| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| 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 question requiring routine graphical methods: interpreting constraints, plotting lines, identifying the feasible region, and using an objective line to find the optimal vertex. All techniques are procedural with no novel problem-solving required, making it easier than average for A-level. |
| 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 |
|---|---|---|
| Answer | Marks | Guidance |
| (Edgar should plant) at least 40 apple trees; (Edgar should plant) at most 50 plum trees | B1 | CAO, both. Must be \(\leq\) and \(\geq\), not \(<\) and \(>\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Line \(3x + 4y = 360\) drawn correctly | B1 | CAO. If extended must go axis to axis within one small square. Must be long enough to form correct feasible region. Lines drawn with ruler |
| Line \(x = 2y\) drawn correctly | B1 | If extended must go through \((0,0)\) and \((120, 60)\) within one small square. Must be long enough to form correct feasible region. Lines drawn with ruler |
| Correct shading | B1ft | ft their lines for correct shading on one of their lines. Implicit if R is correct |
| Region R correct, labelled | B1 | CAO. Must be labelled. 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct feasible region identified | B1 | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Drawing objective line or its reciprocal | M1 A1 | Correct objective line, axis to axis \((0, 30)\) to \((10, 0)\) minimum |
| Finding correct optimal point \((72, 36)\) | DM1 A1 | Depends on 1st M and correct region. CSO. 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P = 60x + 20y\) | B1 | CAO |
| Optimal point \((72, 36)\), maximum profit \(= £5040\) | — | Drawing objective line; calculating optimal point |
# Question 6:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| (Edgar should plant) at least 40 apple trees; (Edgar should plant) at most 50 plum trees | B1 | CAO, both. Must be $\leq$ and $\geq$, not $<$ and $>$ |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line $3x + 4y = 360$ drawn correctly | B1 | CAO. If extended must go axis to axis within one small square. Must be long enough to form correct feasible region. Lines drawn with ruler |
| Line $x = 2y$ drawn correctly | B1 | If extended must go through $(0,0)$ and $(120, 60)$ within one small square. Must be long enough to form correct feasible region. Lines drawn with ruler |
| Correct shading | B1ft | ft their lines for correct shading on one of their lines. Implicit if R is correct |
| Region R correct, labelled | B1 | CAO. Must be labelled. **4 marks total** |
## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct feasible region identified | B1 | CAO |
## Part (d)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Drawing objective line or its reciprocal | M1 A1 | Correct objective line, axis to axis $(0, 30)$ to $(10, 0)$ minimum |
| Finding correct optimal point $(72, 36)$ | DM1 A1 | Depends on 1st M and correct region. CSO. **4 marks total** |
## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P = 60x + 20y$ | B1 | CAO |
| Optimal point $(72, 36)$, maximum profit $= £5040$ | — | Drawing objective line; calculating optimal point |
**Vertices in R:** $(40, 20)$, $(40, 50)$, $(53\frac{1}{3}, 50)$, $(72, 36)$
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e02c4a9a-d2ab-489f-b838-9b4d902c4457-7_2226_1628_299_221}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{center}
\end{figure}
Edgar has recently bought a field in which he intends to plant apple trees and plum trees.
He can use linear programming to determine the number of each type of tree he should plant.
Let $x$ be the number of apple trees he plants and $y$ be the number of plum trees he plants.
Two of the constraints are
$$\begin{aligned}
& x \geqslant 40 \\
& y \leqslant 50
\end{aligned}$$
These are shown on the graph in Figure 6, where the rejected region is shaded out.
\begin{enumerate}[label=(\alph*)]
\item Use these two constraints to write down two statements that describe the number of apple trees and plum trees Edgar can plant.
Two further constraints are
$$\begin{aligned}
3 x + 4 y & \leqslant 360 \\
x & \leqslant 2 y
\end{aligned}$$
\item Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R .
Edgar will make a profit of $\pounds 60$ from each apple tree and $\pounds 20$ from each plum tree. He wishes to maximise his profit, P.
\item Write down the objective function.
\item Use an objective line to determine the optimal point of the feasible region, R . You must make your method clear.
\item Find Edgar's maximum profit.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2012 Q6 [11]}}