7.06d Graphical solution: feasible region, two variables

5 The feasible region of a linear programming problem is determined by the following: $$\begin{aligned} x & \geqslant 1 \\ y & \geqslant 3 \\ x + y & \geqslant 5 \\ x + y & \leqslant 12 \\ 3 x + 8 y & \leqslant 64 \end{aligned}$$

1. On the grid below, draw a suitable diagram to represent the inequalities and indicate the feasible region.
2. Use your diagram to find, on the feasible region:
   1. the maximum value of \(3 x + y\);
   2. the maximum value of \(2 x + 3 y\);
   3. the minimum value of \(- 2 x + y\). In each case, state the coordinates of the point corresponding to your answer.
      [0pt] [6 marks]

9 A company producing chicken food makes three products, Basic, Premium and Supreme, from wheat, maize and barley. A tonne \(( 1000 \mathrm {~kg} )\) of Basic uses 400 kg of wheat, 200 kg of maize and 400 kg of barley.
A tonne of Premium uses 400 kg of wheat, 500 kg of maize and 100 kg of barley.
A tonne of Supreme uses 600 kg of wheat, 200 kg of maize and 200 kg of barley.
The company has 130 tonnes of wheat, 70 tonnes of maize and 72 tonnes of barley available. The company must make at least 75 tonnes of Supreme.
The company makes \(\pounds 50\) profit per tonne of Basic, \(\pounds 100\) per tonne of Premium and \(\pounds 150\) per tonne of Supreme. They plan to make \(x\) tonnes of Basic, \(y\) tonnes of Premium and \(z\) tonnes of Supreme.

1. Write down four inequalities representing the constraints (in addition to \(x , y \geqslant 0\) ).
   [0pt] [4 marks]
2. The company want exactly half the production to be Supreme. Show that the constraints in part (a) become $$\begin{aligned} x + y & \leqslant 130 \\ 4 x + 7 y & \leqslant 700 \\ 2 x + y & \leqslant 240 \\ x + y & \geqslant 75 \\ x & \geqslant 0 \\ y & \geqslant 0 \end{aligned}$$
3. On the grid opposite, illustrate all the constraints and label the feasible region.
4. Write an expression for \(P\), the profit for the whole production, in terms of \(x\) and \(y\) only.
   [0pt] [2 marks]
5. 1. By drawing an objective line on your graph, or otherwise, find the values of \(x\) and \(y\) which give the maximum profit.
      [0pt] [2 marks]
   2. State the maximum profit and the amount of each product that must be made.
      [0pt] [2 marks] \section*{Answer space for question 9}
      \includegraphics[max width=\textwidth, alt={}]{f5890e58-38c3-413c-8762-6f80ce6dcec7-21_1349_1728_310_148}
      QUESTION
      PART
      REFERENCE \includegraphics[max width=\textwidth, alt={}, center]{f5890e58-38c3-413c-8762-6f80ce6dcec7-24_2488_1728_219_141}

8 Nerys runs a cake shop. In November and December she sells Christmas hampers. She makes up the hampers herself, in two sizes: Luxury and Special. Each day, Nerys prepares \(x\) Luxury hampers and \(y\) Special hampers.
It takes Nerys 10 minutes to prepare a Luxury hamper and 15 minutes to prepare a Special hamper. She has 6 hours available, each day, for preparing hampers. From past experience, Nerys knows that each day:

• she will need to prepare at least 5 hampers of each size
• she will prepare at most a total of 32 hampers
• she will prepare at least twice as many Luxury hampers as Special hampers.

Each Luxury hamper that Nerys prepares makes her a profit of \(\pounds 15\); each Special hamper makes a profit of \(\pounds 20\). Nerys wishes to maximise her daily profit, \(\pounds P\).

1. Show that \(x\) and \(y\) must satisfy the inequality \(2 x + 3 y \leqslant 72\).
2. In addition to \(x \geqslant 5\) and \(y \geqslant 5\), write down two more inequalities that model the constraints above.
3. On the grid opposite draw a suitable diagram to enable this problem to be solved graphically. Indicate a feasible region and the direction of an objective line.
4. 1. Use your diagram to find the number of each type of hamper that Nerys should prepare each day to achieve a maximum profit.
   2. Calculate this profit.
      \includegraphics[max width=\textwidth, alt={}]{fb95068f-f76d-492a-b385-bce17b26ae30-27_2490_1719_217_150}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

