Graphical feasible region identification

1 Consider the following LP.
Maximise \(x + y\) subject to \(2 x + y < 44\) \(2 x + 3 y < 60\) \(10 x + 11 y < 244\) Part of a graphical solution is produced below and in your answer book.
Complete this graphical solution in your answer book. \includegraphics[max width=\textwidth, alt={}, center]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-2_1316_1346_916_356}

2 The algorithm gives a method for drawing two straight lines, if certain conditions are met. Start with the equations of the two straight lines
Line 1 is \(a x + b y = c , \quad a , b , c > 0\) Line 2 is \(d x + e y = f , \quad d , e , f > 0\) Let \(X =\) minimum of \(\frac { c } { a }\) and \(\frac { f } { d }\) Let \(Y =\) minimum of \(\frac { c } { b }\) and \(\frac { f } { e }\) If \(X = \frac { c } { a }\) then \(X ^ { * } = \frac { c - b Y } { a }\) and \(Y ^ { * } = \frac { f - d X } { e }\) If \(X = \frac { f } { d }\) then \(X ^ { * } = \frac { f - e Y } { d }\) and \(Y ^ { * } = \frac { c - a X } { b }\) Draw an \(x\)-axis labelled from 0 to \(X\), and a \(y\)-axis labelled from 0 to \(Y\) Join ( \(0 , Y\) ) to ( \(X , Y ^ { * }\) ) with a straight line
Join ( \(X ^ { * } , Y\) ) to ( \(X , 0\) ) with a straight line

1. Apply the algorithm with \(a = 1 , b = 5 , c = 25 , d = 10 , e = 2 , f = 85\).
2. Why might this algorithm be useful in an LP question?

4.
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\section*{Question 4 continued} \section*{Grid 1}

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.

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. \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)

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

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

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

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

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

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

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

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

(Total 11 marks)