day, there are 240 components of type \(A\) available, 300 of type \(B\) and 900 of type \(C\).
Each day, the factory must use at least twice as many components of type \(C\) as type \(B\).
Each day, the factory makes \(x\) basic widgets, \(y\) standard widgets and \(z\) luxury widgets.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find four inequalities in \(x , y\) and \(z\) that model the above constraints, simplifying each inequality.
2. Each day, the factory makes the maximum possible number of widgets. On a particular day, the factory must make the same number of luxury widgets as basic widgets.
   1. Show that your answers in part (a) become $$2 x + y \leqslant 60 , \quad 5 x + y \leqslant 100 , \quad x + y \leqslant 50 , \quad y \geqslant x$$
   2. Using the axes opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region.
   3. Find the total number of widgets made on that day.
   4. Find all possible combinations of the number of each type of widget made that correspond to this maximum number.