Three-variable constraint reduction

6 Each day, a factory makes three types of widget: basic, standard and luxury. The widgets produced need three different components: type \(A\), type \(B\) and type \(C\). Basic widgets need 6 components of type \(A , 6\) components of type \(B\) and 12 components of type \(C\).
Standard widgets need 4 components of type \(A , 3\) components of type \(B\) and 18 components of type \(C\).
Luxury widgets need 2 components of type \(A , 9\) components of type \(B\) and 6 components of type \(C\).
Each day, there are 240 components of type \(A\) available, 300 of type \(B\) and 900 of type \(C\).
Each day, the factory must use at least twice as many components of type \(C\) as type \(B\).
Each day, the factory makes \(x\) basic widgets, \(y\) standard widgets and \(z\) luxury widgets.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find four inequalities in \(x , y\) and \(z\) that model the above constraints, simplifying each inequality.
2. Each day, the factory makes the maximum possible number of widgets. On a particular day, the factory must make the same number of luxury widgets as basic widgets.
   1. Show that your answers in part (a) become $$2 x + y \leqslant 60 , \quad 5 x + y \leqslant 100 , \quad x + y \leqslant 50 , \quad y \geqslant x$$
   2. Using the axes opposite, draw a suitable diagram to enable the problem to be solved graphically, indicating the feasible region.
   3. Find the total number of widgets made on that day.
   4. Find all possible combinations of the number of each type of widget made that correspond to this maximum number.

9 Ollyin is buying new pillows for his hotel. He buys three types of pillow: soft, medium and firm. He must buy at least 100 soft pillows and at least 200 medium pillows.
He must buy at least 400 pillows in total.
Soft pillows cost \(\pounds 4\) each. Medium pillows cost \(\pounds 3\) each. Firm pillows cost \(\pounds 4\) each.
He wishes to spend no more than \(\pounds 1800\) on new pillows.
At least \(40 \%\) of the new pillows must be medium pillows.
Ollyin buys \(x\) soft pillows, \(y\) medium pillows and \(z\) firm pillows.

1. In addition to \(x \geqslant 0 , y \geqslant 0\) and \(z \geqslant 0\), find five inequalities in \(x , y\) and \(z\) that model the above constraints.
2. Ollyin decides to buy twice as many soft pillows as firm pillows.
   1. Show that three of your answers in part (a) become $$\begin{aligned} 3 x + 2 y & \geqslant 800 \\ 2 x + y & \leqslant 600 \\ y & \geqslant x \end{aligned}$$
   2. On the grid opposite, draw a suitable diagram to represent Ollyin's situation, indicating the feasible region.
   3. Use your diagram to find the maximum total number of pillows that Ollyin can buy.
   4. Find the number of each type of pillow that Ollyin can buy that corresponds to your answer to part (b)(iii).
      \includegraphics[max width=\textwidth, alt={}]{1258a6d3-558a-46dc-a916-d71f71b175ff-20_2256_1707_221_153}

7 Paul is a florist. Every day, he makes three types of floral bouquet: gold, silver and bronze. Each gold bouquet has 6 roses, 6 carnations and 6 dahlias.
Each silver bouquet has 4 roses, 6 carnations and 4 dahlias.
Each bronze bouquet has 3 roses, 4 carnations and 4 dahlias.
Each day, Paul must use at least 420 roses and at least 480 carnations, but he can use at most 720 dahlias. Each day, Paul makes \(x\) gold bouquets, \(y\) silver bouquets and \(z\) bronze bouquets.

