Reverse engineering constraints from graph

4 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{f2b85dfb-49df-4ea5-b118-9b95f0b27bad-03_1025_826_374_657}

1. Write down inequalities that define the feasible region.
2. Find the coordinates of the four vertices of the feasible region. The objective is to maximise \(P\), where \(P = x + 2 y\).
3. Find the values of \(x\) and \(y\) that maximise \(P\), and the corresponding maximum value of \(P\). The objective is changed to minimise \(Q\), where \(Q = 2 x - y\).
4. Find the minimum value of \(Q\) and describe the set of feasible points for which \(Q\) takes this value.
5. Show that there are no points in the feasible region for which the value of \(P\) is the same as the value of \(Q\).

4. \begin{figure}[h] \end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

1. Determine the inequalities that define the feasible region.
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P = 2 x + 3 y\).
3. Use point testing at each vertex to find the optimal vertex, \(V\), of the feasible region and state the corresponding value of \(P\) at \(V\).
   (3) The objective is changed to maximise \(Q = 2 x + k y\), where \(k\) is a constant.
4. Find the range of values of \(k\) for which the vertex identified in (c) is still optimal.
   (2)

4. \begin{figure}[h] \end{figure} Figure 4 shows three of the six constraints for a linear programming problem in \(x\) and \(y\) The unshaded region and its boundaries satisfy these three constraints.

1. State these three constraints as simplified inequalities with integer coefficients. The variables \(x\) and \(y\) represent the number of orange fish and the number of blue fish, respectively, that are to be kept in an aquarium. The number of fish in the aquarium is subject to these three further constraints
   • there must be at least one blue fish
   • the orange fish must not outnumber the blue fish by more than ten
   • there must be no more than five blue fish for every orange fish
   • Write each of these three constraints as a simplified inequality with integer coefficients.
   • Represent these three constraints by adding lines and shading to Diagram 1 in the answer book, labelling the feasible region, \(R\)
   The total value (in pounds) of the fish in the aquarium is given by the objective function $$\text { Maximise } P = 3 x + 5 y$$
2. 1. Use the objective line method to determine the optimal point of the feasible region, giving its coordinates as exact fractions.
   2. Hence find the maximum total value of the fish in the aquarium, stating the optimal number of orange fish and the optimal number of blue fish. \begin{table}[h]
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      \end{table} \section*{Pearson Edexcel Level 3 GCE} \section*{Friday 17 May 2024} Afternoon \section*{Further Mathematics} Advanced Subsidiary
      Further Mathematics options
      27: Decision Mathematics 1
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      \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}

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