7.06d Graphical solution: feasible region, two variables

5. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(\quad P = 2 x + 3 y\) subject to $$\begin{aligned} x & \geqslant 25 \\ y & \geqslant 25 \\ 7 x + 8 y & \leqslant 840 \\ 4 y & \leqslant 5 x \\ 5 y & \geqslant 3 x \\ x , y & \geqslant 0 \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it R .
2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.
3. Calculate the exact coordinates of vertex V. Given that an integer solution is required,
4. determine the optimal solution with integer coordinates. You must make your method clear.

8. \begin{figure}[h] \end{figure} The graph in Figure 4 is being used to solve a linear programming problem. The four constraints have been drawn on the graph and the rejected regions have been shaded out. The four vertices of the feasible region \(R\) are labelled \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D .

1. Write down the constraints represented on the graph.
   (2) The objective function, P , is given by $$\mathrm { P } = x + k y$$ where \(k\) is a positive constant. The minimum value of the function P is given by the coordinates of vertex A and the maximum value of the function P is given by the coordinates of vertex D .
2. Find the range of possible values for \(k\). You must make your method clear.
   (Total 8 marks)

6. A linear programming problem in \(x\) and \(y\) is described as follows. Minimise \(C = 2 x + 3 y\) subject to $$\begin{aligned} x + y & \geqslant 8 \\ x & < 8 \\ 4 y & \geqslant x \\ 3 y & \leqslant 9 + 2 x \end{aligned}$$

1. Add lines and shading to Diagram 1 in the answer book to represent these constraints.
2. Hence determine the feasible region and label it R .
3. Use the objective line (ruler) method to find the exact coordinates of the optimal vertex, V, of the feasible region. You must draw and label your objective line clearly.
4. Calculate the corresponding value of \(C\) at V . The objective is now to maximise \(2 x + 3 y\), where \(x\) and \(y\) are integers.
5. Write down the optimal values of \(x\) and \(y\) and the corresponding maximum value of \(2 x + 3 y\). A further constraint, \(y \leqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
6. Determine the least value of \(k\) for which this additional constraint does not affect the feasible region.

8. Charlie needs to buy storage containers. There are two different types of storage container available, standard and deluxe. Standard containers cost \(\pounds 20\) and deluxe containers cost \(\pounds 65\). Let \(x\) be the number of standard containers and \(y\) be the number of deluxe containers. The maximum budget available is \(\pounds 520\)

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Three further constraints are: $$\begin{aligned} x & \geqslant 2 \\ - x + 24 y & \geqslant 24 \\ 7 x + 8 y & \leqslant 112 \end{aligned}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all four constraints. Hence determine the feasible region and label it R . The capacity of a deluxe container is \(50 \%\) greater than the capacity of a standard container. Charlie wishes to maximise the total capacity.
3. State an objective function, in terms of \(x\) and \(y\).
4. Use the objective line method to find the optimal vertex, V, of the feasible region. You must make your objective line clear and label the optimal vertex V.
5. Calculate the exact coordinates of vertex V.
6. Determine the number of each type of container that Charlie should buy. You must make your method clear and calculate the cost of purchasing the storage containers. Write your name here
   Final output \(\_\_\_\_\) (b)
   6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-22_807_1426_121_267} \captionsetup{labelformat=empty} \caption{Figure 5
   [0pt] [The total weight of the network is 384]}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-24_2651_1940_118_121} \includegraphics[max width=\textwidth, alt={}, center]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-25_2261_50_312_36} \section*{Q uestion 7 continued} (c) \(\_\_\_\_\) (d) \section*{Diagram 2} (Total 12 marks)
   □ 8.
   \includegraphics[max width=\textwidth, alt={}]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-26_1570_1591_260_189}
   Diagram 1 \section*{Q uestion 8 continued}
   \includegraphics[max width=\textwidth, alt={}]{22ff916a-4ba8-4e0c-9c53-e82b0aff0b98-28_2646_1833_116_118}

8. \begin{figure}[h] \end{figure} A company produces two products, X and Y .
Let \(x\) and \(y\) be the hourly production, in kgs, of X and Y respectively.
In addition to \(x \geqslant 0\) and \(y \geqslant 0\), two of the constraints governing the production are $$\begin{gathered} 12 x + 7 y \geqslant 840 \\ 4 x + 9 y \geqslant 720 \end{gathered}$$ These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out. Two further constraints are $$\begin{gathered} x \geqslant 20 \\ 3 x + 2 y \leqslant 360 \end{gathered}$$

