Standard +0.3 This is a straightforward linear programming formulation question requiring students to translate word constraints into inequalities. While it involves five constraints including a percentage condition, the mathematical operations are routine: setting up resource constraints (≤ for availability) and minimum usage constraints (≥ for totals). The percentage constraint requires slightly more thought but is still mechanical. This is slightly easier than average as it's pure formulation with no optimization or graphical work required.
8 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges produced need three different components: type \(A\), type \(B\) and type \(C\).
Basic hinges need 2 components of type \(A , 3\) components of type \(B\) and 1 component of type \(C\).
Standard hinges need 4 components of type \(A , 2\) components of type \(B\) and 3 components of type \(C\).
Luxury hinges need 3 components of type \(A\), 4 components of type \(B\) and 5 components of type \(C\).
Each day, there are 360 components of type \(A\) available, 270 of type \(B\) and 450 of type \(C\).
Each day, the factory must use at least 720 components in total.
Each day, the factory must use at least \(40 \%\) of the total components as type \(A\).
Each day, the factory makes \(x\) basic hinges, \(y\) standard hinges and \(z\) luxury hinges.
In addition to \(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\), find five inequalities, each involving \(x , y\) and \(z\), which must be satisfied. Simplify each inequality where possible.
8 Each day, a factory makes three types of hinge: basic, standard and luxury. The hinges produced need three different components: type $A$, type $B$ and type $C$.
Basic hinges need 2 components of type $A , 3$ components of type $B$ and 1 component of type $C$.
Standard hinges need 4 components of type $A , 2$ components of type $B$ and 3 components of type $C$.
Luxury hinges need 3 components of type $A$, 4 components of type $B$ and 5 components of type $C$.
Each day, there are 360 components of type $A$ available, 270 of type $B$ and 450 of type $C$.
Each day, the factory must use at least 720 components in total.\\
Each day, the factory must use at least $40 \%$ of the total components as type $A$.\\
Each day, the factory makes $x$ basic hinges, $y$ standard hinges and $z$ luxury hinges.\\
In addition to $x \geqslant 0 , y \geqslant 0 , z \geqslant 0$, find five inequalities, each involving $x , y$ and $z$, which must be satisfied. Simplify each inequality where possible.
\hfill \mbox{\textit{AQA D1 2008 Q8 [8]}}