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.
2. On a particular day, Paul makes the same number of silver bouquets as bronze bouquets.
   1. Show that \(x\) and \(y\) must satisfy the following inequalities. $$\begin{aligned} & 6 x + 7 y \geqslant 420 \\ & 3 x + 5 y \geqslant 240 \\ & 3 x + 4 y \leqslant 360 \end{aligned}$$
   2. Paul makes a profit of \(\pounds 4\) on each gold bouquet sold, a profit of \(\pounds 2.50\) on each silver bouquet sold and a profit of \(\pounds 2.50\) on each bronze bouquet sold. Each day, Paul sells all the bouquets he makes. Paul wishes to maximise his daily profit, \(\pounds P\). Draw a suitable diagram, on the grid opposite, to enable this problem to be solved graphically, indicating the feasible region and the direction of the objective line.
      (6 marks)
   3. Use your diagram to find Paul's maximum daily profit and the number of each type of bouquet he must make to achieve this maximum.
3. On another day, Paul again makes the same number of silver bouquets as bronze bouquets, but he makes a profit of \(\pounds 2\) on each gold bouquet sold, a profit of \(\pounds 6\) on each silver bouquet sold and a profit of \(\pounds 6\) on each bronze bouquet sold. Find Paul's maximum daily profit, and the number of each type of bouquet he must make to achieve this maximum.
   (3 marks) Turn over -

6 A company is planning its production of "MPowder" for the next three months.

• Over the next 3 months 20 tonnes must be produced.
• Production quantities must not be decreasing. The amount produced in month 2 cannot be less than the amount produced in month 1 , and the amount produced in month 3 cannot be less than the amount produced in month 2.
• No more than 12 tonnes can be produced in total in months 1 and 2.
• Production costs are \(\pounds 2000\) per tonne in month \(1 , \pounds 2200\) per tonne in month 2 and \(\pounds 2500\) per tonne in month 3.

The company planner starts to formulate an LP to find a production plan which minimises the cost of production: $$\begin{array} { l l } \text { Minimise } & 2000 x _ { 1 } + 2200 x _ { 2 } + 2500 x _ { 3 } \\ \text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 x _ { 3 } \geq 0 \\ & x _ { 1 } + x _ { 2 } + x _ { 3 } = 20 \\ & x _ { 1 } \leq x _ { 2 } \\ & \bullet \cdot \cdot \end{array}$$

1. Explain what the variables \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) represent, and write down two more constraints to complete the formulation.
2. Explain how the LP can be reformulated to: $$\begin{array} { l l } \text { Maximise } & 500 x _ { 1 } + 300 x _ { 2 } \\ \text { subject to } & x _ { 1 } \geq 0 x _ { 2 } \geq 0 \\ & x _ { 1 } \leq x _ { 2 } \\ & x _ { 1 } + 2 x _ { 2 } \leq 20 \\ & x _ { 1 } + x _ { 2 } \leq 12 \end{array}$$
3. Use a graphical approach to solve the LP in part (ii). Interpret your solution in terms of the company's production plan, and give the minimum cost.

5 A baker makes three types of jam-and-custard doughnuts.

• Each batch of type X uses 6 units of jam and 4 units of custard.
• Each batch of type Y uses 7 units of jam and 3 units of custard.
• Each batch of type Z uses 8 units of jam and 2 units of custard.

The baker has 360 units of jam and 180 units of custard available. The baker has plenty of doughnut batter, so this does not restrict the number of batches made. From past experience the baker knows that they must make at most 30 batches of type X and at least twice as many batches of type Y as batches of type Z . Let \(x =\) number of batches of type X made \(y =\) number of batches of type Y made \(z =\) number of batches of type Z made.

1. Set up an LP formulation for the problem of maximising the total number of batches of doughnuts made. The baker finds that type Z doughnuts are not popular and decides to make zero batches of type Z .
2. Use a graphical method to find how many batches of each type the baker should make to maximise the total number of batches of doughnuts made.
3. Give a reason why this solution may not be practical. The baker finds that some of the jam has been used so there are only \(k\) units of jam (where \(k < 360\) ).
   There are still 180 units of custard available and the baker still makes zero batches of type Z .
4. Find the values of \(k\) if exactly one of the other (non-trivial) constraints is redundant. Express your answer using inequalities.