5. A leisure company owns boats of each of the following types: 2-person boats which are 4 metres long and weigh 50 kg .
4-person boats which are 3 metres long and weigh 20 kg .
8-person boats which are 14 metres long and weigh 100 kg .
The leisure company is willing to donate boats to a local sports club to accommodate up to 40 people at any one time. However, storage facilities mean that the combined length of the boats must not be more than 75 metres. Also, it must be possible to transport all the boats on a single trailer which has a maximum load capacity of 600 kg . The club intends to hire the boats out to help with the cost of maintaining them. It plans to charge \(\pounds 10 , \pounds 12\) and \(\pounds 8\) per day, for the 2 -, 4 - and 8 -person boats respectively and wishes to maximise its daily revenue ( \(\pounds R\) ). Let \(x , y\) and \(z\) represent the number of 2-, 4- and 8-person boats respectively given to the club.

1. Model this as a linear programming problem. Using the Simplex algorithm the following initial tableau is obtained:

   \(R\)	\(x\)	\(y\)	\(z\)	\(s\)	\(t\)	\(u\)
   1	\({ } ^ { - } 10\)	\({ } ^ { - } 12\)	\({ } ^ { - } 8\)	0	0	0	0
   0	1	2	4	1	0	0	20
   0	4	3	14	0	1	0	75
   0	5	2	10	0	0	1	60

2. Explain the purpose of the variables \(s , t\) and \(u\).
3. By increasing the value of \(y\) first, work out the next two complete tableaus.
4. Explain how you know that your final tableau gives an optimal solution and state this solution in practical terms.

7

1. 1. Complete Figure 4 to identify the feasible region for the problem. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5a826f8b-4751-4589-ad0a-109fc5c821f2-12_922_940_849_552}
      \end{figure} 7
      1. (ii) Determine the maximum value of \(5 x + 4 y\) subject to the constraints.
         7
   2. The simple-connected graph \(G\) has seven vertices. The vertices of \(G\) have degree \(1,2,3 , v , w , x\) and \(y\) 7
   3. 1. Explain why \(x \geq 1\) and \(y \geq 1\) 7
      2. Explain why \(x \leq 6\) and \(y \leq 6\) 7
      3. Explain why \(x + y \leq 11\) 7
      4. State an additional constraint that applies to the values of \(x\) and \(y\) in this context.
         7
      5. The graph \(G\) also has eight edges. The inequalities used in part (a)(i) apply to the graph \(G\).
      7
   4. 1. Given that \(v + w = 4\), find all the feasible values of \(x\) and \(y\).
         7
   5. (ii) It is also given that the graph \(G\) is semi-Eulerian. On Figure 5, draw \(G\). Figure 5

7 Robyn manages a bakery. Each day the bakery bakes 900 rolls, 600 teacakes and 450 croissants.
The bakery sells two types of bakery box which contain rolls, teacakes and croissants, as shown in the table below. Robyn formulates a linear programming problem to find the maximum profit the bakery can make from selling the bakery boxes. 7

1. Part of a graphical method to solve this linear programming problem is shown on Figure 1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-14_1283_1196_1267_424}
   \end{figure} 7
   1. (i) Explain how the line shown on Figure 1 relates to the linear programming problem. Clearly define any variables that you introduce.
      [0pt] [3 marks]
      7
   2. (ii) Use Figure 1 to find the maximum profit that the bakery can make from selling bakery boxes.
      7
   3. State an assumption that you have made in part (a)(ii).
      [0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{21ed3b4e-a089-4607-b5d6-69d8aac03f31-17_2493_1732_214_139}

8 A family business makes and sells two kinds of kitchen table.
Each pine table takes 6 hours to make and the cost of materials is \(\pounds 30\).
Each oak table takes 10 hours to make and the cost of materials is \(\pounds 70\).
Each month, the business has 360 hours available for making the tables and \(\pounds 2100\) available for the materials.
Each month, the business sells all of its tables to a wholesaler.
The wholesaler specifies that it requires at least 10 oak tables per month and at least as many pine tables as oak tables. Each pine table sold gives the business a profit of \(\pounds 40\) and each oak table sold gives the business a profit of \(\pounds 75\). Use a graphical method to find the number of each type of table the business should make each month, in order to maximise its total profit. Show clearly how you obtain your answer.
[0pt] [8 marks]

