| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Easy -1.3 This is a standard textbook linear programming question requiring routine application of well-practiced techniques: formulating inequalities from word problems, sketching constraint lines, identifying the feasible region, and using the objective line method. All steps are algorithmic with no novel insight required, making it easier than average A-level material. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(7x + 5y \leq 350\) | M1 A1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \(y \leq 20\) e.g. make at most 20 small baskets; \(y \leq 4x\) e.g. the number of small (\(y\)) baskets is at most 4 times the number of large baskets (\(x\)). {E.g if \(y = 40\), \(x =10, 11, 12\) etc. or if \(x = 10\), \(y = 40, 39, 38\)} | B1; B1 | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: (see graph next page) Draw three lines correctly. Label R | B3,2,1,0; B1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: \((P=) 2x + 3y\) | B1 | (1) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Profit line or point testing. \(x = 35.7\) \(y = 20\) precise point found. Need integers so optimal point in R is (35, 20); Profit (£)130 | M1 A1; B1; B1;B1 | (5) |
**Part (a)**
Answer: $7x + 5y \leq 350$ | M1 A1 | (2) | 1M1: Coefficients correct (condone swapped $x$ and $y$ coefficients) need 350 and any inequality. 1A1: cso.
**Part (b)**
Answer: $y \leq 20$ e.g. make at most 20 small baskets; $y \leq 4x$ e.g. the number of small ($y$) baskets is at most 4 times the number of large baskets ($x$). {E.g if $y = 40$, $x =10, 11, 12$ etc. or if $x = 10$, $y = 40, 39, 38$} | B1; B1 | (2) | 1B1: cao. 2B1: cao, test their statement, need both = and < aspects.
**Part (c)**
Answer: (see graph next page) Draw three lines correctly. Label R | B3,2,1,0; B1 | (4) | 1B1: One line drawn correctly. 2B1: Two lines drawn correctly. 3B1: Three lines drawn correctly. Check (10, 40) (0, 0) and axes. 4B1: R correct, but allow if one line is slightly out (1 small square).
**Part (d)**
Answer: $(P=) 2x + 3y$ | B1 | (1) | 1B1: cao accept an expression.
**Part (e)**
Answer: Profit line or point testing. $x = 35.7$ $y = 20$ precise point found. Need integers so optimal point in R is (35, 20); Profit (£)130 | M1 A1; B1; B1;B1 | (5) | 1M1: Attempt at profit line or attempt to test at least two vertices in their feasible region. 1A1: Correct profit line or correct testing of at least three vertices. **Point testing**: $(0,0)$ P= 0; $(5,20)$ P = 70; $(50,0)$ P = 100. $\left(35\frac{5}{7}, 20\right) = \left(\frac{250}{7}, 20\right)$ P $= 131\frac{3}{7} = \frac{920}{7}$ also $(35, 20)$ P = 130. Accept $(36,20)$ P = 132 for M but not A. **Objective line**: Accept gradient of 1/m for M mark or line close to correct gradient. 1B1: cao – accept x co-ordinates which round to 35.7. 2B1: cao. 3B1: cao.
---7. Rose makes hanging baskets which she sells at her local market. She makes two types, large and small. Rose makes $x$ large baskets and $y$ small baskets.
Each large basket costs $\pounds 7$ to make and each small basket costs $\pounds 5$ to make. Rose has $\pounds 350$ she can spend on making the baskets.
\begin{enumerate}[label=(\alph*)]
\item Write down an inequality, in terms of $x$ and $y$, to model this constraint.\\
(2)
Two further constraints are
$$\begin{aligned}
& y \leqslant 20 \text { and } \\
& y \leqslant 4 x
\end{aligned}$$
\item Use these two constraints to write down statements that describe the numbers of large and small baskets that Rose can make.
\item On the grid provided, show these three constraints and $x \geqslant 0 , y \geqslant 0$. Hence label the feasible region, R.
Rose makes a profit of $\pounds 2$ on each large basket and $\pounds 3$ on each small basket. Rose wishes to maximise her profit, £P.
\item Write down the objective function.
\item Use your graph to determine the optimal numbers of large and small baskets Rose should make, and state the optimal profit.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 2009 Q7 [14]}}