6 Tamsin is planning how to spend a day off. She will divide her time between walking the coast path, visiting a bird sanctuary and visiting the garden centre. Tamsin has given a value to each hour spent doing each activity. She wants to decide how much time to spend on each activity to maximise the total value of the activities. Tamsin's requirements are that she will spend:

• a total of exactly 6 hours on the three activities
• at most 3.5 hours walking the coast path
• at least as long at the bird sanctuary as at the garden centre
• at least 1 hour at the garden centre.
  1. 1. Explain why the maximum total value of the activities done is achieved when \(3 x + y\) is maximised.
     2. Show how the requirement that she spends at least as long at the bird sanctuary as at the garden centre leads to the constraint \(x + 2 y \geqslant 6\).
     3. Explain why there is no need to require that \(y \geqslant 0\).
  2. Represent the constraints graphically and hence find a solution to Tamsin's problem.

8. A shop sells three types of pen. These are ballpoint pens, rollerball pens and fountain pens. The shop manager knows that each week she should order

• at least 50 pens in total
• at least twice as many rollerball pens as fountain pens

In addition,

• at most \(60 \%\) of the pens she orders must be ballpoint pens
• at least a third of the pens she orders must be rollerball pens

Each ballpoint pen costs \(\pounds 2\), each rollerball pen costs \(\pounds 3\) and each fountain pen costs \(\pounds 5\) The shop manager wants to minimise her costs.
Let \(x\) represent the number of ballpoint pens ordered, let \(y\) represent the number of rollerball pens ordered and let \(z\) represent the number of fountain pens ordered.

1. Formulate this information as a linear programming problem. State the objective and list the constraints as simplified inequalities with integer coefficients. The shop manager decides to order exactly 10 fountain pens. This reduces the problem to the following $$\begin{array} { l r } \text { Minimise } & P = 2 x + 3 y \\ \text { subject to } & x + y \geqslant 40 \\ & 2 x - 3 y \leqslant 30 \\ - x + 2 y \geqslant 10 \\ & y \geqslant 20 \\ & x \geqslant 0 \end{array}$$
2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region R .
3. 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.
4. Write down the number of each type of pen that the shop manager should order. Calculate the cost of this order.
   (Total \(\mathbf { 1 6 }\) marks)

7. Emily is planning to sell three types of milkshake, strawberry, vanilla and chocolate. Emily has completed some market research and has used this to form the following constraints on the number of milkshakes that she will sell each week.

• She will sell fewer than 200 non-vanilla milkshakes in total.
• She will sell at most 2.5 times as many strawberry milkshakes as vanilla milkshakes.
• At most, \(75 \%\) of the milkshakes that she will sell will be vanilla.

The profit on each strawberry milkshake sold is \(\pounds 0.75\), the profit on each vanilla milkshake sold is \(\pounds 1.20\) and the profit on each chocolate milkshake sold is \(\pounds 1.45\) Emily wants to maximise her profit.
Let \(x\) represent the number of strawberry milkshakes sold, let \(y\) represent the number of vanilla milkshakes sold and let \(z\) represent the number of chocolate milkshakes sold.

1. Formulate this as a linear programming problem, stating the objective and listing the constraints as simplified inequalities with integer coefficients. In week 1, Emily sells 100 strawberry milkshakes and 25 chocolate milkshakes.
2. Calculate the maximum possible profit and minimum possible profit, in pounds, for the sale of all milkshakes in week 1. You must show your working.

7. A farmer has 100 acres of land available that can be used for planting three crops: A, B and C . It takes 2 hours to plant each acre of crop A, 1.5 hours to plant each acre of crop B and 45 minutes to plant each acre of crop C . The farmer has 138 hours available for planting. At least one quarter of the total crops planted must be crop A.
For every three acres of crop B planted, at most five acres of crop C will be planted.
The farmer expects a profit of \(\pounds 160\) for each acre of crop A planted, \(\pounds 75\) for each acre of crop B planted and \(\pounds 125\) for each acre of crop C planted. The farmer wishes to maximise the profit from planting these three crops.
Let \(x , y\) and \(z\) represent the number of acres of land used for planting crop A, crop B, and crop C respectively.

