| Exam Board | AQA |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.8 This is a standard textbook linear programming problem requiring routine formulation of constraints, graphical representation, and reading off the optimal vertex. The context is straightforward, the inequalities are simple, and the method is entirely procedural with no novel insight required—easier than average for A-level. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Minimise \(C = 2.5x + 15y\) | B1 | Objective function |
| \(200x + 1000y \geq 5000\) i.e. \(x + 5y \geq 25\) (nails) | B1 | Nails constraint |
| \(200x + 1500y \geq 6000\) i.e. \(2x + 15y \geq 60\) (screws) | B1 | Screws constraint |
| \(100x + 2500y \geq 4000\) i.e. \(x + 25y \geq 40\) (plugs) | B1 | Plugs constraint |
| \(x \geq 0, y \geq 0\) | Non-negativity (may be implied) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Three constraint lines correctly drawn | B3 | B1 each for: \(x+5y=25\), \(2x+15y=60\), \(x+25y=40\) |
| Feasible region correctly identified/shaded | B1 | Must be consistent with lines drawn |
| Objective line drawn with correct gradient | B1 | e.g. \(2.5x+15y=k\) for some \(k\) |
| Direction of optimisation indicated | B1 | Arrow showing direction of decrease |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x=25, y=0\) or intersection point identified from graph | B1 | Follow through from graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Minimum cost \(= 2.5(25)+15(0) = £62.50\) | B1 | Follow through from (b)(ii) |
# Question 7:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Minimise $C = 2.5x + 15y$ | B1 | Objective function |
| $200x + 1000y \geq 5000$ i.e. $x + 5y \geq 25$ (nails) | B1 | Nails constraint |
| $200x + 1500y \geq 6000$ i.e. $2x + 15y \geq 60$ (screws) | B1 | Screws constraint |
| $100x + 2500y \geq 4000$ i.e. $x + 25y \geq 40$ (plugs) | B1 | Plugs constraint |
| $x \geq 0, y \geq 0$ | | Non-negativity (may be implied) |
## Part (b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Three constraint lines correctly drawn | B3 | B1 each for: $x+5y=25$, $2x+15y=60$, $x+25y=40$ |
| Feasible region correctly identified/shaded | B1 | Must be consistent with lines drawn |
| Objective line drawn with correct gradient | B1 | e.g. $2.5x+15y=k$ for some $k$ |
| Direction of optimisation indicated | B1 | Arrow showing direction of decrease |
## Part (b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x=25, y=0$ or intersection point identified from graph | B1 | Follow through from graph |
## Part (b)(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Minimum cost $= 2.5(25)+15(0) = £62.50$ | B1 | Follow through from (b)(ii) |
---7 A builder needs some screws, nails and plugs. At the local store, there are packs containing a mixture of the three items.
A DIY pack contains 200 nails, 200 screws and 100 plugs.\\
A trade pack contains 1000 nails, 1500 screws and 2500 plugs.\\
A DIY pack costs $\pounds 2.50$ and a trade pack costs $\pounds 15$.\\
The builder needs at least 5000 nails, 6000 screws and 4000 plugs.\\
The builder buys $x$ DIY packs and $y$ trade packs and wishes to keep his total cost to a minimum.
\begin{enumerate}[label=(\alph*)]
\item Formulate the builder's situation as a linear programming problem.
\item \begin{enumerate}[label=(\roman*)]
\item On the grid opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of an objective line.
\item Use your diagram to find the number of each type of pack that the builder should buy in order to minimise his cost.
\item Find the builder's minimum cost.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA D1 2011 Q7 [12]}}