| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Easy -1.2 This is a standard textbook linear programming question requiring routine graphical methods: plotting three simple linear constraints, identifying the feasible region, and using the objective line method to find the minimum. All steps are algorithmic with no novel insight required, making it easier than average for A-level. |
| Spec | 7.06a 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 |
| Answer | Marks | Guidance |
|---|---|---|
| He must buy at least 90 boats in total \((x + y \geq 90)\) | B1 (1) | |
| E.g. The number of 2-seater boats(x) must be less than or equal to 1.5 times the number of 4-seater boats (y). (check: \(y = 2, x = 3, 2, 1, ...\)) \((2x \leq 3y)\) | B1 B1 | |
| E.g. The number of 4-seater boats (y) must be greater than or equal to 2/3 the number of 2-seater boats (x). (check: \(x = 3, y = 2, 3, 4, ...\)) | (2) | |
| The correct 3 lines added; \(x + y = 90; 3y = 2x; y = x + 30\) | B1; B1; B1 | (4) |
| Region, R labelled | B1 | |
| (minimise C = ) \(100x + 300y\) | B1 (1) | |
| Method clear – either at least 2 vertices tested or objective line drawn | M1 A1 | (4) |
| \((54, 36)\), so 54 2-seater and 36 4-seater | B1 | |
| At a cost of £16 200 | B1 | |
| 12 marks |
| He must buy at least 90 boats in total $(x + y \geq 90)$ | B1 (1) | |
| E.g. The number of 2-seater boats(x) must be less than or equal to 1.5 times the number of 4-seater boats (y). (check: $y = 2, x = 3, 2, 1, ...$) $(2x \leq 3y)$ | B1 B1 | |
| E.g. The number of 4-seater boats (y) must be greater than or equal to 2/3 the number of 2-seater boats (x). (check: $x = 3, y = 2, 3, 4, ...$) | | (2) |
| | | |
| The correct 3 lines added; $x + y = 90; 3y = 2x; y = x + 30$ | B1; B1; B1 | (4) |
| Region, R labelled | B1 | |
| | | |
| (minimise C = ) $100x + 300y$ | B1 (1) | |
| | | |
| Method clear – either at least 2 vertices tested or objective line drawn | M1 A1 | (4) |
| $(54, 36)$, so 54 2-seater and 36 4-seater | B1 | |
| At a cost of £16 200 | B1 | |
| | | 12 marks |
**Notes for Question 6:**
- a1B1: CAO (must have 'boats', 'least', '90', must be talking about boats not cost)
- b1B1: For a statement in context with either the ratio of coefficients correct (the 2 with the 2-seater and the 3 with the 4-seater) or inequality correct with correct numbers present but not in the correct ratio.
- b2B1: Clear accurate correct statement in context.
- c1B1: $x + y = 90$ correctly drawn. Must pass within one small square of the points of intersection with the axes
- c2B1: $3y = 2x$ correctly drawn. Must pass within one small square of the origin and (90, 60).
- c3B1: $y = x + 30$ correctly drawn. Must pass within one small square of (0, 30) and (60, 90).
- c4B1: Region, R, correctly labelled – not just implied by shading – must have scored all three previous marks in this part.
- d1B1: CAO (isw if $100x + 300y$ 'simplified' to $k(100x + 300y)$ but if $100x + 300y$ not stated then B0)
- e1M1: Line must be correct to within one small square if extended from axis to axis OR attempting to find two vertices of their R by either reading off their graph or using simultaneous equations and testing using their objective function.
- e1A1: Correct objective line (same condition that the line must be correct to within one small square if extended from axis to axis) OR testing (30, 60) correctly (giving 21 000) and testing (54, 36) correctly (giving 16 200).
- e1B1: Correct point identified. (Condone in terms of x and y rather than in terms of boats.)
- e2B1: CAO – condone lack of/incorrect units on the cost.
---6. Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy.
Let $x$ be the number of 2 -seater boats and $y$ be the number of 4 -seater boats.\\
One of the constraints is
$$x + y \geqslant 90$$
\begin{enumerate}[label=(\alph*)]
\item Explain what this constraint means in the context of the question.
Another constraint is
$$2 x \leqslant 3 y$$
\item Explain what this constraint means in the context of the question.
A third constraint is
$$y \leqslant x + 30$$
\item Represent these three constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R .
Each 2 -seater boat costs $\pounds 100$ and each 4 -seater boat costs $\pounds 300$ to buy. Harry wishes to minimise the total cost of buying the boats.
\item Write down the objective function, C , in terms of $x$ and $y$.
\item Determine the number of each type of boat that Harry should buy. You must make your method clear and state the minimum cost.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2013 Q6 [12]}}