1. Add two lines and shading to Figure 6 in your answer book to represent these inequalities.
2. Hence determine and label the feasible region, R. The company makes a profit of 70 p and 20 p per kilogram of X and Y respectively.
3. Write down an expression, in terms of \(x\) and \(y\), for the hourly profit, £P.
4. Mark points A and B on your graph where A and B represent the maximum and minimum values of P respectively. Make your method clear.
   (4)

7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is \(\pounds 40\). Each hard drive takes 2 hours to repair and the cost of components is \(\pounds 20\). Each keyboard takes 1 hour to repair and the cost of components is \(\pounds 5\). Each month, the business has 360 hours available for repairs and \(\pounds 2500\) available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of \(\pounds 80 , \pounds 35\) and \(\pounds 15\) respectively. The company repairs and sells \(x\) monitors, \(y\) hard drives and \(z\) keyboards each month. The company wishes to maximise its total profit. 7

1. Find five inequalities involving \(x , y\) and \(z\) for the company's problem.
   [0pt] [3 marks]
   7
2. (i) Find how many of each type of computer hardware the company should repair and sell each month.
   7 (b) (ii) Explain how you know that you had reached the optimal solution in part (b) (i).
   7 (b) (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
   [0pt] [1 mark]

4. The manager of a factory is planning the production schedule for the next three weeks for a range of cabinets. The following constraints apply to the production schedule.

• The total number of cabinets produced in week 3 cannot be fewer than the total number produced in weeks 1 and 2
• At most twice as many cabinets must be produced in week 3 as in week 2
• The number of cabinets produced in weeks 2 and 3 must, in total, be at most 125

The production cost for each cabinet produced in weeks 1,2 and 3 is \(\pounds 250 , \pounds 275\) and \(\pounds 200\) respectively.
The factory manager decides to formulate a linear programming problem to find a production schedule that minimises the total cost of production. The objective is to minimise \(250 x + 275 y + 200 z\)

1. Explain what the variables \(x , y\) and \(z\) represent.
2. Write down the constraints of the linear programming problem in terms of \(x , y\) and \(z\). Due to demand, exactly 150 cabinets must be produced during these three weeks. This reduces the constraints to $$\begin{gathered} x + y \leqslant 75 \\ x + 3 y \geqslant 150 \\ x \geqslant 25 \\ y \geqslant 0 \end{gathered}$$ which are shown in Diagram 1 in the answer book.
   Given that the manager does not want any cabinets left unfinished at the end of a week,
3. 1. use a graphical approach to solve the linear programming problem and hence determine the production schedule which minimises the cost of production. You should make your method and working clear.
   2. Find the minimum total cost of the production schedule.

4. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a maximisation linear programming problem in \(x\) and \(y\), where \(x \geqslant 0\) and \(y \geqslant 0\). The unshaded area, including its boundaries, forms the feasible region, \(R\). An objective line has been drawn and labelled on the graph.

1. List the constraints as simplified inequalities with integer coefficients. The optimal value of the objective function is 216
2. 1. Calculate the exact coordinates of the optimal vertex.
   2. Hence derive the objective function. Given that \(x\) represents the number of small flower pots and \(y\) represents the number of large flower pots supplied to a customer,
3. deduce the optimal solution to the problem. TOTAL FOR DECISION MATHEMATICS 1 IS 40 MARKS END

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]
      \captionsetup{labelformat=empty} \caption{Please check the examination details below before entering your candidate information}

      Candidate surname								Other names
      Centre Number				Candidate Number
      □			□

      \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
      (Part of options D, F, H and K) \section*{D1 Answer Book} Do not return the question paper with the answer book.
      1. \(\begin{array} { l l l l l l l l l l l } 4 & 6.5 & 7 & 1.3 & 2 & 5 & 1.5 & 6 & 4.5 & 6 & 1 \end{array}\) 2.
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_435_815_392_463}
      \section*{Diagram 1} Use this diagram only if you need to redraw your activity network. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-12_442_820_2043_458} Copy of Diagram 1

      VJYV SIHI NI JIIYM ION OC

      V346 SIHI NI JLIYM ION OC

      V34V SIHI NI IIIIM ION OC

      Key: \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ca57c64b-0b33-4179-be7f-684bd6ea2162-13_1217_1783_451_236} \captionsetup{labelformat=empty} \caption{Diagram 2}
      \end{figure} 3. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-14_2463_1240_339_465}
      Shortest route from A to M:
      Length of shortest route from A to M:
      
