| Exam Board | Edexcel |
|---|---|
| Module | D1 (Decision Mathematics 1) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear Programming |
| Type | Graphical optimization with objective line |
| Difficulty | Moderate -0.5 This is a standard textbook linear programming problem requiring routine formulation of constraints, graphical representation, and finding the optimal vertex. Part (d) adds a simple sensitivity analysis. While multi-step, it follows a completely standard D1 template with no novel problem-solving required, making it slightly easier than average A-level difficulty. |
| Spec | 7.06a LP formulation: variables, constraints, objective function7.06b Slack variables: converting inequalities to equations7.06c Working with constraints: algebra and ad hoc methods7.06d Graphical solution: feasible region, two variables7.06e Sensitivity analysis: effect of changing coefficients |
| \(\frac { \text { Participants } } { \text { Children } }\) | 7 hours windsurfing, 2 hours sailing | \(\frac { \text { Revenue } } { \pounds 50 }\) |
| Adults | 5 hours windsurfing, 6 hours sailing | \(\pounds 100\) |
| Answer | Marks |
|---|---|
| Subject to \(x + y \leq 90\), \(y \leq 40\), \(7x + 5y \leq 600\), \(2x + 6y \leq 300\) (or \(x + 3y \leq 150\)), \(x \geq 0, y \geq 0\) | M2 A2 |
| Answer | Marks |
|---|---|
| [Graph showing feasible region with constraints plotted, including lines for \(7x + 5y = 600\), \(x + y = 90\), \(y = 40\), \(x + 3y = 150\)] | B4 |
| Answer | Marks |
|---|---|
| Considering vertices/lines of constant revenue: maximum \(R\) where \(x + y = 90\) meets \(x + 3y = 150\) giving \(x = 60, y = 30\). Therefore should accept 60 children and 30 adults giving revenue of £6000 | M2 A2 |
| Answer | Marks | Guidance |
|---|---|---|
| No, as the windsurfing restriction is not a factor in optimal solution | B2 | (14) |
**Part (a)**
Let $x$ = no. of children and $y$ = no. of adults
Maximise $R = 50x + 100y$
Subject to $x + y \leq 90$, $y \leq 40$, $7x + 5y \leq 600$, $2x + 6y \leq 300$ (or $x + 3y \leq 150$), $x \geq 0, y \geq 0$ | M2 A2 |
**Part (b)**
[Graph showing feasible region with constraints plotted, including lines for $7x + 5y = 600$, $x + y = 90$, $y = 40$, $x + 3y = 150$] | B4 |
**Part (c)**
Considering vertices/lines of constant revenue: maximum $R$ where $x + y = 90$ meets $x + 3y = 150$ giving $x = 60, y = 30$. Therefore should accept 60 children and 30 adults giving revenue of £6000 | M2 A2 |
**Part (d)**
No, as the windsurfing restriction is not a factor in optimal solution | B2 | (14) |
---6. The manager of a new leisure complex needs to maximise the Revenue $( \pounds R )$ from providing the following two weekend programmes.
\begin{center}
\begin{tabular}{ c c c }
$\frac { \text { Participants } } { \text { Children } }$ & 7 hours windsurfing, 2 hours sailing & $\frac { \text { Revenue } } { \pounds 50 }$ \\
Adults & 5 hours windsurfing, 6 hours sailing & $\pounds 100$ \\
\end{tabular}
\end{center}
The following restrictions apply to each weekend.\\
No more than 90 participants can be accommodated.\\
There must be at most 40 adults.\\
A maximum of 600 person-hours of windsurfing can be offered.\\
A maximum of 300 person-hours of sailing can be offered.
\begin{enumerate}[label=(\alph*)]
\item Formulate the above information as a linear programming problem, listing the constraints as inequalities and stating the objective function $R$.
\item On graph paper, illustrate the constraints, indicating clearly the feasible region.
\item Solve the problem graphically, stating how many adults and how many children should be accepted each weekend and what the revenue will be.
The manager is considering buying more windsurfing equipment at a cost of $\pounds 2000$. This would increase windsurfing provision by $10 \%$.
\item State, with a reason, whether such a purchase would be cost effective.
\end{enumerate}
\hfill \mbox{\textit{Edexcel D1 Q6 [14]}}