| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a standard D1 linear programming question requiring routine formulation of constraints from a word problem, graphical solution, and optimization. The constraints are straightforward to extract, and the graphical method is a well-practiced technique. While multi-step, it requires no novel insight—just systematic application of textbook methods. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| All inequalities must be as below | B1 | Both |
| \(x \leq 100\), \(y \leq 80\) | B1 | |
| \(x + y \geq 60\) | B1 | |
| \(x < y\) | B1 | |
| \(2x + 8y \geq 320\) | B1 | 5 |
| (minimise \(C =\)) \(1.5x + 3y\) |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | \(x = 100, y = 80\) within \(\frac{1}{2}\) square from (0,0) to (80,80) | |
| B1 × 3 | Other lines | |
| B1 | Feasible Region CAO (must have scored B4 for drawing lines) (condone \(x = y\) as solid line) | |
| B1 | 6 | An Objective Line with gradient –0.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Considering an extreme point in their region. Min at intersect of \(x + y = 60\) and \(x + 4y = 160\) | M1 | PI by indication on diagram or \(x = 26\frac{2}{3}\) \(y = 33\frac{1}{3}\) |
| Considering a pair of integer values where \(26 \leq x \leq 28\), \(32 \leq y \leq 34\) | M1 | |
| (\(C =\)) £141 at (26, 34) or £141 at (28, 33) | A1 | 4 |
| Total | 15 |
## 6(a)
| All inequalities must be as below | B1 | Both |
| $x \leq 100$, $y \leq 80$ | B1 | |
| $x + y \geq 60$ | B1 | |
| $x < y$ | B1 | |
| $2x + 8y \geq 320$ | B1 | 5 | OE |
| (minimise $C =$) $1.5x + 3y$ | | |
## 6(b)
| | B1 | $x = 100, y = 80$ within $\frac{1}{2}$ square from (0,0) to (80,80) |
| | B1 × 3 | Other lines |
| | B1 | Feasible Region CAO (must have scored B4 for drawing lines) (condone $x = y$ as solid line) |
| | B1 | 6 | An Objective Line with gradient –0.5 |
## 6(c)
| Considering an extreme point in their region. Min at intersect of $x + y = 60$ and $x + 4y = 160$ | M1 | PI by indication on diagram or $x = 26\frac{2}{3}$ $y = 33\frac{1}{3}$ |
| Considering a pair of integer values where $26 \leq x \leq 28$, $32 \leq y \leq 34$ | M1 | |
| ($C =$) £141 at (26, 34) or £141 at (28, 33) | A1 | 4 |
| Total | | 15 |6 [Figure 1, printed on the insert, is provided for use in this question.]\\
A factory makes two types of lock, standard and large, on a particular day.\\
On that day:\\
the maximum number of standard locks that the factory can make is 100 ;\\
the maximum number of large locks that the factory can make is 80 ;\\
the factory must make at least 60 locks in total;\\
the factory must make more large locks than standard locks.\\
Each standard lock requires 2 screws and each large lock requires 8 screws, and on that day the factory must use at least 320 screws.
On that day, the factory makes $x$ standard locks and $y$ large locks.\\
Each standard lock costs $\pounds 1.50$ to make and each large lock costs $\pounds 3$ to make.\\
The manager of the factory wishes to minimise the cost of making the locks.
\begin{enumerate}[label=(\alph*)]
\item Formulate the manager's situation as a linear programming problem.
\item On Figure 1, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
\item Find the values of $x$ and $y$ that correspond to the minimum cost. Hence find this minimum cost.
\end{enumerate}
\hfill \mbox{\textit{AQA D1 2008 Q6 [15]}}