7 An engineering company makes brake kits and clutch kits to sell to motorsport teams. The table below summarises the time taken and costs involved in making the two different types of kit. The workers at the engineering company have a combined 2500 hours available to make the kits every month. The engineering company has \(\pounds 200000\) available to cover the costs of making the kits every month. To meet the minimum demands of the motorsport teams, the engineering company must make at least 100 of each type of kit every month. 7

1. Using a graphical method on the grid opposite, find the number of each type of kit that the engineering company should make every month, in order to maximise its total monthly profit. Show clearly how you obtain your answer. \includegraphics[max width=\textwidth, alt={}, center]{c297a67f-65fd-47e0-a60c-d38fd86c6081-13_2486_1709_221_153} Do not write outside the box 7
2. Give a reason why the engineering company may not be able to make the number of each kit that you found in part (a). 7
3. During one particular month the engineering company removes the need to make at least 100 of each type of kit. Explain whether or not this has an effect on your answer to part (a).

3 A student is solving a maximising linear programming problem. The graph below shows the constraints, feasible region and objective line for the student's linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-03_1248_1184_502_427} Which vertex is the optimal vertex? Circle your answer. \(A\) B
C
D

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. Figure 3 also shows an objective line for the problem and the optimal vertex, which is labelled as \(V\). The value of the objective at \(V\) is 556
Express the linear programming problem in algebraic form. List the constraints as simplified inequalities with integer coefficients and determine the objective. Please check the examination details below before entering your candidate information
Candidate surname
Other names Pearson Edexcel
Centre Number
Candidate Number Level 3 GCE \includegraphics[max width=\textwidth, alt={}, center]{a2a6e659-aab5-4eec-9af4-ca6ab895f1c8-09_122_433_356_991}
□
□
□
□ \section*{Thursday 14 May 2020} Afternoon
Paper Reference 8FMO/27 \section*{Further Mathematics} Advanced Subsidiary
Further Mathematics options
27: Decision Mathematics 1
(Part of options D, F, H and K) \section*{Answer Book} Do not return the question paper with the answer book.
1. \(\begin{array} { l l l l l l l l l l } 3.7 & 2.5 & 5.4 & 1.9 & 2.7 & 3.2 & 3.1 & 2.7 & 4.2 & 2.0 \end{array}\)

1. (a)

\begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} 3. \begin{figure}[h] \end{figure} [The weight of the network is \(5 x + 246\) ] 4. \begin{figure}[h] \end{figure}

5. A standard transportation problem is described in the linear programming formulation below. Let \(X _ { i j }\) be the number of units transported from \(i\) to \(j\) where \(i \in \{ \mathrm {~A} , \mathrm {~B} , \mathrm { C } , \mathrm { D } \}\) $$j \in \{ \mathrm { R } , \mathrm {~S} , \mathrm {~T} \} \text { and } x _ { i j } \geqslant 0$$ Minimise \(P = 23 x _ { \mathrm { AR } } + 17 x _ { \mathrm { AS } } + 24 x _ { \mathrm { AT } } + 15 x _ { \mathrm { BR } } + 29 x _ { \mathrm { BS } } + 32 x _ { \mathrm { BT } }\) $$+ 25 x _ { \mathrm { CR } } + 25 x _ { \mathrm { CS } } + 27 x _ { \mathrm { CT } } + 19 x _ { \mathrm { DR } } + 20 x _ { \mathrm { DS } } + 25 x _ { \mathrm { DT } }$$ subject to $$\begin{aligned} & \sum x _ { \mathrm { A } j } \leqslant 34 \\ & \sum x _ { \mathrm { B } j } \leqslant 27 \\ & \sum x _ { \mathrm { C } j } \leqslant 41 \\ & \sum x _ { \mathrm { D } j } \leqslant 18 \\ & \sum x _ { i \mathrm { R } } \geqslant 44 \\ & \sum x _ { i \mathrm {~S} } \geqslant 37 \\ & \sum x _ { i \mathrm {~T} } \geqslant k \end{aligned}$$ Given that the problem is balanced,

1. state the value of \(k\).
2. Explain precisely what the constraint \(\sum x _ { i \mathrm { R } } \geqslant 44\) means in the transportation problem.
3. Use the north-west corner method to obtain the cost of an initial solution to this transportation problem.
4. Perform one iteration of the stepping-stone method to obtain an improved solution. You must make your method clear by showing the route and the

A linear programming problem in \(x\) and \(y\) is described as follows. Maximise P = \(5x + 3y\) subject to: \(x \geqslant 3\) $$x + y \leqslant 9$$ $$15x + 22y \leqslant 165$$ $$26x - 50y \leqslant 325$$

1. Add lines and shading to Diagram 2 in the answer book to represent these constraints. Hence determine the feasible region and label it R. \hfill [4]
2. Use the objective line method to find the optimal vertex, V, of the feasible region. You must draw and label your objective line and label vertex V clearly. \hfill [2]
3. Calculate the exact coordinates of vertex V and hence calculate the corresponding value of P at V. \hfill [3]

The objective is now to minimise \(5x + 3y\), where \(x\) and \(y\) are integers.

