| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Constraint derivation verification |
| Difficulty | Moderate -0.8 This is a straightforward linear programming question requiring translation of verbal constraints into inequalities and basic graphical work. Part (a) is given, part (b) involves direct translation of simple conditions (x≥2, y≥5, y≥2x, y≤3x), and parts (c)-(d) are standard graphical LP techniques taught in D1. No novel problem-solving or complex reasoning required—purely routine application of a well-practiced method. |
| Spec | 7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| \(1000x + 500y \leq 9000\) \((2x + y \leq 18)\) | B1 | (1 mark total) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x \geq 2\), \(y \geq 5\); \(y \geq 2x\); \(y \leq 3x\) | B1, B1, B1 | −1 for strict inequalities; −1 for 'w's and 'f's (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2, y = 5\); \(2x + y = 18\); Line \(y = mx\); \(y = 2x\); \(y = 3x\) | B1, B1, M1, A1, A1 | Feasible region (6 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Considering an extreme point on their f.r. \(x = 4.5\), \(y = 9\) | M1, A1, A1 | Extreme point - vertex (3 marks total) |
**6(a)**
| $1000x + 500y \leq 9000$ $(2x + y \leq 18)$ | B1 | (1 mark total) |
**6(b)**
| $x \geq 2$, $y \geq 5$; $y \geq 2x$; $y \leq 3x$ | B1, B1, B1 | −1 for strict inequalities; −1 for 'w's and 'f's (3 marks total) |
**6(c)**
| $x = 2, y = 5$; $2x + y = 18$; Line $y = mx$; $y = 2x$; $y = 3x$ | B1, B1, M1, A1, A1 | Feasible region (6 marks total) |
**6(d)**
| Considering an extreme point on their f.r. $x = 4.5$, $y = 9$ | M1, A1, A1 | Extreme point - vertex (3 marks total) |6 [Figure 1, printed on the insert, is provided for use in this question.]\\
Dino is to have a rectangular swimming pool at his villa.\\
He wants its width to be at least 2 metres and its length to be at least 5 metres.\\
He wants its length to be at least twice its width.\\
He wants its length to be no more than three times its width.\\
Each metre of the width of the pool costs $\pounds 1000$ and each metre of the length of the pool costs $\pounds 500$.
He has $\pounds 9000$ available.
Let the width of the pool be $x$ metres and the length of the pool be $y$ metres.
\begin{enumerate}[label=(\alph*)]
\item Show that one of the constraints leads to the inequality
$$2 x + y \leqslant 18$$
\item Find four further inequalities.
\item On Figure 1, draw a suitable diagram to show the feasible region.
\item Use your diagram to find the maximum width of the pool. State the corresponding length of the pool.
\end{enumerate}
\hfill \mbox{\textit{AQA D1 2007 Q6 [13]}}