Edexcel D1 2014 June — Question 5 13 marks

Exam BoardEdexcel
ModuleD1 (Decision Mathematics 1)
Year2014
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicLinear Programming
TypeGraphical optimization with objective line
DifficultyModerate -0.8 This is a standard textbook linear programming question requiring routine application of well-practiced techniques: formulating inequalities from word problems, graphing constraints, identifying the feasible region, and using the vertex method to optimize. All steps are algorithmic with no novel insight required, making it easier than average for A-level.
Spec7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables
5. Michael and his team are making toys to give to children at a summer fair. They make two types of toy, a soft toy and a craft set. Let \(x\) be the number of soft toys they make and \(y\) be the number of craft sets they make.
Each soft toy costs \(\pounds 3\) to make and each craft set costs \(\pounds 5\) to make. Michael and his team have a budget of \(\pounds 1000\) to spend on making the toys for the summer fair.
  1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Two further constraints are: $$\begin{gathered} y \leqslant 2 x \\ 4 y - x \geqslant 210 \end{gathered}$$
  2. Add lines and shading to Diagram 1 in the answer book to represent all of these constraints. Hence determine the feasible region and label it R . Michael's objective is to make as many toys as possible.
  3. State the objective function.
  4. Determine the exact coordinates of each of the vertices of the feasible region, and hence use the vertex method to find the optimal number of soft toys and craft sets Michael and his team should make. You should make your method clear.
Question 5:
Part (a)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(3x + 5y \leq 1000\)B1 (1)
Part (b)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Line \(3x + 5y = 1000\) drawn correctlyB1 —
Line \(y = 2x\) drawn correctlyB1 —
Line \(4y - x = 210\) drawn correctlyB1 —
Feasible region R correctly identified/shadedB1 (4)
Part (c)
AnswerMarks Guidance
Answer/WorkingMark Guidance
Objective: maximise \((P =)\ x + y\)B1 (1)
Part (d)
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((A=)(30, 60)\)M1 —
\((B=)\!\left(76\tfrac{12}{13},\, 153\tfrac{11}{13}\right)\)A1 —
\((C=)\!\left(173\tfrac{9}{17},\, 95\tfrac{15}{17}\right)\)A1 —
At A, \(P = 90\); at B, \(P = 230\tfrac{10}{13}\); at C, \(P = 269\tfrac{7}{17}\)M1, A1 —
C is optimal pointM1 —
Test integer solutions around C: \(x=173\), \(y=96\) is optimal integer solution; make 173 soft toys and 96 craft setsA1 (7)
Question 5:
Part a
AnswerMarks Guidance
AnswerMark Guidance
CAOB1 Correct answer only
Part b1
AnswerMarks Guidance
AnswerMark Guidance
\(3x + 5y = 1000\) passing through one small square of \((0, 200)\), \((200, 80)\), \((333\frac{1}{3}, 0)\)B1 Line must pass through one of these points
Part b2
AnswerMarks Guidance
AnswerMark Guidance
\(y = 2x\) passing through one small square of \((0, 0)\), \((100, 200)\), \((150, 300)\)B1 Line must pass through one of these points
Part b3
AnswerMarks Guidance
AnswerMark Guidance
\(4y - x = 210\) passing through one small square of \((0, 52.5)\), \((200, 102.5)\), \((400, 152.5)\)B1 Line must pass through one of these points
Part b4
AnswerMarks Guidance
AnswerMark Guidance
Region R correctly labelledB1 Must be labelled not just implied by shading; must have scored all three previous marks
Part c
AnswerMarks Guidance
AnswerMark Guidance
CAO – correct expressionB1 Correct answer only
Part d1
AnswerMarks Guidance
AnswerMark Guidance
Attempt to solve two correct equations simultaneously, up to \(x = \ldots\) or \(y = \ldots\)M1
At least 1 correct R vertex found correct to at least 2dp: \((30, 60)\), \((76.923\ldots, 153.846\ldots)\), \((173.529\ldots, 95.705\ldots)\)A1 If any vertex stated correctly (with or without working) scores M1A1
Part d2
AnswerMarks Guidance
AnswerMark Guidance
All correct R vertices found exactly: \(B\left(\frac{1000}{13}, \frac{2000}{13}\right)\), \(C\left(\frac{2950}{17}, \frac{1630}{17}\right)\)A1 Must be working for determining points B and C
Evaluating the correct objective function at at least two of their points for their feasible regionM1 Allow if vertices have been read off graph; condone testing 'nearest' integer values e.g. if state \((76.9, 153.8)\) allow testing \((76, 153)\), \((77, 153)\), \((76, 154)\) or \((77, 154)\) only
Part d3
AnswerMarks Guidance
AnswerMark Guidance
All three correct P values either given exactly \(\left\{90, \frac{3000}{13}, \frac{4580}{17}\right\}\) or to at least 1dp: \(\{90, 230.769\ldots, 269.411\ldots\}\)A1 Must be testing the exact coordinates for this mark
Testing the correct inequalities for at least two of \((173, 95)\), \((173, 96)\), \((174, 95)\), \((174, 96)\)M1
Part d4
AnswerMarks Guidance
AnswerMark Guidance
\(x = 173\) and \(y = 96\) or as coordinatesA1 CSO – all previous marks in (d) must have been awarded 
# Question 5:

## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3x + 5y \leq 1000$ | B1 | **(1)** |

## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Line $3x + 5y = 1000$ drawn correctly | B1 | — |
| Line $y = 2x$ drawn correctly | B1 | — |
| Line $4y - x = 210$ drawn correctly | B1 | — |
| Feasible region R correctly identified/shaded | B1 | **(4)** |

## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Objective: maximise $(P =)\ x + y$ | B1 | **(1)** |

## Part (d)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(A=)(30, 60)$ | M1 | — |
| $(B=)\!\left(76\tfrac{12}{13},\, 153\tfrac{11}{13}\right)$ | A1 | — |
| $(C=)\!\left(173\tfrac{9}{17},\, 95\tfrac{15}{17}\right)$ | A1 | — |
| At A, $P = 90$; at B, $P = 230\tfrac{10}{13}$; at C, $P = 269\tfrac{7}{17}$ | M1, A1 | — |
| C is optimal point | M1 | — |
| Test integer solutions around C: $x=173$, $y=96$ is optimal integer solution; make **173 soft toys** and **96 craft sets** | A1 | **(7)** |

# Question 5:

## Part a
| Answer | Mark | Guidance |
|--------|------|----------|
| CAO | B1 | Correct answer only |

## Part b1
| Answer | Mark | Guidance |
|--------|------|----------|
| $3x + 5y = 1000$ passing through one small square of $(0, 200)$, $(200, 80)$, $(333\frac{1}{3}, 0)$ | B1 | Line must pass through one of these points |

## Part b2
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 2x$ passing through one small square of $(0, 0)$, $(100, 200)$, $(150, 300)$ | B1 | Line must pass through one of these points |

## Part b3
| Answer | Mark | Guidance |
|--------|------|----------|
| $4y - x = 210$ passing through one small square of $(0, 52.5)$, $(200, 102.5)$, $(400, 152.5)$ | B1 | Line must pass through one of these points |

## Part b4
| Answer | Mark | Guidance |
|--------|------|----------|
| Region R correctly labelled | B1 | Must be labelled not just implied by shading; must have scored all three previous marks |

## Part c
| Answer | Mark | Guidance |
|--------|------|----------|
| CAO – correct expression | B1 | Correct answer only |

## Part d1
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to solve two correct equations simultaneously, up to $x = \ldots$ or $y = \ldots$ | M1 | |
| At least 1 correct R vertex found correct to at least 2dp: $(30, 60)$, $(76.923\ldots, 153.846\ldots)$, $(173.529\ldots, 95.705\ldots)$ | A1 | If **any** vertex stated correctly (with or without working) scores M1A1 |

## Part d2
| Answer | Mark | Guidance |
|--------|------|----------|
| All correct R vertices found **exactly**: $B\left(\frac{1000}{13}, \frac{2000}{13}\right)$, $C\left(\frac{2950}{17}, \frac{1630}{17}\right)$ | A1 | **Must** be working for determining points B and C |
| Evaluating the correct objective function at at least two of their points for their feasible region | M1 | Allow if vertices have been read off graph; condone testing 'nearest' integer values e.g. if state $(76.9, 153.8)$ allow testing $(76, 153)$, $(77, 153)$, $(76, 154)$ or $(77, 154)$ **only** |

## Part d3
| Answer | Mark | Guidance |
|--------|------|----------|
| All three correct P values either given exactly $\left\{90, \frac{3000}{13}, \frac{4580}{17}\right\}$ or to at least 1dp: $\{90, 230.769\ldots, 269.411\ldots\}$ | A1 | Must be testing the exact coordinates for this mark |
| Testing the correct inequalities for at least two of $(173, 95)$, $(173, 96)$, $(174, 95)$, $(174, 96)$ | M1 | |

## Part d4
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 173$ and $y = 96$ or as coordinates | A1 | CSO – all previous marks in (d) must have been awarded |

---
5. Michael and his team are making toys to give to children at a summer fair. They make two types of toy, a soft toy and a craft set.

Let $x$ be the number of soft toys they make and $y$ be the number of craft sets they make.\\
Each soft toy costs $\pounds 3$ to make and each craft set costs $\pounds 5$ to make.

Michael and his team have a budget of $\pounds 1000$ to spend on making the toys for the summer fair.
\begin{enumerate}[label=(\alph*)]
\item Write down an inequality, in terms of $x$ and $y$, to model this constraint.

Two further constraints are:

$$\begin{gathered}
y \leqslant 2 x \\
4 y - x \geqslant 210
\end{gathered}$$
\item Add lines and shading to Diagram 1 in the answer book to represent all of these constraints. Hence determine the feasible region and label it R .

Michael's objective is to make as many toys as possible.
\item State the objective function.
\item Determine the exact coordinates of each of the vertices of the feasible region, and hence use the vertex method to find the optimal number of soft toys and craft sets Michael and his team should make. You should make your method clear.
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2014 Q5 [13]}}
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