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-16_3038_2264_0_0}
      \includegraphics[max width=\textwidth, alt={}]{ca57c64b-0b33-4179-be7f-684bd6ea2162-17_1103_1247_397_512}
      \section*{Diagram 1} \section*{There is a copy of Diagram 1 on page 11 if you need to redraw your graph.}

      VJYV SIHI NI JIIIM ION OC

      V341 S1H1 NI JLIYM ION OA

      V34V SIHI NI IIIVM ION OC

      Use this diagram only if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{ca57c64b-0b33-4179-be7f-684bd6ea2162-19_1108_1252_1606_509} Copy of Diagram 1

2. \begin{figure}[h] \end{figure} A teacher buys pens and pencils. The number of pens, \(x\), and the number of pencils, \(y\), that he buys can be represented by a linear programming problem as shown in Figure 2, which models the following constraints: $$\begin{aligned} 8 x + 3 y & \leqslant 480 \\ 8 x + 7 y & \geqslant 560 \\ y & \geqslant 4 x \\ x , y & \geqslant 0 \end{aligned}$$ The total cost, in pence, of buying the pens and pencils is given by $$C = 12 x + 15 y$$ Determine the number of pens and the number of pencils which should be bought in order to minimise the total cost. You should make your method and working clear.

4. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region.

1. Write down the inequalities that define \(R\). The objective is to maximise \(P\), where \(P = 3 x + y\)
2. Obtain the exact value of \(P\) at each of the three vertices of \(R\) and hence find the optimal vertex, \(V\). The objective is changed to maximise \(Q\), where \(Q = 3 x + a y\). Given that \(a\) is a constant and the optimal vertex is still \(V\),
3. find the range of possible values of \(a\).

7. \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. The objective is to maximise \(P = x + k y\), where \(k\) is a positive constant.
The optimal vertex of \(R\) is to be found using the Simplex algorithm.

1. Set up an initial tableau for solving this linear programming problem using the Simplex algorithm. After two iterations of the Simplex algorithm a possible tableau \(T\) is

   b.v.	\(x\)	\(y\)	\(S _ { 1 }\)	\(s _ { 2 }\)	\(S _ { 3 }\)	\(s _ { 4 }\)	Value
   \(s _ { 1 }\)	0	0	1	\(- \frac { 3 } { 5 }\)	0	\(\frac { 1 } { 5 }\)	1
   \(x\)	1	0	0	\(\frac { 1 } { 5 }\)	0	\(- \frac { 2 } { 5 }\)	2
   \(S _ { 3 }\)	0	0	0	\(- \frac { 11 } { 5 }\)	1	\(\frac { 12 } { 5 }\)	22
   \(y\)	0	1	0	\(\frac { 2 } { 5 }\)	0	\(\frac { 1 } { 5 }\)	5
   \(P\)	0	0	0	\(\frac { 1 } { 5 } + \frac { 2 } { 5 } k\)	0	\(- \frac { 2 } { 5 } + \frac { 1 } { 5 } k\)	\(5 k + 2\)

2. State the value of each variable after the second iteration.
   (1) It is given that \(T\) does not give an optimal solution to the linear programming problem.
   After a third iteration of the Simplex algorithm the resulting tableau does give an optimal solution to the problem.
3. Perform the third iteration of the Simplex algorithm and hence determine the range of possible values for \(P\). You should state the row operations you use and make your method and working clear.
   (9)

6 An online magazine consists of an editorial, articles, reviews and advertisements.
The magazine must have a total of at least 12 pages. The editorial always takes up exactly half a page. There must be at least 3 pages of articles and at most 1.5 pages of reviews. At least a quarter but fewer than half of the pages in the magazine must be used for advertisements. Let \(x\) be the number of pages used for articles, \(y\) be the number of pages used for reviews and \(z\) be the number of pages used for advertisements. The constraints on the values of \(x , y\) and \(z\) are: $$\begin{aligned} & x + y + z \geqslant 11.5 \\ & x \geqslant 3 \\ & y \leqslant 1.5 \\ & 2 x + 2 y - 2 z + 1 \geqslant 0 \\ & 2 x + 2 y - 6 z + 1 \leqslant 0 \\ & y \geqslant 0 \end{aligned}$$

