Standard +0.8 This linear programming formulation requires students to translate multiple constraint types (resource limits, comparative constraints, and a percentage constraint) into algebraic inequalities. The percentage constraint requiring algebraic manipulation to eliminate fractions and the need to carefully track coefficients across three variables makes this more challenging than standard LP formulations, though it remains within D1 scope.
8 A factory packs three different kinds of novelty box: red, blue and green. Each box contains three different types of toy: \(\mathrm { A } , \mathrm { B }\) and C .
Each red box has 2 type A toys, 3 type B toys and 4 type C toys.
Each blue box has 3 type A toys, 1 type B toy and 3 type C toys.
Each green box has 4 type A toys, 5 type B toys and 2 type C toys.
Each day, the maximum number of each type of toy available to be packed is 360 type A, 300 type B and 400 type C.
Each day, the factory must pack more type A toys than type B toys.
Each day, the total number of type A and type B toys that are packed must together be at least as many as the number of type C toys that are packed.
Each day, at least \(40 \%\) of the total toys that are packed must be type C toys.
Each day, the factory packs \(x\) red boxes, \(y\) blue boxes and \(z\) green boxes.
Formulate the above situation as 6 inequalities, in addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), simplifying your answers.
8 A factory packs three different kinds of novelty box: red, blue and green. Each box contains three different types of toy: $\mathrm { A } , \mathrm { B }$ and C .
Each red box has 2 type A toys, 3 type B toys and 4 type C toys.\\
Each blue box has 3 type A toys, 1 type B toy and 3 type C toys.\\
Each green box has 4 type A toys, 5 type B toys and 2 type C toys.\\
Each day, the maximum number of each type of toy available to be packed is 360 type A, 300 type B and 400 type C.
Each day, the factory must pack more type A toys than type B toys.\\
Each day, the total number of type A and type B toys that are packed must together be at least as many as the number of type C toys that are packed.
Each day, at least $40 \%$ of the total toys that are packed must be type C toys.\\
Each day, the factory packs $x$ red boxes, $y$ blue boxes and $z$ green boxes.\\
Formulate the above situation as 6 inequalities, in addition to $x \geqslant 0 , y \geqslant 0$ and $z \geqslant 0$, simplifying your answers.
\hfill \mbox{\textit{AQA D1 2010 Q8 [8]}}