Standard +0.3 This is a straightforward linear programming formulation question requiring translation of verbal constraints into inequalities. The machine time constraints are routine, the ordering constraints (x>y, y>z) are direct translations, and the percentage constraint (x≥0.4(x+y+z)) requires only basic algebraic manipulation. All steps are mechanical with no novel problem-solving required, making it slightly easier than average.
9 A factory makes three different types of widget: plain, bland and ordinary. Each widget is made using three different machines: \(A , B\) and \(C\).
Each plain widget needs 5 minutes on machine \(A , 12\) minutes on machine \(B\) and 24 minutes on machine \(C\).
Each bland widget needs 4 minutes on machine \(A , 8\) minutes on machine \(B\) and 12 minutes on machine \(C\).
Each ordinary widget needs 3 minutes on machine \(A\), 10 minutes on machine \(B\) and 18 minutes on machine \(C\).
Machine \(A\) is available for 3 hours a day, machine \(B\) for 4 hours a day and machine \(C\) for 9 hours a day.
The factory must make:
more plain widgets than bland widgets;
more bland widgets than ordinary widgets.
At least \(40 \%\) of the total production must be plain widgets.
Each day, the factory makes \(x\) plain, \(y\) bland and \(z\) ordinary widgets.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), writing your answers with simplified integer coefficients.
(8 marks)
9 A factory makes three different types of widget: plain, bland and ordinary. Each widget is made using three different machines: $A , B$ and $C$.
Each plain widget needs 5 minutes on machine $A , 12$ minutes on machine $B$ and 24 minutes on machine $C$.
Each bland widget needs 4 minutes on machine $A , 8$ minutes on machine $B$ and 12 minutes on machine $C$.
Each ordinary widget needs 3 minutes on machine $A$, 10 minutes on machine $B$ and 18 minutes on machine $C$.
Machine $A$ is available for 3 hours a day, machine $B$ for 4 hours a day and machine $C$ for 9 hours a day.
The factory must make:\\
more plain widgets than bland widgets;\\
more bland widgets than ordinary widgets.\\
At least $40 \%$ of the total production must be plain widgets.\\
Each day, the factory makes $x$ plain, $y$ bland and $z$ ordinary widgets.\\
Formulate the above situation as 6 inequalities, in addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, writing your answers with simplified integer coefficients.\\
(8 marks)
\hfill \mbox{\textit{AQA D1 2006 Q9 [8]}}