1. Write down the minimum value of \(5x + 3y\) and the corresponding value of \(x\) and corresponding value of \(y\). \hfill [2]

Two fertilizers are available, a liquid \(X\) and a powder \(Y\). A bottle of \(X\) contains 5 units of chemical \(A\), 2 units of chemical \(B\) and \(\frac{1}{2}\) unit of chemical \(C\). A packet of \(Y\) contains 1 unit of \(A\), 2 units of \(B\) and 2 units of \(C\). A professional gardener makes her own fertilizer. She requires at least 10 units of \(A\), at least 12 units of \(B\) and at least 6 units of \(C\). She buys \(x\) bottles of \(X\) and \(y\) packets of \(Y\).

1. Write down the inequalities which model this situation. [4]
2. On the grid provided construct and label the feasible region. [3]

A bottle of \(X\) costs £2 and a packet of \(Y\) costs £3.

1. Write down an expression, in terms of \(x\) and \(y\), for the total cost \(£T\). [1]
2. Using your graph, obtain the values of \(x\) and \(y\) that give the minimum value of \(T\). Make your method clear and calculate the minimum value of \(T\). [4]
3. Suggest how the situation might be changed so that it could no longer be represented graphically. [2]

Becky's bird food company makes two types of bird food. One type is for bird feeders and the other for bird tables. Let \(x\) represent the quantity of food made for bird feeders and \(y\) represent the quantity of food made for bird tables. Due to restrictions in the production process, and known demand, the following constraints apply. $$x + y \leq 12,$$ $$y < 2x,$$ $$2y \geq 7,$$ $$y + 3x \geq 15.$$

1. On the axes provided, show these constraints and label the feasible region \(R\). [5]

The objective is to minimise \(C = 2x + 5y\).

1. Solve this problem, making your method clear. Give, as fractions, the value of \(C\) and the amount of each type of food that should be produced. [4]

Another objective (for the same constraints given above) is to maximise \(P = 3x + 2y\), where the variables must take integer values.

1. Solve this problem, making your method clear. State the value of \(P\) and the amount of each type of food that should be produced. [4]

A company produces two types of party bag, Infant and Junior. Both types of bag contain a balloon, a toy and a whistle. In addition the Infant bag contains 3 sweets and 3 stickers and the Junior bag contains 10 sweets and 2 stickers. The sweets and stickers are produced in the company's factory. The factory can produce up to 3000 sweets per hour and 1200 stickers per hour. The company buys a large supply of balloons, toys and whistles. Market research indicates that at least twice as many Infant bags as Junior bags should be produced. Both types of party bag are sold at a profit of 15p per bag. All the bags are sold. The company wishes to maximise its profit. Let \(x\) be the number of Infant bags produced and \(y\) be the number of Junior bags produced per hour.

1. Formulate the above situation as a linear programming problem. [5]
2. Represent your inequalities graphically, indicating clearly the feasible region. [6]
3. Find the number of Infant bags and Junior bags that should be produced each hour and the maximum hourly profit. Make your method clear. [3]

In order to increase the profit further, the company decides to buy additional equipment. It can buy equipment to increase the production of either sweets or stickers, but not both.

1. Using your graph, explain which equipment should be bought, giving your reasoning. [2]

The manager of the company does not understand why the balloons, toys and whistles have not been considered in the above calculations.

1. Explain briefly why they do not need to be considered. [2]

\includegraphics{figure_6} The captain of the Malde Mare takes passengers on trips across the lake in her boat. The number of children is represented by \(x\) and the number of adults by \(y\). Two of the constraints limiting the number of people she can take on each trip are $$x < 10$$ and $$2 \leq y \leq 10$$ These are shown on the graph in Figure 6, where the rejected regions are shaded out.

