| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Three-variable constraint reduction |
| Difficulty | Moderate -0.8 This is a standard linear programming question requiring routine application of D1 techniques: identifying variables, forming constraints from word problems, writing an objective function, eliminating a variable through substitution, and graphing the feasible region. All steps are mechanical with no novel problem-solving required, making it easier than average but not trivial due to the multi-step nature and algebraic manipulation needed. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06c Working with constraints: algebra and ad hoc methods |
| Answer | Marks |
|---|---|
| \(x =\) area of wall to be panelled (m²) | B1 |
| \(y =\) area to be painted | B1 |
| \(z =\) area to be covered with pinboard | |
| Reference to area or m² (at least once); Identifying \(x\) as panelling, \(y\) as paint and \(z\) as pinboard, in any way | B1 |
| [2] |
| Answer | Marks |
|---|---|
| Cost \(\leq £150\) | B1 |
| \(\Rightarrow 8x + 4y + 10z \leq 150\) | B1 |
| \(\Rightarrow 4x + 2y + 5z \leq 75\) (given) | |
| Use of word 'cost' or equivalent; \(8x + 4y + 10z \leq 150\) seen or explicitly referred to | B1 |
| [2] |
| Answer | Marks |
|---|---|
| (Minimise \(P =\)) \(15x + 30y + 20z\) | B1 ft |
| Any positive multiple of this, e.g. \(3x + 6y + 4z\) or \(\frac{1}{5}x + \frac{1}{5}y + \frac{1}{5}z\) | B1 ft |
| [1] |
| Answer | Marks |
|---|---|
| (Minimise \(P = 480 +) -5x + 10y\) | B1 ft |
| Subject to \(x + 3y \geq 45\) | B1 |
| \(x \geq 10\) | B1 |
| \(y \geq 0\) | B1 |
| \(x + y \leq 22\) | B1 |
| Any positive multiple of this, e.g. \(2y - x(+c)\) | B1 ft |
| Any equivalent simplified form; \(x \geq 10\) may be implied; \(y \geq 0\) may be implied; \(x + y \leq 22\), any equivalent simplified form | B1 |
| [3] |
| Answer | Marks |
|---|---|
| \(x = 10\) drawn accurately with a sensible scale | M1 |
| \(x + y = 22\) drawn accurately with a sensible scale | M1 |
| Their \(x + 3y = 45\) drawn accurately with a sensible scale | M1 |
| Shading correct or identification of the feasible region (triangle with \((10, 11\frac{2}{3})\), \((10, 12)\) and \((10\frac{1}{2}, 11\frac{1}{2})\) as vertices) | A1 |
| [4] |
## (i)
$x =$ area of wall to be panelled (m²) | B1 |
$y =$ area to be painted | B1 |
$z =$ area to be covered with pinboard | |
Reference to area or m² (at least once); Identifying $x$ as panelling, $y$ as paint and $z$ as pinboard, in any way | B1 |
| [2] |
## (ii)
Cost $\leq £150$ | B1 |
$\Rightarrow 8x + 4y + 10z \leq 150$ | B1 |
$\Rightarrow 4x + 2y + 5z \leq 75$ (given) | |
Use of word 'cost' or equivalent; $8x + 4y + 10z \leq 150$ seen or explicitly referred to | B1 |
| [2] |
## (iii)
(Minimise $P =$) $15x + 30y + 20z$ | B1 ft |
Any positive multiple of this, e.g. $3x + 6y + 4z$ or $\frac{1}{5}x + \frac{1}{5}y + \frac{1}{5}z$ | B1 ft |
| [1] |
## (iv)
(Minimise $P = 480 +) -5x + 10y$ | B1 ft |
Subject to $x + 3y \geq 45$ | B1 |
$x \geq 10$ | B1 |
$y \geq 0$ | B1 |
$x + y \leq 22$ | B1 |
Any positive multiple of this, e.g. $2y - x(+c)$ | B1 ft |
Any equivalent simplified form; $x \geq 10$ may be implied; $y \geq 0$ may be implied; $x + y \leq 22$, any equivalent simplified form | B1 |
| [3] |
## (v)
$x = 10$ drawn accurately with a sensible scale | M1 |
$x + y = 22$ drawn accurately with a sensible scale | M1 |
Their $x + 3y = 45$ drawn accurately with a sensible scale | M1 |
Shading correct or identification of the feasible region (triangle with $(10, 11\frac{2}{3})$, $(10, 12)$ and $(10\frac{1}{2}, 11\frac{1}{2})$ as vertices) | A1 |
| [4] |
---Mark wants to decorate the walls of his study. The total wall area is 24 m$^2$. Mark can cover the walls using any combination of three materials: panelling, paint and pinboard. He wants at least 2 m$^2$ of pinboard and at least 10 m$^2$ of panelling.
Panelling costs £8 per m$^2$ and it will take Mark 15 minutes to put up 1 m$^2$ of panelling. Paint costs £4 per m$^2$ and it will take Mark 30 minutes to paint 1 m$^2$. Pinboard costs £10 per m$^2$ and it will take Mark 20 minutes to put up 1 m$^2$ of pinboard. He has all the equipment that he will need for the decorating jobs.
Mark is able to spend up to £150 on the materials for the decorating. He wants to know what area should be covered with each material to enable him to complete the whole job in the shortest time possible.
Mark models the problem as an LP with five constraints. His constraints are:
$$x + y + z = 24,$$
$$4x + 2y + 5z \leqslant 75,$$
$$x \geqslant 10,$$
$$y \geqslant 0,$$
$$z \geqslant 2.$$
\begin{enumerate}[label=(\roman*)]
\item Identify the meaning of each of the variables $x$, $y$ and $z$. [2]
\item Show how the constraint $4x + 2y + 5z \leqslant 75$ was formed. [2]
\item Write down an objective function, to be minimised. [1]
\end{enumerate}
Mark rewrites the first constraint as $z = 24 - x - y$ and uses this to eliminate $z$ from the problem.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Rewrite and simplify the objective and the remaining four constraints as functions of $x$ and $y$ only. [3]
\item Represent your constraints from part (iv) graphically and identify the feasible region. Your graph should show $x$ and $y$ values from 0 to 15 only. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR D1 2008 Q5 [12]}}