Standard +0.3 This is a straightforward linear programming question requiring standard constraint formulation and feasibility checking. Part (a) is trivial identification, part (b) uses the same method as the given example, part (c) is simple substitution, and part (d) requires systematic checking of integer solutions—all routine techniques for Further Maths Decision students with no novel problem-solving required.
7 Sam is making pies.
There is exactly enough pastry to make 7 large pies or 20 medium pies or 36 small pies, or some mixture of large, medium and small pies.
This is represented as a constraint \(180 x + 63 y + 35 z \leqslant 1260\).
Write down what \(\mathrm { X } , \mathrm { Y }\) and Z represent.
There is exactly enough filling to make 5 large pies or 12 medium pies or 18 small pies, or some mixture of large, medium and small pies.
Express this as a constraint of the form \(a x + b y + c z \leqslant d\), where \(a , b , c\) and \(d\) are integers.
The number of small pies must equal the total number of large and medium pies.
Show that making exactly 9 small pies is inconsistent with the constraints.
Determine the maximum number of large pies that can be made.
7 Sam is making pies.\\
There is exactly enough pastry to make 7 large pies or 20 medium pies or 36 small pies, or some mixture of large, medium and small pies.
This is represented as a constraint $180 x + 63 y + 35 z \leqslant 1260$.
\begin{enumerate}[label=(\alph*)]
\item Write down what $\mathrm { X } , \mathrm { Y }$ and Z represent.
There is exactly enough filling to make 5 large pies or 12 medium pies or 18 small pies, or some mixture of large, medium and small pies.
\item Express this as a constraint of the form $a x + b y + c z \leqslant d$, where $a , b , c$ and $d$ are integers.
The number of small pies must equal the total number of large and medium pies.
\item Show that making exactly 9 small pies is inconsistent with the constraints.
\item Determine the maximum number of large pies that can be made.
\begin{itemize}
\item Your reasoning should be in the form of words, calculations or algebra.
\item You must check that your solution is feasible.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2019 Q7 [12]}}