Graphical optimization with objective line

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.

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).

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. 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)

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.

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.

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. \(F = x + 3 y\).

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. \(\quad P = 5 x + y\).

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.

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

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]

10 x + 7 y & \leqslant 140
& x + y \leqslant 15
& 2 x + 3 y \geqslant 36
& x \geqslant 0 , \quad y \geqslant 0 \end{aligned} \end{array}$$ (c) Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, \(R\).
(d) Use the objective line method to find the optimal number of each type of cake that Martin should make, and the amount of sugar used.
(e) Determine how much flour and how many eggs Martin will have left over after making the optimal number of cakes. BLANK PAGE \end{document}