Moderate -0.8 This is a straightforward linear programming formulation with clear constraints and objective. Students need only define two variables (number of each box type), write three inequalities from the egg constraints, and state the objective to maximize total boxes. It's a standard textbook-style question requiring no novel insight or complex reasoning.
3 Ben is packing eggs into boxes, labelled Town Box or Country Box.
Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs.
Ben has 253 chicken eggs and 151 duck eggs.
Ben wants to pack as many boxes as possible.
Formulate Ben's situation as a linear programming problem, defining any variables you introduce.
\(x =\) number of town boxes, \(y =\) number of country boxes
B1
Defines two variables representing the number of town boxes and the number of country boxes
Correct non-trivial constraint for chicken eggs or duck eggs, e.g. \(10x + 4y \leq 253\) or \(2x + 8y \leq 151\)
M1
Condone strict inequality
Both correct non-trivial constraints: \(10x + 4y \leq 253\) and \(2x + 8y \leq 151\)
A1
Both constraints fully correct
Maximise \(P = x + y\), subject to all constraints including \(x \geq 0,\ y \geq 0\), \(x\) and \(y\) are integers
A1
Formulates the full linear programming problem with statement of maximising \(x + y\) and all constraints fully correct
Question total: 4 marks
## Question 3:
| Answer | Mark | Guidance |
|--------|------|----------|
| $x =$ number of town boxes, $y =$ number of country boxes | B1 | Defines two variables representing the **number of** town boxes and the **number of** country boxes |
| Correct non-trivial constraint for chicken eggs or duck eggs, e.g. $10x + 4y \leq 253$ or $2x + 8y \leq 151$ | M1 | Condone strict inequality |
| Both correct non-trivial constraints: $10x + 4y \leq 253$ and $2x + 8y \leq 151$ | A1 | Both constraints fully correct |
| Maximise $P = x + y$, subject to all constraints including $x \geq 0,\ y \geq 0$, $x$ and $y$ are integers | A1 | Formulates the full linear programming problem with statement of maximising $x + y$ and all constraints fully correct |
**Question total: 4 marks**
3 Ben is packing eggs into boxes, labelled Town Box or Country Box.
Each Town Box must contain 10 chicken eggs and 2 duck eggs. Each Country Box must contain 4 chicken eggs and 8 duck eggs.
Ben has 253 chicken eggs and 151 duck eggs.
Ben wants to pack as many boxes as possible.
Formulate Ben's situation as a linear programming problem, defining any variables you introduce.\\
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2023 Q3 [4]}}