| Exam Board | OCR MEI |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2011 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.3 This is a standard linear programming formulation question with straightforward constraints and a clear objective function. While it requires multiple steps (formulation, graphing, optimization), each component follows textbook procedures with no novel insight needed. The context is accessible and the mathematical operations are routine for D1 level. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables |
| Answer | Marks | Guidance |
|---|---|---|
| Part | Answer/Working | Marks |
| (i) | Let \(x\) = number of large houses; \(y\) = number of standard houses; Land: \(200x + 120y \le 120000\) or equivalent; Cash: \(60x + 50y \le 42200\) or equivalent; Market: \(x \le 0.5y\) or equivalent | M1, A1, B1, B1, B1 |
| (ii) | Graph showing three constraint lines with feasible region shaded | B1, B1, B1 |
| (iii) | Intersection of \(y=2x\) and \(6x+5y=4240\): \((265, 530)\); Objective function value: \(2650\) | M1, A1 |
| (iv) | Their \(60x + 50y \le 45000\) or line from their \((0, 900)\) to \((750, 0)\) | B1 |
| Best point is at the intersection of the land constraint and the new cash constraint, and not on \(y=2x\) | M1, A1 | Not just ringing points; their identified best point is not on \(y=2x\) or an axis |
| \((214, 643)\); Objective function value: \(2785\) | M1, A1 | Identification, coordinates not required here; Bedrooms—their \(4x+3y\) from \((200–220, 620–660)\) |
| **Part** | **Answer/Working** | **Marks** | **Guidance** |
|----------|-------------------|-----------|------------|
| (i) | Let $x$ = number of large houses; $y$ = number of standard houses; Land: $200x + 120y \le 120000$ or equivalent; Cash: $60x + 50y \le 42200$ or equivalent; Market: $x \le 0.5y$ or equivalent | M1, A1, B1, B1, B1 | M1 for variables for large and for standard; A1 for "number"; Use "isw" for incorrect simplifications; -1 once only for any "$<$" |
| (ii) | Graph showing three constraint lines with feasible region shaded | B1, B1, B1 | For instance, if $x \le 2y$ in part (i), allow correct graph of $x \le 0.5y$ or $x \le 2y$. Plotting tolerance on axis intersection points within correct small square. |
| (iii) | Intersection of $y=2x$ and $6x+5y=4240$: $(265, 530)$; Objective function value: $2650$ | M1, A1 | Identification only—coordinates not required here; their $4x+3y$ from $(260–280, 520–540)$ |
| (iv) | Their $60x + 50y \le 45000$ or line from their $(0, 900)$ to $(750, 0)$ | B1 | Can be implied from final M1 working. |
| | Best point is at the intersection of the land constraint and the new cash constraint, and not on $y=2x$ | M1, A1 | Not just ringing points; their identified best point is not on $y=2x$ or an axis |
| | $(214, 643)$; Objective function value: $2785$ | M1, A1 | Identification, coordinates not required here; Bedrooms—their $4x+3y$ from $(200–220, 620–660)$ |
---4 An eco-village is to be constructed consisting of large houses and standard houses.\\
Each large house has 4 bedrooms, needs a plot size of $200 \mathrm {~m} ^ { 2 }$ and costs $\pounds 60000$ to build.\\
Each standard house has 3 bedrooms, needs a plot size of $120 \mathrm {~m} ^ { 2 }$ and costs $\pounds 50000$ to build.\\
The area of land available for houses is $120000 \mathrm {~m} ^ { 2 }$. The project has been allocated a construction budget of $\pounds 42.4$ million.
The market will not sustain more than half as many large houses as standard houses. So, for instance, if there are 500 standard houses then there must be no more than 250 large houses.\\
(i) Define two variables so that the three constraints can be formulated in terms of your variables. Formulate the three constraints in terms of your variables.\\
(ii) Graph your three inequalities from part (i), indicating the feasible region.\\
(iii) Find the maximum number of bedrooms which can be provided, and the corresponding numbers of each type of house.\\
(iv) Modify your solution if the construction budget is increased to $\pounds 45$ million.
\hfill \mbox{\textit{OCR MEI D1 2011 Q4 [16]}}