AQA Further AS Paper 2 Discrete Specimen — Question 8 8 marks

Exam BoardAQA
ModuleFurther AS Paper 2 Discrete (Further AS Paper 2 Discrete)
SessionSpecimen
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeGraphical optimization with objective line
DifficultyModerate -0.3 This is a standard linear programming question requiring formulation of constraints, graphical representation, and identification of the optimal vertex. While it has multiple constraints and requires careful plotting, it follows a completely routine template taught in Decision Maths with no novel problem-solving required. The 8 marks reflect the multiple steps (formulating inequalities, plotting lines, testing vertices) rather than conceptual difficulty.
Spec7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables
8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is \(\pounds 30\).
Each oak table takes 10 hours to make and the cost of materials is \(\pounds 70\).
Each month, the business has 360 hours available for making the tables and \(\pounds 2100\) available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of \(\pounds 40\) and each oak table sold gives the business a profit of \(\pounds 75\). Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]
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Question 8:
AnswerMarks Guidance
Answer/WorkingMark Guidance Notes
\(x =\) number of pine tables, \(y =\) number of oak tablesB1 Introduces two variables, defines at least one as 'number of'
\(30x + 70y \leq 2100\)B1 Correct inequality considering costs (OE)
\(6x + 10y \leq 360\)B1 Correct inequality considering labour (OE)
\(y \leq x,\ y \geq 10,\ x \geq 0\)B1 Correct three inequalities considering wholesaler
Line \(6x + 10y = 360\) through \((0, 36)\) and \((60, 0)\) AND line \(30x + 70y = 2100\) through \((0, 30)\) and \((70, 0)\)B1 Both lines correctly plotted (OE) — *see diagram*
\(y = x\) and \(y = 10\) both correctly and accurately plottedB1 Both lines plotted — *see diagram*
Feasible region correctly identified and labelled (F)B1 *See diagram*
35 Pine Tables and 15 Oak Tables (Profit \(= £2525\))A1 CAO Uses objective line with gradient \(-\frac{8}{15}\); uses optimal vertex of feasible region and states solution in context
Total: 8 marks
## Question 8:

| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $x =$ number of pine tables, $y =$ number of oak tables | B1 | Introduces two variables, defines at least one as 'number of' |
| $30x + 70y \leq 2100$ | B1 | Correct inequality considering costs (OE) |
| $6x + 10y \leq 360$ | B1 | Correct inequality considering labour (OE) |
| $y \leq x,\ y \geq 10,\ x \geq 0$ | B1 | Correct three inequalities considering wholesaler |
| Line $6x + 10y = 360$ through $(0, 36)$ and $(60, 0)$ **AND** line $30x + 70y = 2100$ through $(0, 30)$ and $(70, 0)$ | B1 | Both lines correctly plotted (OE) — *see diagram* |
| $y = x$ and $y = 10$ both correctly and accurately plotted | B1 | Both lines plotted — *see diagram* |
| Feasible region correctly identified and labelled (F) | B1 | *See diagram* |
| 35 Pine Tables and 15 Oak Tables (Profit $= £2525$) | A1 CAO | Uses objective line with gradient $-\frac{8}{15}$; uses optimal vertex of feasible region and states solution in context |

**Total: 8 marks**
8 A family business makes and sells two kinds of kitchen table.\\
Each pine table takes 6 hours to make and the cost of materials is $\pounds 30$.\\
Each oak table takes 10 hours to make and the cost of materials is $\pounds 70$.\\
Each month, the business has 360 hours available for making the tables and $\pounds 2100$ available for the materials.\\
Each month, the business sells all of its tables to a wholesaler.\\
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables.

Each pine table sold gives the business a profit of $\pounds 40$ and each oak table sold gives the business a profit of $\pounds 75$.

Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit.

Show clearly how you obtain your answer.\\[0pt]
[8 marks]

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\end{center}

\hfill \mbox{\textit{AQA Further AS Paper 2 Discrete  Q8 [8]}}
This paper (8 questions)
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Q1 1 Q2 1 Q3 2 Q4 6 Q5 8 Q6 11 Q7 4 Q8 8