7.06e Sensitivity analysis: effect of changing coefficients

4. \begin{figure}[h] \end{figure} Figure 2 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines are shown on the graph.

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 \(P\), where \(P = 2 x + k y\)
3. For the case \(k = 3\), use the point testing method to find the optimal vertex of the feasible region and state the corresponding value of \(P\).
4. Determine the range of values for \(k\) for which the optimal vertex found in (c) is still optimal.

6. A linear programming problem in \(x\) and \(y\) is described as follows. Maximise \(P = k x + y\), where \(k\) is a constant
subject to: \(\quad 3 y \geqslant x\) $$\begin{aligned} x + 2 y & \leqslant 130 \\ 4 x + y & \geqslant 100 \\ 4 x + 3 y & \leqslant 300 \end{aligned}$$

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. For the case when \(k = 0.8\)
   1. 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.
   2. calculate the coordinates of \(V\) and hence calculate the corresponding value of \(P\) at \(V\). Given that for a different value of \(k , V\) is not the optimal vertex of \(R\),
3. determine the range of possible values for \(k\). You must make your method and working clear.

7. You are in charge of buying new cupboards for a school laboratory. The cupboards are available in two different sizes, standard and large.
The maximum budget available is \(\pounds 1800\). Standard cupboards cost \(\pounds 150\) and large cupboards cost \(\pounds 300\).
Let \(x\) be the number of standard cupboards and \(y\) be the number of large cupboards.

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint.
   (2) The cupboards will be fitted along a wall 9 m long. Standard cupboards are 90 cm long and large cupboards are 120 cm long.
2. Show that this constraint can be modelled by $$3 x + 4 y \leqslant 30$$ You must make your reasoning clear. Given also that \(y \geqslant 2\),
3. explain what this constraint means in the context of the question. The capacity of a large cupboard is \(40 \%\) greater than the capacity of a standard cupboard. You wish to maximise the total capacity.
4. Show that your objective can be expressed as $$\text { maximise } 5 x + 7 y$$
5. Represent your inequalities graphically, on the axes in your answer booklet, indicating clearly the feasible region, R.
6. Find the number of standard cupboards and large cupboards that need to be purchased. Make your method clear.

6. \begin{figure}[h] \end{figure} Edgar has recently bought a field in which he intends to plant apple trees and plum trees. He can use linear programming to determine the number of each type of tree he should plant. Let \(x\) be the number of apple trees he plants and \(y\) be the number of plum trees he plants. Two of the constraints are $$\begin{aligned} & x \geqslant 40 \\ & y \leqslant 50 \end{aligned}$$ These are shown on the graph in Figure 6, where the rejected region is shaded out.

1. Use these two constraints to write down two statements that describe the number of apple trees and plum trees Edgar can plant. Two further constraints are $$\begin{aligned} 3 x + 4 y & \leqslant 360 \\ x & \leqslant 2 y \end{aligned}$$
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 . Edgar will make a profit of \(\pounds 60\) from each apple tree and \(\pounds 20\) from each plum tree. He wishes to maximise his profit, P.
3. Write down the objective function.
4. Use an objective line to determine the optimal point of the feasible region, R . You must make your method clear.
5. Find Edgar's maximum profit.

6. \begin{figure}[h] \end{figure} Lethna is producing floral arrangements for an awards ceremony.
She will produce two types of arrangement, Celebration and Party.
Let \(x\) be the number of Celebration arrangements made.
Let \(y\) be the number of Party arrangements made.
Figure 6 shows three constraints, other than \(x , y \geqslant 0\) The rejected region has been shaded.
Given that two of the three constraints are \(y \leqslant 30\) and \(x \leqslant 60\),

1. write down, as an inequality, the third constraint shown in Figure 6. Each Celebration arrangement includes 2 white roses and 4 red roses.
   Each Party arrangement includes 1 white rose and 5 red roses.
   Lethna wishes to use at least 70 white roses and at least 200 red roses.
2. Write down two further inequalities to represent this information.
   (3)
3. Add two lines and shading to Diagram 1 in the answer book to represent these two inequalities.
4. Hence determine the feasible region and label it R . The times taken to produce each Celebration arrangement and each Party arrangement are 10 minutes and 4 minutes respectively. Lethna wishes to minimise the total time taken to produce the arrangements.
5. Write down the objective function, T , in terms of \(x\) and \(y\).
6. Use point testing to find the optimal number of each type of arrangement Lethna should produce, and find the total time she will take.

8. A chemical company produces two products \(X\) and \(Y\). Based on potential demand, the total production each week must be at least 380 gallons. A major customer's weekly order for 125 gallons of \(Y\) must be satisfied. Product \(X\) requires 2 hours of processing time for each gallon and product \(Y\) requires 4 hours of processing time for each gallon. There are 1200 hours of processing time available each week. Let \(x\) be the number of gallons of \(X\) produced and \(y\) be the number of gallons of \(Y\) produced each week.