1. Formulate this information as a linear programming problem. State the objective, and list the constraints as simplified inequalities with integer coefficients. The farmer decides that all 100 acres of available land will be used for planting the three crops.
2. Explain why the maximum total profit is achieved when \(- 7 x + 10 y\) is minimised. The farmer's decision to use all 100 acres reduces the constraints of the problem to the following: $$\begin{aligned} x & \geqslant 25 \\ 3 x + 8 y & \geqslant 300 \\ x + y & \leqslant 100 \\ 5 x + 3 y & \leqslant 252 \\ y & \geqslant 0 \end{aligned}$$
3. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region, \(R\).
4. 1. Determine the exact coordinates of each of the vertices of \(R\).
   2. Apply the vertex method to determine how the 100 acres should be used for planting the three crops.
   3. Hence find the corresponding maximum expected profit.

5. A clothing shop sells a particular brand of shirt, which comes in three different sizes, small, medium and large. Each month the manager of the shop orders \(x\) small shirts, \(y\) medium shirts and \(z\) large shirts.
The manager forms constraints on the number of each size of shirts he will have to order.
One constraint is that for every 3 medium shirts he will order at least 5 large shirts.

1. Write down an inequality, with integer coefficients, to model this constraint. Two further constraints are $$x + y + z \geqslant 250 \text { and } x \leqslant 0.2 ( x + y + z )$$
2. Use these two constraints to write down statements, in context, that describe the number of different sizes of shirt the manager will order. The cost of each small shirt is \(\pounds 6\), the cost of each medium shirt is \(\pounds 10\) and the cost of each large shirt is \(\pounds 15\) The manager must minimise the total cost of all the shirts he will order.
3. Write down the objective function. Initially, the manager decides to order exactly 150 large shirts.
4. 1. Rewrite the constraints, as simplified inequalities with integer coefficients, in terms of \(x\) and \(y\) only.
   2. Represent these constraints on Diagram 1 in the answer book. Hence determine, and label, the feasible region \(R\).
5. Use the objective line method to find the optimal vertex, \(V\), of the feasible region. You must make your objective line clear and label \(V\).
6. Write down the number of each size of shirt the manager should order. Calculate the total cost of this order. Later, the manager decides to order exactly 50 small shirts and exactly 75 medium shirts instead of 150 large shirts.
7. Find the minimum number of large shirts the manager should order and show that this leads to a lower cost than the cost found in (f).

8. A bakery makes three types of doughnut. These are ring, jam and custard. The bakery has the following constraints on the number of doughnuts it must make each day.

• The total number of doughnuts made must be at least 200
• They must make at least three times as many ring doughnuts as jam doughnuts
• At most \(70 \%\) of the doughnuts the bakery makes must be ring doughnuts
• At least a fifth of the doughnuts the bakery makes must be jam doughnuts

It costs 8 pence to make each ring doughnut, 10 pence to make each jam doughnut and 14 pence to make each custard doughnut. The bakery wants to minimise the total daily costs of making the required doughnuts. Let \(x\) represent the number of ring doughnuts, let \(y\) represent the number of jam doughnuts and let z represent the number of custard doughnuts the bakery makes each day.

1. Formulate this as a linear programming problem stating the objective and listing the constraints as simplified inequalities with integer coefficients. On a given day, instead of making at least 200 doughnuts, the bakery requires that exactly 200 doughnuts are made. Furthermore, the bakery decides to make the minimum number of jam doughnuts which satisfy all the remaining constraints. Given that the bakery still wants to minimise the total cost of making the required doughnuts, use algebra to
2. 1. calculate the number of each type of doughnut the bakery will make on that day,
   2. calculate the corresponding total cost of making all the doughnuts. \section*{END}

4. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(\quad P = - x + y\) subject to $$\begin{gathered} x + 2 y + z \leqslant 15 \\ 3 x - 4 y + 2 z \geqslant 1 \\ 2 x + y + z = 14 \\ x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$

1. 1. Eliminate \(z\) from the first two inequality constraints, simplifying your answers.
   2. Hence state the maximum possible value of \(P\) Given that \(P\) takes the maximum possible value found in (a)(ii),
2. 1. determine the maximum possible value of \(x\)
   2. Hence find a solution to the linear programming problem.

8. A headteacher is deciding how to allocate prizes to the students who are leaving at the end of the school year. There are three categories of prize: academic, sport, and leadership.