1. Explain why the line \(x = 10\) is shown as a dotted line. (1)
2. Use the constraints to write down statements that describe the number of children and the number of adults that can be taken on each trip. (3)

For each trip she charges £2 per child and £3 per adult. She must take at least £24 per trip to cover costs. The number of children must not exceed twice the number of adults.

1. Use this information to write down two inequalities. (2)
2. Add two lines and shading to Diagram 1 in your answer book to represent these inequalities. Hence determine the feasible region and label it R. (4)
3. Use your graph to determine how many children and adults would be on the trip if the captain takes:
   1. the minimum number of passengers,
   2. the maximum number of passengers.
   (4)

(Total 14 marks)

A company produces two types of self-assembly wooden bedroom suites, the 'Oxford' and the 'York'. After the pieces of wood have been cut and finished, all the materials have to be packaged. The table below shows the time, in hours, needed to complete each stage of the process and the profit made, in pounds, on each type of suite.

	Oxford	York
Cutting	4	6
Finishing	3.5	4
Packaging	2	4
Profit (£)	300	500

The times available each week for cutting, finishing and packaging are 66, 56 and 40 hours respectively. The company wishes to maximise its profit. Let \(x\) be the number of Oxford, and \(y\) be the number of York suites made each week.

1. Write down the objective function. [1]
2. In addition to $$2x + 3y \leq 33,$$ $$x \geq 0,$$ $$y \geq 0,$$ find two further inequalities to model the company's situation. [2]
3. On the grid in the answer booklet, illustrate all the inequalities, indicating clearly the feasible region. [4]
4. Explain how you would locate the optimal point. [2]
5. Determine the number of Oxford and York suites that should be made each week and the maximum profit gained. [3]

It is noticed that when the optimal solution is adopted, the time needed for one of the three stages of the process is less than that available.

1. Identify this stage and state by how many hours the time may be reduced. [3]

The Young Enterprise Company "Decide", is going to produce badges to sell to decision maths students. It will produce two types of badges. Badge 1 reads "I made the decision to do maths" and Badge 2 reads "Maths is the right decision". "Decide" must produce at least 200 badges and has enough material for 500 badges. Market research suggests that the number produced of Badge 1 should be between 20% and 40% of the total number of badges made. The company makes a profit of 30p on each Badge 1 sold and 40p on each Badge 2. It will sell all that it produced, and wishes to maximise its profit. Let \(x\) be the number produced of Badge 1 and \(y\) be the number of Badge 2.

1. Formulate this situation as a linear programming problem, simplifying your inequalities so that all the coefficients are integers. [6]
2. On the grid provided in the answer book, construct and clearly label the feasible region. [5]
3. Using your graph, advise the company on the number of each badge it should produce. State the maximum profit "Decide" will make. [3]

\includegraphics{figure_6} Keith organises two types of children's activity, 'Sports Mad' and 'Circus Fun'. He needs to determine the number of times each type of activity is to be offered. Let \(x\) be the number of times he offers the 'Sports Mad' activity. Let \(y\) be the number of times he offers the 'Circus Fun' activity. Two constraints are $$x \leq 15$$ and $$y > 6$$ These constraints are shown on the graph in Figure 6, where the rejected regions are shaded out.

1. Explain why \(y = 6\) is shown as a dotted line. [1] Two further constraints are $$3x \geq 2y$$ and $$5x + 4y \geq 80$$
2. Add two lines and shading to Diagram 1 in the answer book to represent these inequalities. Hence determine the feasible region and label it R. [3] Each 'Sports Mad' activity costs £500. Each 'Circus Fun' activity costs £800. Keith wishes to minimise the total cost.
3. Write down the objective function, C, in terms of \(x\) and \(y\). [2]
4. Use your graph to determine the number of times each type of activity should be offered and the total cost. You must show sufficient working to make your method clear. [5]

(Total 11 marks)

Herman is packing some hampers. Each day, he packs three types of hamper: basic, standard and luxury. Each basic hamper has 6 tins, 9 packets and 6 bottles. Each standard hamper has 9 tins, 6 packets and 12 bottles. Each luxury hamper has 9 tins, 9 packets and 18 bottles. Each day, Herman has 600 tins and 600 packets available, and he must use at least 480 bottles. Each day, Herman packs \(x\) basic hampers, \(y\) standard hampers and \(z\) luxury hampers.

