Question 7 - A-Level Maths

Edexcel D1 2008 January — Question 7 16 marks

Exam Board	Edexcel
Module	D1 (Decision Mathematics 1)
Year	2008
Session	January
Marks	16
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear Programming
Type	Graphical optimization with objective line
Difficulty	Moderate -0.8 This is a standard D1 linear programming question following a routine template: read constraints from a graph, plot additional lines, identify feasible region, and optimize using the objective line method. All steps are algorithmic with no novel problem-solving required—significantly easier than average A-level maths questions.
Spec	7.06d Graphical solution: feasible region, two variables

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-8_2158_1803_239_137} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure}

Phil sells boxed lunches to travellers at railway stations. Customers can select either the vegetarian box or the non-vegetarian box.

Phil decides to use graphical linear programming to help him optimise the numbers of each type of box he should produce each day. Each day Phil produces $x$ vegetarian boxes and $y$ non-vegetarian boxes.
One of the constraints limiting the number of boxes is $$x + y \geqslant 70$$ This, together with $x \geqslant 0 , y \geqslant 0$ and a fourth constraint, has been represented in Figure 7. The rejected region has been shaded.

Write down the inequality represented by the fourth constraint. Two further constraints are: $$\begin{aligned} & x + 2 y \leqslant 160 \\ & \text { and } y > 60 \end{aligned}$$
Add two lines and shading to Diagram 4 in your answer book to represent these inequalities.
Hence determine and label the feasible region, R .
Use your graph to determine the minimum total number of boxes he needs to prepare each day. Make your method clear. Phil makes a profit of $\pounds 1.20$ on each vegetarian box and $\pounds 1.40$ on each non-vegetarian box. He wishes to maximise his profit.
Write down the objective function.
Use your graph to obtain the optimal number of vegetarian and non-vegetarian boxes he should produce each day. You must make your method clear.
Find Phil's maximum daily profit.

Show mark scheme Show mark scheme source

Question 7:

Part (a):

Answer	Marks	Guidance
Answer	Marks	Guidance
$y \geq 2x$	B1	2 (or $\frac{1}{2}$) one correct side
cao	B1	Condone any inequality or equals, or bod

Part (b):

Answer	Marks	Guidance
Answer	Marks	Guidance
$x + 2y = 160$ correctly drawn	B1, B1	Lines correct to $\leq 1$ small square at axis; errors to look for: $y = 60$ distinct in some way
$y \leq 60$ correctly drawn and distinctive (strict inequality)	B1, B1	$-1$ e.e.; labels on lines
Shading correct	B1	Ruler required

Part (c):

Answer	Marks	Guidance
Answer	Marks	Guidance
$R$ correct	B1ft	$R$ 'correct'; ft their lines, but shading needs to be correct

Part (d):

Answer	Marks	Guidance
Answer	Marks	Guidance
Profit line added or point testing seen	M1	Attempt at profit line (axis to axis) or point testing 2 points
Correctly done	A1	Profit line correct (within 1 small square) or three points tested correctly
70 boxes identified	A1	cao

Part (e):

Answer	Marks	Guidance
Answer	Marks	Guidance
$(P=)\ 1.2x + 1.4y$	B1	cao

Part (f):

Answer	Marks	Guidance
Answer	Marks	Guidance
Profit line added or point testing seen	M1	Attempt at profit line (axis to axis) or point testing 2 points
Correctly done	A1ft/A1	Correct but ft their $R$ and their (e) for profit line and 3 point testing; correct (so a mark for correct with no need to ft)
$(32, 64)$ identified	A1	cao $(32, 64)$ only

Part (g):

Answer	Marks	Guidance
Answer	Marks	Guidance
$£128.00$	A1	cao follow through (ignore units)

Total: 16 marks

# Question 7:

## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y \geq 2x$ | B1 | 2 (or $\frac{1}{2}$) one correct side |
| cao | B1 | Condone any inequality or equals, or bod |

## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x + 2y = 160$ correctly drawn | B1, B1 | Lines correct to $\leq 1$ small square at axis; errors to look for: $y = 60$ distinct in some way |
| $y \leq 60$ correctly drawn and distinctive (strict inequality) | B1, B1 | $-1$ e.e.; labels on lines |
| Shading correct | B1 | Ruler required |

## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $R$ correct | B1ft | $R$ 'correct'; ft their lines, but shading needs to be correct |

## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Profit line added or point testing seen | M1 | Attempt at profit line (axis to axis) or point testing 2 points |
| Correctly done | A1 | Profit line correct (within 1 small square) or three points tested correctly |
| 70 boxes identified | A1 | cao |

## Part (e):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(P=)\ 1.2x + 1.4y$ | B1 | cao |

## Part (f):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Profit line added or point testing seen | M1 | Attempt at profit line (axis to axis) or point testing 2 points |
| Correctly done | A1ft/A1 | Correct but ft their $R$ and their (e) for profit line and 3 point testing; correct (so a mark for correct with no need to ft) |
| $(32, 64)$ identified | A1 | cao $(32, 64)$ only |

## Part (g):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $£128.00$ | A1 | cao follow through (ignore units) |

**Total: 16 marks**

Show LaTeX source

7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-8_2158_1803_239_137}
\captionsetup{labelformat=empty}
\caption{Figure 7}
\end{center}
\end{figure}

\begin{enumerate}
  \item Phil sells boxed lunches to travellers at railway stations. Customers can select either the vegetarian box or the non-vegetarian box.
\end{enumerate}

Phil decides to use graphical linear programming to help him optimise the numbers of each type of box he should produce each day.

Each day Phil produces $x$ vegetarian boxes and $y$ non-vegetarian boxes.\\
One of the constraints limiting the number of boxes is

$$x + y \geqslant 70$$

This, together with $x \geqslant 0 , y \geqslant 0$ and a fourth constraint, has been represented in Figure 7. The rejected region has been shaded.\\
(a) Write down the inequality represented by the fourth constraint.

Two further constraints are:

$$\begin{aligned}
& x + 2 y \leqslant 160 \\
& \text { and } y > 60
\end{aligned}$$

(b) Add two lines and shading to Diagram 4 in your answer book to represent these inequalities.\\
(c) Hence determine and label the feasible region, R .\\
(d) Use your graph to determine the minimum total number of boxes he needs to prepare each day. Make your method clear.

Phil makes a profit of $\pounds 1.20$ on each vegetarian box and $\pounds 1.40$ on each non-vegetarian box. He wishes to maximise his profit.\\
(e) Write down the objective function.\\
(f) Use your graph to obtain the optimal number of vegetarian and non-vegetarian boxes he should produce each day. You must make your method clear.\\
(g) Find Phil's maximum daily profit.

\hfill \mbox{\textit{Edexcel D1 2008 Q7 [16]}}

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