| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a standard linear programming formulation and graphical solution question typical of D1 modules. Parts (i) and (ii) require straightforward translation of constraints into inequalities with no conceptual difficulty. Part (iii) involves routine graphical methods taught explicitly in the specification. The problem is multi-part but each step follows textbook procedures with no novel insight required. Slightly easier than average due to the structured guidance and standard D1 content. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06d Graphical solution: feasible region, two variables |
| Stroke | Style marks | Time taken |
| Breaststroke | 2 marks per length | 2 minutes per length |
| Backstroke | 1 mark per length | 0.5 minutes per length |
| Butterfly | 5 marks per length | 1 minute per length |
| Answer | Marks |
|---|---|
| B1 | For defining variables as 'number of lengths swum' using each stroke, or equivalent |
| Answer | Marks |
|---|---|
| B1 (2) | For a correct expression using their variables |
| B1 | For a correct expression using their variables |
| Answer | Marks |
|---|---|
| B1 | For a correct expression using their variables |
| Answer | Marks |
|---|---|
| B1 (3) | For correct expressions using their variables |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) [Graph with shaded feasible region shown correctly] | M1 | For plotting the sloping lines correctly |
| A1 | For completely correct shading |
| Answer | Marks |
|---|---|
| M1 | For two correct vertices from their graph |
| A1 | For all three vertices correct to at least 1 dp |
| Answer | Marks |
|---|---|
| M1 | For calculating \(P\) at vertices or using a valid line of constant profit or writing down their max point |
| Answer | Marks |
|---|---|
| A1 (6) | For the correct values |
| Answer | Marks |
|---|---|
| B1 | For interpreting their solution in the context of the original problem (at least for \(x\) and \(y\)) |
| B1 (2) | For calculating the number of marks for their solution |
| \(\mathbf{13}\) |
**(i)** $x$ = number of lengths swum using breaststroke
$y$ = number of lengths swum using backstroke
$z$ = number of lengths swum using butterfly
| B1 | For defining variables as 'number of lengths swum' using each stroke, or equivalent |
**(ii)** Maximise $2x + y + 5z$
| B1 (2) | For a correct expression using their variables |
| B1 | For a correct expression using their variables |
$x + y + z \geq 8$
| B1 | For a correct expression using their variables |
$2x + 0.5y + z \leq 10$
| B1 (3) | For correct expressions using their variables |
$x \geq 2, y \geq 2, z \geq 2$
**(iii)** [Graph with shaded feasible region shown correctly] | M1 | For plotting the sloping lines correctly |
| A1 | For completely correct shading |
Vertices at $(2, 4), (2, 8), (3.3, 2.7)$
| M1 | For two correct vertices from their graph |
| A1 | For all three vertices correct to at least 1 dp |
$2x2 + 8 = 12$
$2x3.33 + 2.67 = 9.33$
| M1 | For calculating $P$ at vertices or using a valid line of constant profit or writing down their max point |
So maximum is when $x = 2$ and $y = 8$
| A1 (6) | For the correct values |
**(iv)** Swim 2 lengths using breaststroke, 8 lengths using backstroke and 2 lengths using butterfly
| B1 | For interpreting their solution in the context of the original problem (at least for $x$ and $y$) |
| B1 (2) | For calculating the number of marks for their solution |
| | $\mathbf{13}$ |5 Findlay is trying to get into his local swimming team. The coach will watch him swim and will then make his decision. Findlay must swim at least two lengths using each stroke and must swim at least 8 lengths in total, taking at most 10 minutes.
Findlay needs to put together a routine that includes breaststroke, backstroke and butterfly. The table shows how Findlay expects to perform with each stroke.
\begin{center}
\begin{tabular}{ | l | l | c | }
\hline
\multicolumn{1}{|c|}{Stroke} & \multicolumn{1}{c|}{Style marks} & Time taken \\
\hline
Breaststroke & 2 marks per length & 2 minutes per length \\
\hline
Backstroke & 1 mark per length & 0.5 minutes per length \\
\hline
Butterfly & 5 marks per length & 1 minute per length \\
\hline
\end{tabular}
\end{center}
Findlay needs to work out how many lengths to swim using each stroke to maximise his expected total number of style marks.\\
(i) Identify appropriate variables for Findlay's problem and write down the objective function, to be maximised, in terms of these variables.\\
(ii) Formulate a constraint for the total number of lengths swum, a constraint for the time spent swimming and constraints on the number of lengths swum using each stroke.
Findlay decides that he will swim two lengths using butterfly. This reduces his problem to the following LP formulation:
$$\begin{array} { l c }
\text { maximise } & P = 2 x + y , \\
\text { subject to } & x + y \geqslant 6 , \\
& 4 x + y \leqslant 16 , \\
& x \geqslant 2 , y \geqslant 2 ,
\end{array}$$
with $x$ and $y$ both integers.\\
(iii) Use a graphical method to identify the feasible region for this problem. Write down the coordinates of the vertices of the feasible region and hence find the integer values of $x$ and $y$ that maximise $P$.\\
(iv) Interpret your solution for Findlay.
\hfill \mbox{\textit{OCR D1 2006 Q5 [13]}}