| Exam Board | OCR |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Formulation from word problem |
| Difficulty | Moderate -0.8 This is a straightforward linear programming formulation question requiring students to identify variables, write constraints from given information, and solve graphically. The setup is clearly structured with explicit time allocations, and the constraints are direct translations requiring only basic arithmetic (e.g., 30x + 10y ≤ 180). This is a standard D1 textbook exercise with no novel problem-solving required beyond applying the standard LP method. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\) = number of clients on programme \(X\); \(y\) = number of clients on programme \(Y\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Spin cycle: \(30x + 10y \leq 180 \Rightarrow 3x + y \leq 18\) | B1 | |
| Rower: \(10x \leq 40 \Rightarrow x \leq 4\) | B1 | |
| Free weights: \(20x + 30y \leq 300 \Rightarrow 2x + 3y \leq 30\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x \geq 0\), \(y \geq 0\), \(x+y \leq 12\), \(x\) and \(y\) integer | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct lines drawn for all constraints | M1 A1 | |
| Feasible region correctly identified | A1 | |
| Optimal point found by vertex testing or objective line | M1 | |
| \(x=4\), \(y=6\), \(P = 9(4)+6(6)=72\) | A1 A1 |
# Question 3:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$ = number of clients on programme $X$; $y$ = number of clients on programme $Y$ | B1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Spin cycle: $30x + 10y \leq 180 \Rightarrow 3x + y \leq 18$ | B1 | |
| Rower: $10x \leq 40 \Rightarrow x \leq 4$ | B1 | |
| Free weights: $20x + 30y \leq 300 \Rightarrow 2x + 3y \leq 30$ | B1 | |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x \geq 0$, $y \geq 0$, $x+y \leq 12$, $x$ and $y$ integer | B1 | |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct lines drawn for all constraints | M1 A1 | |
| Feasible region correctly identified | A1 | |
| Optimal point found by vertex testing or objective line | M1 | |
| $x=4$, $y=6$, $P = 9(4)+6(6)=72$ | A1 A1 | |
---3 Maggie is a personal trainer. She has twelve clients who want to lose weight. She decides to put some of her clients on weight loss programme $X$, some on programme $Y$ and the rest on programme $Z$. Each programme involves a strict diet; in addition programmes $X$ and $Y$ involve regular exercise at Maggie's home gym. The programmes each last for one month.
In addition to the diet, clients on programme $X$ spend 30 minutes each day on the spin cycle, 10 minutes each day on the rower and 20 minutes each day on free weights. At the end of one month they can each expect to have lost 9 kg more than a client on just the diet.
In addition to the diet, clients on programme $Y$ spend 10 minutes each day on the spin cycle and 30 minutes each day on free weights; they do not use the rower. At the end of one month they can each expect to have lost 6 kg more than a client on just the diet.
Because of other clients who use Maggie's home gym, the spin cycle is available for the weight loss clients for 180 minutes each day, the rower for 40 minutes each day and the free weights for 300 minutes each day. Only one client can use each piece of apparatus at any one time.
Maggie wants to decide how many clients to put on each programme to maximise the total expected weight loss at the end of the month. She models the objective as follows.
$$\text { Maximise } P = 9 x + 6 y$$
(i) What do the variables $x$ and $y$ represent?\\
(ii) Write down and simplify the constraints on the values of $x$ and $y$ from the availability of each of the pieces of apparatus.\\
(iii) What other constraints and restrictions apply to the values of $x$ and $y$ ?\\
(iv) Use a graphical method to represent the feasible region for Maggie's problem. You should use graph paper and choose scales so that the feasible region can be clearly seen. Hence determine how many clients should be put on each programme.
\hfill \mbox{\textit{OCR D1 2010 Q3 [11]}}