7.06d Graphical solution: feasible region, two variables

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)

4. \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. 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 \(P = 2 x + 3 y\).
3. Use point testing at each vertex to find the optimal vertex, \(V\), of the feasible region and state the corresponding value of \(P\) at \(V\).
   (3) The objective is changed to maximise \(Q = 2 x + k y\), where \(k\) is a constant.
4. Find the range of values of \(k\) for which the vertex identified in (c) is still optimal.
   (2)

7. \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. The equations of two of the lines have been shown in Figure 3. Given that \(k\) is a positive constant,

1. determine, in terms of \(k\) where necessary, the inequalities that define \(R\). The objective is to maximise \(P = 5 x + k y\) Given that the value of \(P\) is 38 at the optimal vertex of \(R\),
2. determine the possible value(s) of \(k\). You must show algebraic working and make your method clear.
   (Total 11 marks)

5. \begin{figure}[h] \end{figure} Figure 3 shows the constraints of a linear programming problem in \(x\) and \(y\). The unshaded area, including its boundaries, forms the feasible region, \(R\). The four vertices of \(R\) are \(A ( 6,8 ) , B ( 13,12 ) , C ( 9,22 )\) and \(D ( 5,18 )\).
An objective line has been drawn and labelled on the graph.
When the objective function, \(P\), is maximised, the value of \(P\) is 540
When the objective function, \(P\), is minimised, the value of \(P\) is \(k\) Determine the value of \(k\). You must make your method and working clear.
(You may assume that the objective function, \(P\), takes the form \(a x + b y\) where \(a\) and \(b\) are constants.)

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. Michael and his team are making toys to give to children at a summer fair. They make two types of toy, a soft toy and a craft set. Let \(x\) be the number of soft toys they make and \(y\) be the number of craft sets they make.
Each soft toy costs \(\pounds 3\) to make and each craft set costs \(\pounds 5\) to make. Michael and his team have a budget of \(\pounds 1000\) to spend on making the toys for the summer fair.

1. Write down an inequality, in terms of \(x\) and \(y\), to model this constraint. Two further constraints are: $$\begin{gathered} y \leqslant 2 x \\ 4 y - x \geqslant 210 \end{gathered}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all of these constraints. Hence determine the feasible region and label it R . Michael's objective is to make as many toys as possible.
3. State the objective function.
4. Determine the exact coordinates of each of the vertices of the feasible region, and hence use the vertex method to find the optimal number of soft toys and craft sets Michael and his team should make. You should make your method clear.

7. Ian plans to produce two types of book, hardbacks and paperbacks. He will use linear programming to determine the number of each type of book he should produce. Let \(x\) represent the number of hardbacks Ian will produce. Let \(y\) represent the number of paperbacks Ian will produce. Each hardback takes 1 hour to print and 15 minutes to bind.
Each paperback takes 35 minutes to print and 24 minutes to bind.
The printing machine must be used for at least 14 hours. The binding machine must be used for at most 8 hours.

1. 1. Show that the printing time restriction leads to the constraint \(12 x + 7 y \geqslant k\), where \(k\) is a constant to be determined.
   2. Write the binding time restriction in a similar simplified form. Ian decides to produce at most twice as many hardbacks as paperbacks.
2. Write down an inequality to model this constraint in terms of \(x\) and \(y\).
3. Add lines and shading to Diagram 1 in the answer book to represent the constraints found in (a) and (b). Hence determine, and label, the feasible region R. Ian wishes to maximise \(\mathrm { P } = 60 x + 36 y\), where P is the total profit in pounds.
4. 1. Use the objective line (ruler) method to find the optimal vertex, V, of the feasible region. You must draw and clearly label your objective line and the vertex V .
   2. Determine the exact coordinates of V. You must show your working.
5. Given that P is Ian's expected total profit, in pounds, find the number of each type of book that he should produce and his maximum expected profit.

3. \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. The equations of two of the lines have been given.

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 \(P = k x + y\).
3. For the case \(k = 2\), use point testing to find the optimal vertex of the feasible region.
4. For the case \(k = 2.5\), find the set of points for which \(P\) takes its maximum value.

5. A school awards two types of prize, junior and senior. The school decides that it will award at least 25 junior prizes and at most 60 senior prizes.
Let \(x\) be the number of junior prizes that the school awards and let \(y\) be the number of senior prizes that the school awards.

1. Write down two inequalities to model these constraints.
   (2) Two further constraints are $$\begin{aligned} & 2 x + 5 y \geqslant 250 \\ & 5 x - 3 y \leqslant 150 \end{aligned}$$
2. Add lines and shading to Diagram 1 in the answer book to represent all four of these constraints. Hence determine the feasible region and label it \(R\). The cost of a senior prize is three times the cost of a junior prize. The school wishes to minimise the cost of the prizes.
3. State the objective function, giving your answer in terms of \(x\) and \(y\).
4. Determine the exact coordinates of the vertices of the feasible region. Hence use the vertex method to find the number of junior prizes and the number of senior prizes that the school should award. You should make your working clear.