1. (a) Explain why \(x + y + z \geqslant 11.5\).
   (b) Explain why only one non-negativity constraint is needed.
   (c) Show that the requirement that at least one quarter of the pages in the magazine must be used for advertisements leads to the constraint \(2 x + 2 y - 6 z + 1 \leqslant 0\). Advertisements bring in money but are not popular with the subscribers. The editor decides to limit the number of pages of advertisements to at most four.
2. Graph the feasible region in the case when \(z = 4\) using the axes in the Printed Answer Booklet. To be successful the magazine needs to maximise the number of subscribers.
   The editor has found that when \(z \leqslant 4\) the expected number of subscribers is given by \(P = 300 x + 400 y\).
3. (a) What is the maximum expected number of subscribers when \(z = 4\) ?
   (b) By first considering the feasible region for \(z = k\), where \(k \leqslant 4\), find an expression for the maximum number of subscribers in terms of \(k\). \section*{END OF QUESTION PAPER} \section*{OCR} \section*{Oxford Cambridge and RSA}

6 Jack is making pizzas for a party. He can make three types of pizza:

• There is enough dough to make 30 pizzas.
• Type Z pizzas use vegan cheese. Jack only has enough vegan cheese to make 2 type Z pizzas.
• At least half the pizzas made must be suitable for vegetarians.
• Jack has enough onions to make 50 type X pizzas or 20 type Y pizzas or 20 type Z pizzas or some mixture of the three types.

Suppose that Jack makes \(x\) type X pizzas, \(y\) type Y pizzas and \(z\) type Z pizzas.

1. Formulate the constraints above in terms of the non-negative, integer valued variables \(x , y\) and \(z\), together with non-negative slack variables \(s , t , u\) and \(v\). Jack wants to find out the maximum total number of pizzas that he can make.
2. 1. Set up an initial simplex tableau for Jack's problem.
   2. Carry out one iteration of the simplex algorithm, choosing your pivot so that \(x\) becomes a basic variable. When Jack carries out the simplex algorithm his final tableau is:

      \(P\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)	\(v\)	RHS
      1	0	0	0	0	0	\(\frac { 3 } { 7 }\)	\(\frac { 2 } { 7 }\)	\(28 \frac { 4 } { 7 }\)
      0	0	0	0	1	0	\(- \frac { 3 } { 7 }\)	\(- \frac { 2 } { 7 }\)	\(1 \frac { 3 } { 7 }\)
      0	0	0	1	0	1	0	0	2
      0	1	0	0	0	0	\(\frac { 5 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)
      0	0	1	1	0	0	\(- \frac { 2 } { 7 }\)	\(\frac { 1 } { 7 }\)	\(14 \frac { 2 } { 7 }\)

3. Use this final tableau to deduce how many pizzas of each type Jack should make. Jack knows that some of the guests are vegans. He decides to make 2 pizzas of type \(Z\).
4. 1. Plot the feasible region for \(x\) and \(y\).
   2. Complete the branch-and-bound formulation in the Printed Answer Booklet to find the number of pizzas of each type that Jack should make.
      You should branch on \(x\) first. \section*{END OF QUESTION PAPER}

7. A tailor makes two types of garment, \(A\) and \(B\). He has available \(70 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(90 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(A\) requires \(1 \mathrm {~m} ^ { 2 }\) of cotton fabric and \(3 \mathrm {~m} ^ { 2 }\) of woollen fabric. Garment \(B\) requires \(2 \mathrm {~m} ^ { 2 }\) of each fabric. The tailor makes \(x\) garments of type \(A\) and \(y\) garments of type \(B\).

1. Explain why this can be modelled by the inequalities $$\begin{aligned} & x + 2 y \leq 70 \\ & 3 x + 2 y \leq 90 \\ & x \geq 0 , y \geq 0 \end{aligned}$$ The tailor sells type \(A\) for \(\pounds 30\) and type \(B\) for \(\pounds 40\). All garments made are sold. The tailor wishes to maximise his total income.
2. Set up an initial Simplex tableau for this problem.
3. Solve the problem using the Simplex algorithm. Figure 4 shows a graphical representation of the feasible region for this problem. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-008_686_1277_1319_453} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
4. Obtain the coordinates of the points A, \(C\) and \(D\).
5. Relate each stage of the Simplex algorithm to the corresponding point in Fig. 4. 6689 Decision Mathematics 1 (New Syllabus) Order of selecting edges
   Final tree
   (b) Minimum total length of cable
   Question 4 to be answered on this page
   (a) \(A\)
   • Monday (M) \(B\) ◯
   • Tuesday (Tu) \(C \odot\)
   • Wednesday (W) \(D\) ◯
   • Thursday (Th) \(E\) -
   • Friday (F)
     (b)
     (c)
   Question 5 to be answered on this page
   Key
   (a) Early
   Time
   Late
   Time \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_433_227_534_201} \(F ( 3 )\) \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_117_222_1016_992}
   H(4) K(6)
   (b) Critical activities
   Length of critical path \(\_\_\_\_\) (c) \includegraphics[max width=\textwidth, alt={}, center]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-011_492_1604_1925_266} Question 6 to be answered on pages 4 and 5
   (a)
   1. SAET \(\_\_\_\_\)
   2. SBDT \(\_\_\_\_\)
   3. SCFT \(\_\_\_\_\)
   