• Each academic prize costs \(\pounds 14\), each sport prize costs \(\pounds 8\), and each leadership prize costs \(\pounds 12\). The total amount available to spend on all prizes is \(\pounds 976\)
• For every 5 academic prizes there must be at least 2 leadership prizes
• At least half the prizes must be academic
• \(20 \%\) of the prizes must be for sport

The headteacher wishes to maximise the total number of prizes.
Let \(x , y\) and \(z\) represent the number of academic, sport and leadership prizes respectively.

1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective and listing the constraints as simplified inequalities with integer coefficients. Given that the headteacher awards 16 sport prizes,
2. calculate the corresponding number of leadership prizes that the headteacher awards. You must show your working.

5. The head of a Mathematics department needs to order three types of paper. The three types of paper are plain, lined and graph. All three types of paper are sold in reams. (A ream is 500 sheets of paper.)
Based on the last academic year the head of department formed the following constraints.

• At least half the paper must be lined
• No more than \(15 \%\) of the paper must be graph paper
• The ratio of plain paper to graph paper must be \(5 : 2\)

The cost of each ream of plain, lined and graph paper is \(\pounds 5 , \pounds 12\) and \(\pounds 15\) respectively. The head of department has at most \(\pounds 834\) to spend on paper. The head of department wants to maximise the total number of reams of paper ordered.
Let \(x , y\) and \(z\) represent the number of reams of plain paper, lined paper and graph paper ordered respectively.

1. Formulate this information as a linear programming problem in \(x\) and \(y\) only, stating the objective and listing the constraints as simplified inequalities with integer coefficients. The head of department decides to order exactly 42 reams of lined paper and still wishes to maximise the total number of reams of paper ordered.
2. Determine
   1. the total number of reams of paper to be ordered,
   2. the number of reams of graph paper to be ordered.

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 [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)

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}

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]

Mark wants to decorate the walls of his study. The total wall area is 24 m\(^2\). Mark can cover the walls using any combination of three materials: panelling, paint and pinboard. He wants at least 2 m\(^2\) of pinboard and at least 10 m\(^2\) of panelling. Panelling costs £8 per m\(^2\) and it will take Mark 15 minutes to put up 1 m\(^2\) of panelling. Paint costs £4 per m\(^2\) and it will take Mark 30 minutes to paint 1 m\(^2\). Pinboard costs £10 per m\(^2\) and it will take Mark 20 minutes to put up 1 m\(^2\) of pinboard. He has all the equipment that he will need for the decorating jobs. Mark is able to spend up to £150 on the materials for the decorating. He wants to know what area should be covered with each material to enable him to complete the whole job in the shortest time possible. Mark models the problem as an LP with five constraints. His constraints are: $$x + y + z = 24,$$ $$4x + 2y + 5z \leqslant 75,$$ $$x \geqslant 10,$$ $$y \geqslant 0,$$ $$z \geqslant 2.$$

1. Identify the meaning of each of the variables \(x\), \(y\) and \(z\). [2]
2. Show how the constraint \(4x + 2y + 5z \leqslant 75\) was formed. [2]
3. Write down an objective function, to be minimised. [1]

Mark rewrites the first constraint as \(z = 24 - x - y\) and uses this to eliminate \(z\) from the problem.

1. Rewrite and simplify the objective and the remaining four constraints as functions of \(x\) and \(y\) only. [3]
2. Represent your constraints from part (iv) graphically and identify the feasible region. Your graph should show \(x\) and \(y\) values from 0 to 15 only. [4]

Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order. • At least three-fifths of all the flowers must be roses. • For every 2 hydrangeas there must be at most 3 peonies. • The total number of flowers must be exactly 1000 The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4 Ben wants to minimise the cost of the flowers. Let \(x\) represent the number of roses, let \(y\) represent the number of hydrangeas and let \(z\) represent the number of peonies that he will order.

1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. [7]

Ben decides to order the minimum number of roses that satisfy his constraints.

1. 1. Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
   2. Calculate the corresponding total cost of this order. [3]