| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 2 Discrete (Further AS Paper 2 Discrete) |
| Year | 2020 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.3 This is a standard linear programming question requiring interpretation of a given feasible region diagram and use of the objective line method. Students must identify constraints from the table, explain the objective line, read off the optimal vertex, and state a standard assumption. While it involves multiple steps (3+2+1 marks), each component is routine for Further Maths students who have studied Decision Maths, requiring no novel insight beyond textbook techniques. |
| Spec | 7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
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| Standard | 12 | 6 | 3 | \(\pounds 2.50\) | ||||||||||
| Luxury | 6 | 6 | 9 | \(\pounds 2.00\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x =\) number of standard boxes, \(y =\) number of luxury boxes | B1 | Introduces two variables, defines at least one as 'number of' |
| Line drawn on graph is \(y = -2x + 150\); constraint for total number of rolls: \(12x + 6y \leq 900\) | B1 | Finds equation/inequality for number of rolls |
| The line forms the boundary of the region \(12x + 6y \leq 900\), which is the constraint for total number of rolls | E1 | Explains clearly how the line represents the boundary of the constraint for total number of rolls |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Teacakes: \(6x + 6y \leq 600\); Croissants: \(3x + 9y \leq 450\) | M1 | Finds at least one non-trivial inequality considering total numbers of teacakes or croissants |
| Correct straight lines for the two non-trivial constraints with consistent use of variables | A1 | Produces correct straight lines |
| Identifies feasible region | A1F | Correct feasible region identified |
| Uses an objective line or uses a vertex of feasible region | M1 | |
| Optimal vertex at \((60, 30)\) | A1 | Obtains correct coordinates of optimal vertex |
| Maximum profit \(= 60 \times 2.5 + 30 \times 2 = £210\) | A1 | Calculates correct daily profit of £210 |
| Answer | Marks | Guidance |
|---|---|---|
| Graph showing feasible region (FR) with two constraint lines plotted, one from approximately (0, 140) to (80, 0) and another from (0, 45) to (100, 0) | See graph | Lines correctly drawn with feasible region indicated |
| Answer | Marks | Guidance |
|---|---|---|
| States a plausible assumption that recognises a limitation of the model (e.g. refers to demand, capacity, resources) | 3.5b | B1 |
## Question 7(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x =$ number of standard boxes, $y =$ number of luxury boxes | B1 | Introduces two variables, defines at least one as 'number of' |
| Line drawn on graph is $y = -2x + 150$; constraint for total number of rolls: $12x + 6y \leq 900$ | B1 | Finds equation/inequality for number of rolls |
| The line forms the boundary of the region $12x + 6y \leq 900$, which is the constraint for total number of rolls | E1 | Explains clearly how the line represents the boundary of the constraint for total number of rolls |
## Question 7(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Teacakes: $6x + 6y \leq 600$; Croissants: $3x + 9y \leq 450$ | M1 | Finds at least one non-trivial inequality considering total numbers of teacakes or croissants |
| Correct straight lines for the two non-trivial constraints with consistent use of variables | A1 | Produces correct straight lines |
| Identifies feasible region | A1F | Correct feasible region identified |
| Uses an objective line or uses a vertex of feasible region | M1 | |
| Optimal vertex at $(60, 30)$ | A1 | Obtains correct coordinates of optimal vertex |
| Maximum profit $= 60 \times 2.5 + 30 \times 2 = £210$ | A1 | Calculates correct daily profit of £210 |
# Question 7(a)(ii) (cont):
Graph showing feasible region (FR) with two constraint lines plotted, one from approximately (0, 140) to (80, 0) and another from (0, 45) to (100, 0) | See graph | Lines correctly drawn with feasible region indicated
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# Question 7(b):
States a plausible assumption that recognises a limitation of the model (e.g. refers to demand, capacity, resources) | 3.5b | B1 | To maximise the profit, the bakery must sell all the bakery boxes.
**Total: 10**
---7 Robyn manages a bakery.
Each day the bakery bakes 900 rolls, 600 teacakes and 450 croissants.\\
The bakery sells two types of bakery box which contain rolls, teacakes and croissants, as shown in the table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
\begin{tabular}{ l }
Type of \\
bakery box \\
\end{tabular} & \begin{tabular}{ c }
Number of \\
rolls \\
\end{tabular} & \begin{tabular}{ c }
Number of \\
teacakes \\
\end{tabular} & \begin{tabular}{ c }
Number of \\
croissants \\
\end{tabular} & \begin{tabular}{ c }
Profit per \\
box sold \\
\end{tabular} \\
\hline
Standard & 12 & 6 & 3 & $\pounds 2.50$ \\
\hline
Luxury & 6 & 6 & 9 & $\pounds 2.00$ \\
\hline
\end{tabular}
\end{center}
Robyn formulates a linear programming problem to find the maximum profit the bakery can make from selling the bakery boxes.
7
\begin{enumerate}[label=(\alph*)]
\item Part of a graphical method to solve this linear programming problem is shown on Figure 1.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-14_1283_1196_1267_424}
\end{center}
\end{figure}
7 (a) (i) Explain how the line shown on Figure 1 relates to the linear programming problem.
Clearly define any variables that you introduce.\\[0pt]
[3 marks]\\
7 (a) (ii) Use Figure 1 to find the maximum profit that the bakery can make from selling bakery boxes.\\
7
\item State an assumption that you have made in part (a)(ii).\\[0pt]
[1 mark]\\
\includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-17_2493_1732_214_139}
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete 2020 Q7 [10]}}