1. In addition to \(x \geqslant 0\), \(y \geqslant 0\) and \(z \geqslant 0\), find three inequalities in \(x\), \(y\) and \(z\) that model the above constraints, simplifying each inequality. [4]
2. On a particular day, Herman packs the same number of standard hampers as luxury hampers.
   1. Show that your answers in part (a) become \(x + 3y \leqslant 100\) \(3x + 5y \leqslant 200\) \(x + 5y \geqslant 80\) [2]
   2. On the grid opposite, draw a suitable diagram to represent Herman's situation, indicating the feasible region. [4]
   3. Use your diagram to find the maximum total number of hampers that Herman can pack on that day. [2]
   4. Find the number of each type of hamper that Herman packs that corresponds to your answer to part (b)(iii). [1]

Phil is to buy some squash balls for his club. There are three different types of ball that he can buy: slow, medium and fast. He must buy at least 190 slow balls, at least 50 medium balls and at least 50 fast balls. He must buy at least 300 balls in total. Each slow ball costs £2.50, each medium ball costs £2.00 and each fast ball costs £2.00. He must spend no more than £1000 in total. At least 60% of the balls that he buys must be slow balls. Phil buys \(x\) slow balls, \(y\) medium balls and \(z\) fast balls.

1. Find six inequalities that model Phil's situation. [4 marks]
2. Phil decides to buy the same number of medium balls as fast balls.
   1. Show that the inequalities found in part (a) simplify to give $$x \geq 190, \quad y \geq 50, \quad x + 2y \geq 300, \quad 5x + 8y \leq 2000, \quad y \leq \frac{1}{3}x$$ [2 marks]
   2. Phil sells all the balls that he buys to members of the club. He sells each slow ball for £3.00, each medium ball for £2.25 and each fast ball for £2.25. He wishes to maximise his profit. On Figure 1 on page 14, draw a diagram to enable this problem to be solved graphically, indicating the feasible region and the direction of an objective line. [7 marks]
   3. Find Phil's maximum possible profit and state the number of each type of ball that he must buy to obtain this maximum profit. [4 marks]

The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics{figure_3}

1. Write down the inequalities that define the feasible region. [4]
2. Write down the coordinates of the three vertices of the feasible region. [2]

The objective is to maximise \(2x + 3y\).

1. Find the values of \(x\) and \(y\) at the optimal point, and the corresponding maximum value of \(2x + 3y\). [3]

The objective is changed to maximise \(2x + ky\), where \(k\) is positive.

1. Find the range of values of \(k\) for which the optimal point is the same as in part (iii). [2]

Leone is designing her new garden. She wants to have at least 1000 m\(^2\), split between lawn and flower beds. Initial costs are £0.80 per m\(^2\) for lawn and £0.40 per m\(^2\) for flowerbeds. Leone's budget is £500. Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least 200 m\(^2\) of lawn. Maintenance costs each year are £0.15 per m\(^2\) for lawn and £0.25 per m\(^2\) for flower beds. Leone wants to minimize the maintenance costs of her garden.

1. Formulate Leone's problem as a linear programming problem. [7]
2. Produce a graph to illustrate the inequalities. [6]
3. Solve Leone's problem. [2]
4. If Leone had more than £500 available initially, how much extra could she spend to minimize maintenance costs? [1]

Kassi and Theo are discussing how much oil and how much vinegar to use to dress their salad. They agree to use between 5 and 10ml of oil and between 3 and 6ml of vinegar and that the amount of oil should not exceed twice the amount of vinegar. Theo prefers to have more oil than vinegar. He formulates the following problem to maximise the proportion of oil: Maximise \(\frac{x}{x + y}\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\).

1. Explain why this problem is not an LP. [1]
2. Use the simplex method to solve the following LP. Maximise \(x - y\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\). [7]
3. Kassi prefers to have more vinegar than oil. She formulates the following LP. Maximise \(y - x\) subject to \(5 \leq x \leq 10\), \(3 \leq y \leq 6\), \(x - 2y \leq 0\). Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii). [5]
4. Explain why the formulation in part (ii) produced a solution for Theo's problem, and why it is more difficult to use the simplex method to solve Kassi's problem in part (iii). [2]
5. Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed. [5]

A linear programming problem is set up to maximise \(P = ax + y\) where \(a\) is a constant. \(P\) is maximised subject to three linear constraints which form the triangular feasible region shown in the diagram below. \includegraphics{figure_8} The vertices of the triangle are \((1, 6)\), \((5, 11)\) and \((13, 9)\) \(P\) is maximised at \((5, 11)\) Find the range of possible values for \(P\) [5 marks]