1. Write down the inequalities that \(x\) and \(y\) must satisfy.
   (3) It costs \(\pounds 3\) to produce 1 gallon of \(X\) and \(\pounds 2\) to produce 1 gallon of \(Y\). Given that the total cost of production is \(\pounds C\),
2. express \(C\) in terms of \(x\) and \(y\).
   (1) The company wishes to minimise the total cost.
3. Using the graphical method, solve the resulting Linear Programming problem. Find the optimal values of \(x\) and \(y\) and the resulting total cost.
4. Find the maximum cost of production for all possible choices of \(x\) and \(y\) which satisfy the inequalities you wrote down in part (a).

7. \begin{figure}[h] \end{figure} A company is going to hire out two types of car, standard and luxury. Let \(x\) be the number of standard cars it should buy.
Let \(y\) be the number of luxury cars it should buy. Figure 6 shows three constraints, other than \(x , y \geqslant 0\) Two of these are \(x \geqslant 20\) and \(y \geqslant 8\)

1. Write, as an inequality, the third constraint shown in Figure 6. The company decides that at least \(\frac { 1 } { 6 }\) of the cars must be luxury cars.
2. Express this information as an inequality and show that it simplifies to $$5 y \geqslant x$$ You must make the steps in your working clear. Each time the cars are hired they need to be prepared. It takes 5 hours to prepare a standard car and it takes 6 hours to prepare a luxury car. There are 300 hours available each week to prepare the cars.
3. Express this information as an inequality.
4. Add two lines and shading to Diagram 1 in the answer book to illustrate the constraints found in parts (b) and (c).
5. Hence determine the feasible region and label it R . The company expects to make \(\pounds 80\) profit per week on each car.
   It therefore wishes to maximise \(\mathrm { P } = 80 x + 80 y\), where P is the profit per week.
6. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must clearly draw and label your objective line and the vertex V.
7. Given that P is the expected profit, in pounds, per week, find the number of each type of car that the company should buy and the maximum expected profit.

8. \begin{figure}[h] \end{figure} A company makes two types of garden bench, the 'Rustic' and the 'Contemporary'. The company wishes to maximise its profit and decides to use linear programming. Let \(x\) be the number of 'Rustic' benches made each week and \(y\) be the number of 'Contemporary' benches made each week. The graph in Figure 6 is being used to solve this linear programming problem.
Two of the constraints have been drawn on the graph and the rejected region shaded out.

1. Write down the constraints shown on the graph giving your answers as inequalities in terms of \(x\) and \(y\). It takes 4 working hours to make one 'Rustic' bench and 3 working hours to make one 'Contemporary' bench. There are 120 working hours available in each week.
2. Write down an inequality to represent this information. Market research shows that 'Rustic' benches should be at most \(\frac { 3 } { 4 }\) of the total benches made each week.
3. Write down, and simplify, an inequality to represent this information. Your inequality must have integer coefficients.
4. Add two lines and shading to Diagram 1 in your answer book to represent the inequalities of (b) and (c). Hence determine and label the feasible region, R. The profit on each 'Rustic' bench and each 'Contemporary' bench is \(\pounds 45\) and \(\pounds 30\) respectively.
5. Write down the objective function, P , in terms of \(x\) and \(y\).
6. Determine the coordinates of each of the vertices of the feasible region and hence use the vertex method to determine the optimal point.
7. State the maximum weekly profit the company could make.
   (Total 16 marks)

6. Harry wants to rent out boats at his local park. He can use linear programming to determine the number of each type of boat he should buy. Let \(x\) be the number of 2 -seater boats and \(y\) be the number of 4 -seater boats.
One of the constraints is $$x + y \geqslant 90$$

1. Explain what this constraint means in the context of the question. Another constraint is $$2 x \leqslant 3 y$$
2. Explain what this constraint means in the context of the question. A third constraint is $$y \leqslant x + 30$$
3. Represent these three constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R . Each 2 -seater boat costs \(\pounds 100\) and each 4 -seater boat costs \(\pounds 300\) to buy. Harry wishes to minimise the total cost of buying the boats.
4. Write down the objective function, C , in terms of \(x\) and \(y\).
5. Determine the number of each type of boat that Harry should buy. You must make your method clear and state the minimum cost.

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}

5. \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. Write down the inequalities that form region \(R\).
2. Find the exact coordinates of the vertices of the feasible region. The objective is to maximise \(P\), where \(P = 2 x + 3 y\)
3. Use point testing to find the optimal vertex, V, of the feasible region. The objective is changed to maximise \(Q\), where \(Q = 2 x + \lambda y\) Given that \(\lambda\) is a constant and V is still the only optimal vertex of the feasible region,
4. find the range of possible values of \(\lambda\).

6. \begin{figure}[h] \end{figure} Figure 4 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The vertices of the feasible region are \(A ( 4,7 ) , B ( 5,3 ) , C ( - 1,5 )\) and \(D ( - 2,1 )\).

1. Determine the inequality that defines the boundary of \(R\) that passes through vertices \(A\) and \(C\), leaving your answer with integer coefficients only. The objective is to maximise \(P = 5 x + y\)
2. Find the coordinates of the optimal vertex and the corresponding value of \(P\). The objective is changed to maximise \(Q = k x + y\)
3. If \(k\) can take any value, find the range of values of \(k\) for which \(A\) is the only optimal vertex.

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)

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. 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) \(A\)
   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) (i) SAET \(\_\_\_\_\) (ii) SBDT \(\_\_\_\_\) (iii) 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.

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 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}

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}

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]