Edexcel D1 2015 June — Question 7 16 marks

Exam BoardEdexcel
ModuleD1 (Decision Mathematics 1)
Year2015
SessionJune
Marks16
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeConstraint derivation verification
DifficultyModerate -0.8 This is a standard D1 linear programming question requiring routine conversion of word problems into inequalities (converting minutes to hours and simplifying), plotting constraints, and finding optimal vertices. The 'show that' in part (a)(i) makes it even more straightforward as the form is given. All techniques are textbook exercises with no novel problem-solving required.
Spec7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients
7. Ian plans to produce two types of book, hardbacks and paperbacks. He will use linear programming to determine the number of each type of book he should produce. Let \(x\) represent the number of hardbacks Ian will produce. Let \(y\) represent the number of paperbacks Ian will produce. Each hardback takes 1 hour to print and 15 minutes to bind.
Each paperback takes 35 minutes to print and 24 minutes to bind.
The printing machine must be used for at least 14 hours. The binding machine must be used for at most 8 hours.
    1. Show that the printing time restriction leads to the constraint \(12 x + 7 y \geqslant k\), where \(k\) is a constant to be determined.
    2. Write the binding time restriction in a similar simplified form. Ian decides to produce at most twice as many hardbacks as paperbacks.
  2. Write down an inequality to model this constraint in terms of \(x\) and \(y\).
  3. Add lines and shading to Diagram 1 in the answer book to represent the constraints found in (a) and (b). Hence determine, and label, the feasible region R. Ian wishes to maximise \(\mathrm { P } = 60 x + 36 y\), where P is the total profit in pounds.
    1. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must draw and clearly label your objective line and the vertex V .
    2. Determine the exact coordinates of V. You must show your working.
  5. Given that P is Ian's expected total profit, in pounds, find the number of each type of book that he should produce and his maximum expected profit.
Question 7:
Part (a)
AnswerMarks Guidance
AnswerMarks Guidance
\(60x+35y \geq 840\) or \(x+\frac{7}{12}y \geq 14 \Rightarrow 12x+7y \geq 168\)M1 A1
\(15x+24y \leq 480\) or \(\frac{1}{4}x+\frac{2}{5}y \leq 8 \Rightarrow 5x+8y \leq 160\)M1 A1
Part (b)
AnswerMarks Guidance
AnswerMarks Guidance
\(2y \geq x\)M1 A1
Part (c)
AnswerMarks Guidance
AnswerMarks Guidance
Line \(12x+7y=168\) drawn correctlyB1
Line \(5x+8y=160\) drawn correctlyB1
Line \(2y=x\) drawn correctlyB1
Region R correctB1
Part (d)(i)
AnswerMarks Guidance
AnswerMarks Guidance
Objective line correctly drawn and labelledB1
Optimal vertex labelledDB1
Part (d)(ii)
AnswerMarks Guidance
AnswerMarks Guidance
\(V\!\left(\dfrac{160}{9},\, \dfrac{80}{9}\right)\)M1 A1
Part (e)
AnswerMarks Guidance
AnswerMarks Guidance
Make 17 hardbacks and 9 paperbacksB1
Expected profit £1344B1
Question 7:
Part (a)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Two of three coefficients correct with correct inequality sign in unsimplified form or all three coefficients correct with any sign \((=, <, >, \leq, \geq)\)M1 Unsimplified form acceptable
CAO (correct answer)A1 Correct answer with no working can imply M1 only
Two of three coefficients correct with correct inequality sign in either unsimplified or simplified form or all three coefficients correct with any sign \((=, <, >, \leq, \geq)\)M1
CAOA1 Correct answer with no working can imply M1A1
Part (b)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Both coefficients correct (accept \(=, <, >, \leq, \geq\)) or \(y \geq 2x\)M1
CAOA1
Part (c) — Graph
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(12x + 7y = 168\) drawn correctlyB1 Does not pass outside small square of \((0, 24)\) and \((14, 0)\); ignore shading
\(5x + 8y = 160\) drawn correctlyB1 Does not pass outside small square of \((0, 20)\) and \((32, 0)\); ignore shading
\(2y = x\) drawn correctlyB1 Does not pass outside small square of \((0,0)\), \((16, 8)\); sufficiently long to define feasible region; ignore shading
R labelled correctB1 Not just implied by shading; must have earned all previous marks in this part
Part (d)(i)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Correct objective line drawn on graphB1 Line must not pass outside a small square if extended from axis to axis
V labelled clearly on graphDB1 Dependent on both correct three line segments defining feasible region boundary and correct objective line
Part (d)(ii)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Simultaneous equations \(5x + 8y = 160\) and \(x = 2y\) used to find VM1 Must get to \(x = \cdots\) or \(y = \cdots\) (condone one error in solving)
\(\left(\dfrac{160}{9}, \dfrac{80}{9}\right)\) or \(\left(17\dfrac{7}{9},\ 8\dfrac{8}{9}\right)\)A1 CAO; coordinates must be exact; correct answer with no working can imply M1A1
Part (e)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((17, 9)\) — accept \(x = 17,\ y = 9\)B1 CAO
\((\pounds)1344\)B1 CAO 
# Question 7:

## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $60x+35y \geq 840$ or $x+\frac{7}{12}y \geq 14 \Rightarrow 12x+7y \geq 168$ | M1 A1 | |
| $15x+24y \leq 480$ or $\frac{1}{4}x+\frac{2}{5}y \leq 8 \Rightarrow 5x+8y \leq 160$ | M1 A1 | |

## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2y \geq x$ | M1 A1 | |

## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line $12x+7y=168$ drawn correctly | B1 | |
| Line $5x+8y=160$ drawn correctly | B1 | |
| Line $2y=x$ drawn correctly | B1 | |
| Region R correct | B1 | |

## Part (d)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Objective line correctly drawn and labelled | B1 | |
| Optimal vertex labelled | DB1 | |

## Part (d)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V\!\left(\dfrac{160}{9},\, \dfrac{80}{9}\right)$ | M1 A1 | |

## Part (e)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Make 17 hardbacks and 9 paperbacks | B1 | |
| Expected profit £1344 | B1 | |

# Question 7:

## Part (a)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Two of three coefficients correct with correct inequality sign in unsimplified form **or** all three coefficients correct with any sign $(=, <, >, \leq, \geq)$ | M1 | Unsimplified form acceptable |
| CAO (correct answer) | A1 | Correct answer with no working can imply M1 only |
| Two of three coefficients correct with correct inequality sign in either unsimplified or simplified form **or** all three coefficients correct with any sign $(=, <, >, \leq, \geq)$ | M1 | |
| CAO | A1 | Correct answer with no working can imply M1A1 |

## Part (b)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Both coefficients correct (accept $=, <, >, \leq, \geq$) **or** $y \geq 2x$ | M1 | |
| CAO | A1 | |

## Part (c) — Graph

| Answer/Working | Mark | Guidance |
|---|---|---|
| $12x + 7y = 168$ drawn correctly | B1 | Does not pass outside small square of $(0, 24)$ and $(14, 0)$; ignore shading |
| $5x + 8y = 160$ drawn correctly | B1 | Does not pass outside small square of $(0, 20)$ and $(32, 0)$; ignore shading |
| $2y = x$ drawn correctly | B1 | Does not pass outside small square of $(0,0)$, $(16, 8)$; sufficiently long to define feasible region; ignore shading |
| R labelled correct | B1 | Not just implied by shading; must have earned all previous marks in this part |

## Part (d)(i)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct objective line drawn on graph | B1 | Line must not pass outside a small square if extended from axis to axis |
| V labelled clearly on graph | DB1 | Dependent on **both** correct three line segments defining feasible region boundary **and** correct objective line |

## Part (d)(ii)

| Answer/Working | Mark | Guidance |
|---|---|---|
| Simultaneous equations $5x + 8y = 160$ and $x = 2y$ used to find V | M1 | Must get to $x = \cdots$ or $y = \cdots$ (condone one error in solving) |
| $\left(\dfrac{160}{9}, \dfrac{80}{9}\right)$ or $\left(17\dfrac{7}{9},\ 8\dfrac{8}{9}\right)$ | A1 | CAO; coordinates must be exact; correct answer with no working can imply M1A1 |

## Part (e)

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(17, 9)$ — accept $x = 17,\ y = 9$ | B1 | CAO |
| $(\pounds)1344$ | B1 | CAO |
7. Ian plans to produce two types of book, hardbacks and paperbacks. He will use linear programming to determine the number of each type of book he should produce.

Let $x$ represent the number of hardbacks Ian will produce.

Let $y$ represent the number of paperbacks Ian will produce.

Each hardback takes 1 hour to print and 15 minutes to bind.\\
Each paperback takes 35 minutes to print and 24 minutes to bind.\\
The printing machine must be used for at least 14 hours. The binding machine must be used for at most 8 hours.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the printing time restriction leads to the constraint $12 x + 7 y \geqslant k$, where $k$ is a constant to be determined.
\item Write the binding time restriction in a similar simplified form.

Ian decides to produce at most twice as many hardbacks as paperbacks.
\end{enumerate}\item Write down an inequality to model this constraint in terms of $x$ and $y$.
\item Add lines and shading to Diagram 1 in the answer book to represent the constraints found in (a) and (b). Hence determine, and label, the feasible region R.

Ian wishes to maximise $\mathrm { P } = 60 x + 36 y$, where P is the total profit in pounds.
\item \begin{enumerate}[label=(\roman*)]
\item Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must draw and clearly label your objective line and the vertex V .
\item Determine the exact coordinates of V. You must show your working.
\end{enumerate}\item Given that P is Ian's expected total profit, in pounds, find the number of each type of book that he should produce and his maximum expected profit.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2015 Q7 [16]}}
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