   (b) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-012_691_1307_893_384} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure} (c) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_699_1314_167_382} \captionsetup{labelformat=empty} \caption{Diagram 2}
   \end{figure} Flow augmenting routes
   (d) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{12f9ae59-b2ff-4a03-9ac9-c61dbaf8c9f5-013_693_1314_1368_382} \captionsetup{labelformat=empty} \caption{Diagram 3}
   \end{figure} (e) \(\_\_\_\_\)

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.

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint.
   (2) Two further constraints are $$\begin{aligned} & y \leqslant 20 \text { and } \\ & y \leqslant 4 x \end{aligned}$$
2. Use these two constraints to write down statements that describe the numbers of large and small baskets that Rose can make.
3. 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.
4. Write down the objective function.
5. Use your graph to determine the optimal numbers of large and small baskets Rose should make, and state the optimal profit.

4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{194d16e0-8e05-45c0-8948-99808440ed2a-005_1349_1395_408_294}

1. On the feasible region, find:
   1. the maximum value of \(2 x + 3 y\);
   2. the maximum value of \(3 x + 2 y\);
   3. the minimum value of \(- 2 x + y\).
2. Find the 5 inequalities that define the feasible region.

4 The diagram shows the feasible region of a linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{4a186c87-5f84-4ec3-8cc3-a0ed8721b040-04_1349_1395_408_294}

1. On the feasible region, find:
   1. the maximum value of \(2 x + 3 y\);
   2. the maximum value of \(3 x + 2 y\);
   3. the minimum value of \(- 2 x + y\).
2. Find the 5 inequalities that define the feasible region.

6 [Figure 1, printed on the insert, is provided for use in this question.]
Dino is to have a rectangular swimming pool at his villa.
He wants its width to be at least 2 metres and its length to be at least 5 metres.
He wants its length to be at least twice its width.
He wants its length to be no more than three times its width.
Each metre of the width of the pool costs \(\pounds 1000\) and each metre of the length of the pool costs \(\pounds 500\). He has \(\pounds 9000\) available. Let the width of the pool be \(x\) metres and the length of the pool be \(y\) metres.

1. Show that one of the constraints leads to the inequality $$2 x + y \leqslant 18$$
2. Find four further inequalities.
3. On Figure 1, draw a suitable diagram to show the feasible region.
4. Use your diagram to find the maximum width of the pool. State the corresponding length of the pool.

2 [Figure 1, printed on the insert, is provided for use in this question.]
The feasible region of a linear programming problem is represented by $$\begin{aligned} x + y & \leqslant 30 \\ 2 x + y & \leqslant 40 \\ y & \geqslant 5 \\ x & \geqslant 4 \\ y & \geqslant \frac { 1 } { 2 } x \end{aligned}$$

1. On Figure 1, draw a suitable diagram to represent these inequalities and indicate the feasible region.
2. Use your diagram to find the maximum value of \(F\), on the feasible region, in the case where:
   1. \(F = 3 x + y\);
   2. \(F = x + 3 y\).

4 [Figure 2, printed on the insert, is provided for use in this question.]
Each year, farmer Giles buys some goats, pigs and sheep.
He must buy at least 110 animals.
He must buy at least as many pigs as goats.
The total of the number of pigs and the number of sheep that he buys must not be greater than 150 .
Each goat costs \(\pounds 16\), each pig costs \(\pounds 8\) and each sheep costs \(\pounds 24\).
He has \(\pounds 3120\) to spend on the animals.
At the end of the year, Giles sells all of the animals. He makes a profit of \(\pounds 70\) on each goat, \(\pounds 30\) on each pig and \(\pounds 50\) on each sheep. Giles wishes to maximize his total profit, \(\pounds P\). Each year, Giles buys \(x\) goats, \(y\) pigs and \(z\) sheep.