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).

6. \begin{figure}[h] \end{figure} The graph in Figure 2 is being used to solve a linear programming problem in \(x\) and \(y\). The three constraints have been drawn on the graph and the rejected regions have been shaded out. The three vertices of the feasible region \(R\) are labelled \(\mathrm { A } , \mathrm { B }\) and C .

1. Determine the inequalities that define \(R\).
   (2) The objective function, \(P\), is given by $$P = a x + b y$$ where \(a\) and \(b\) are positive constants.
   The minimum value of \(P\) is 8 and the maximum value of \(P\) occurs at C .
2. Find the range of possible values of \(a\). You must make your method clear.
   (5)

7. \begin{figure}[h] \end{figure} Figure 5 shows the constraints of a linear programming problem in \(x\) and \(y\), where \(R\) is the feasible region. The equations of two of the lines and the three intersection points, \(A\), \(B\) and \(C\), are shown. The coordinates of \(C\) are \(\left( \frac { 35 } { 4 } , \frac { 15 } { 4 } \right)\) The objective function is \(P = x + 3 y\) When the objective is to maximise \(x + 3 y\), the value of \(P\) is 24
When the objective is to minimise \(x + 3 y\), the value of \(P\) is 10

1. 1. Find the coordinates of \(A\) and \(B\).
   2. Determine the inequalities that define \(R\). An additional constraint, \(y \geqslant k x\), where \(k\) is a positive constant, is added to the linear programming problem.
2. Determine the greatest value of \(k\) for which this additional constraint does not affect the feasible region.

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. \begin{figure}[h] \end{figure} 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 $$\text { and } \quad \begin{aligned} & x \leqslant 15 \\ & y > 6 \end{aligned}$$ 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. Two further constraints are $$\begin{aligned} & 3 x \geqslant 2 y \\ \text { and } \quad 5 x + 4 y & \geqslant 80 \end{aligned}$$
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 . Each 'Sports Mad' activity costs \(\pounds 500\).
   Each 'Circus Fun' activity costs \(\pounds 800\).
   Keith wishes to minimise the total cost.
3. Write down the objective function, C , in terms of \(x\) and \(y\).
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.

7. \begin{figure}[h] \end{figure}

1. Phil sells boxed lunches to travellers at railway stations. Customers can select either the vegetarian box or the non-vegetarian box.

Phil decides to use graphical linear programming to help him optimise the numbers of each type of box he should produce each day. Each day Phil produces \(x\) vegetarian boxes and \(y\) non-vegetarian boxes.
One of the constraints limiting the number of boxes is $$x + y \geqslant 70$$ This, together with \(x \geqslant 0 , y \geqslant 0\) and a fourth constraint, has been represented in Figure 7. The rejected region has been shaded.

1. Write down the inequality represented by the fourth constraint. Two further constraints are: $$\begin{aligned} & x + 2 y \leqslant 160 \\ & \text { and } y > 60 \end{aligned}$$
2. Add two lines and shading to Diagram 4 in your answer book to represent these inequalities.
3. Hence determine and label the feasible region, R .
4. Use your graph to determine the minimum total number of boxes he needs to prepare each day. Make your method clear. Phil makes a profit of \(\pounds 1.20\) on each vegetarian box and \(\pounds 1.40\) on each non-vegetarian box. He wishes to maximise his profit.
5. Write down the objective function.
6. Use your graph to obtain the optimal number of vegetarian and non-vegetarian boxes he should produce each day. You must make your method clear.
7. Find Phil's maximum daily profit.

7. A linear programming problem is modelled by the following constraints $$\begin{aligned} 8 x + 3 y & \leqslant 480 \\ 8 x + 7 y & \geqslant 560 \\ y & \geqslant 4 x \\ x , y & \geqslant 0 \end{aligned}$$

1. Use the grid provided in your answer book to represent these inequalities graphically. Hence determine the feasible region and label it R . The objective function, \(F\), is given by $$F = 3 x + y$$
2. Making your method clear, determine
   1. the minimum value of the function \(F\) and the coordinates of the optimal point,
   2. the maximum value of the function \(F\) and the coordinates of the optimal point.

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} The graph in Figure 6 is being used to solve a linear programming problem. Two of the constraints have been drawn on the graph and the rejected regions shaded out.

1. Write down the constraints shown on the graph. Two further constraints are $$\begin{aligned} x + y & \geqslant 30 \\ \text { and } \quad 5 x + 8 y & \leqslant 400 \end{aligned}$$
2. Add two lines and shading to Graph 1 in your answer book to represent these constraints. Hence determine the feasible region and label it R . The objective is to $$\text { minimise } 15 x + 10 y$$
3. Draw a profit line on Graph 1 and use it to find the optimal solution. You must label your profit line clearly.
   (3)

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.