1. Formulate Giles's situation as a linear programming problem.
2. One year, Giles buys 30 sheep.
   1. Show that the constraints for Giles's situation for this year can be modelled by $$y \geqslant x , \quad 2 x + y \leqslant 300 , \quad x + y \geqslant 80 , \quad y \leqslant 120$$ (2 marks)
   2. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
      (8 marks)
   3. Find Giles's maximum profit for this year and the number of each animal that he must buy to obtain this maximum profit.
      (3 marks)

3 [Figure 1, printed on the insert, is provided for use in this question.]
The feasible region of a linear programming problem is represented by the following: $$\begin{aligned} x \geqslant 0 , y & \geqslant 0 \\ x + 4 y & \leqslant 36 \\ 4 x + y & \leqslant 68 \\ y & \leqslant 2 x \\ y & \geqslant \frac { 1 } { 4 } x \end{aligned}$$

1. On Figure 1, draw a suitable diagram to represent the inequalities and indicate the feasible region.
2. Use your diagram to find the maximum value of \(P\), stating the corresponding coordinates, on the feasible region, in the case where:
   1. \(P = x + 5 y\);
   2. \(\quad P = 5 x + y\).

8 [Figure 2, printed on the insert, is provided for use in this question.]
A company makes two types of boxes of chocolates, executive and luxury.
Every hour the company must make at least 15 of each type and at least 35 in total.
Each executive box contains 20 dark chocolates and 12 milky chocolates.
Each luxury box contains 10 dark chocolates and 18 milky chocolates.
Every hour the company has 600 dark chocolates and 600 milky chocolates available.
The company makes a profit of \(\pounds 1.50\) on each executive box and \(\pounds 1\) on each luxury box.
The company makes and sells \(x\) executive boxes and \(y\) luxury boxes every hour.
The company wishes to maximise its hourly profit, \(\pounds P\).

1. Show that one of the constraints leads to the inequality \(2 x + 3 y \leqslant 100\).
2. Formulate the company's situation as a linear programming problem.
3. On Figure 2, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and an objective line.
4. Use your diagram to find the maximum hourly profit.

6 [Figure 3, printed on the insert, is provided for use in this question.]
Ernesto is to plant a garden with two types of tree: palms and conifers.
He is to plant at least 10, but not more than 80 palms.
He is to plant at least 5 , but not more than 40 conifers.
He cannot plant more than 100 trees in total. Each palm needs 20 litres of water each day and each conifer needs 60 litres of water each day. There are 3000 litres of water available each day. Ernesto makes a profit of \(\pounds 2\) on each palm and \(\pounds 1\) on each conifer that he plants and he wishes to maximise his profit. Ernesto plants \(x\) palms and \(y\) conifers.

1. Formulate Ernesto's situation as a linear programming problem.
2. On Figure 3, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region and the direction of the objective line.
3. Find the maximum profit for Ernesto.
4. Ernesto introduces a new pricing structure in which he makes a profit of \(\pounds 1\) on each palm and \(\pounds 4\) on each conifer. Find Ernesto's new maximum profit and the number of each type of tree that he should plant to obtain this maximum profit.

5 [Figure 2, printed on the insert, is provided for use in this question.]
The Jolly Company sells two types of party pack: excellent and luxury.
Each excellent pack has five balloons and each luxury pack has ten balloons.
Each excellent pack has 32 sweets and each luxury pack has 8 sweets.
The company has 1500 balloons and 4000 sweets available.
The company sells at least 50 of each type of pack and at least 140 packs in total.
The company sells \(x\) excellent packs and \(y\) luxury packs.

1. Show that the above information can be modelled by the following inequalities. $$x + 2 y \leqslant 300 , \quad 4 x + y \leqslant 500 , \quad x \geqslant 50 , \quad y \geqslant 50 , \quad x + y \geqslant 140$$ (4 marks)
2. The company sells each excellent pack for 80 p and each luxury pack for \(\pounds 1.20\). The company needs to find its minimum and maximum total income.
   1. On Figure 2, draw a suitable diagram to enable this linear programming problem to be solved graphically, indicating the feasible region and an objective line.
   2. Find the company's maximum total income and state the corresponding number of each type of pack that needs to be sold.
   3. Find the company's minimum total income and state the corresponding number of each type of pack that needs to be sold.