Questions - Edexcel D1 - A-Level Maths

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Edexcel D1 2024 June Q5

10 marks Standard +0.8

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.

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.
Determine
1. the total number of reams of paper to be ordered,
2. the number of reams of graph paper to be ordered.

Edexcel D1 2024 June Q6

10 marks Moderate -0.3

Activity	Immediately preceding activities
A	-
B	-
C	A
D	-
E	A, B, D
F	D
G	A, B, D
H	F, G
I	A
J	F, G
K	C, E, H, I
L	I
M	C, E, H, I

Draw the activity network for the project described in the precedence table, using activity on arc and the minimum number of dummies. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ba9337bf-7a3c-49aa-b395-dd7818cf1d13-10_880_1154_1464_452} \captionsetup{labelformat=empty} \caption{Grid 1}
\end{figure} A cascade chart for all the activities of the project, except activity $\mathbf { L }$, is shown on Grid 1. The time taken to complete each activity is given in hours and each activity requires one worker. The project is to be completed in the minimum time using as few workers as possible.
State the critical activities of the project.
Use the cascade chart to determine the minimum number of workers needed to complete the project in the shortest possible time. You must make specific reference to time and activities. (You do not need to provide a schedule of the activities.) The duration of activity L is $x$ hours. Given that the total float of activity L is at most 7 hours,
determine the range of possible values for $\chi$.

Edexcel D1 2021 October Q1

10 marks Moderate -0.8

(a) Explain what is meant by the term 'path'.
(2)

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-02_812_1262_427_404} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents a network of roads. The number on each arc represents the length, in km, of the corresponding road. Piatrice wishes to travel from A to J.
(b) Use Dijkstra's algorithm to find the shortest path Piatrice could take from A to J. State your path and its length.
(6) Piatrice needs to return from J to A via G.
(c) Find the shortest path Piatrice could take from J to A via G and state its length.

Edexcel D1 2021 October Q2

6 marks Moderate -0.3

2. Chris has been asked to design a badge in the shape of a triangle XYZ subject to the following constraints.

Angle $Y$ should be at least three times the size of angle $X$
Angle $Z$ should be at least $50 ^ { \circ }$ larger than angle $X$
Angle $Y$ must be at most $120 ^ { \circ }$

Chris has been asked to maximise the sum of the angles $X$ and $Y$.
Let $x$ be the size of angle $X$ in degrees.
Let $y$ be the size of angle $Y$ in degrees.
Let z be the size of angle $Z$ in degrees.
Formulate this information as a linear programming problem in $x$ and $y$ only. State the objective and list the constraints as simplified inequalities with integer coefficients. You are not required to solve this problem.

Edexcel D1 2021 October Q3

15 marks Moderate -0.3

3. The table below represents a complete network that shows the least costs of travelling between eight cities, A, B, C, D, E, F, G and H.

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

Srinjoy must visit each city at least once. He will start and finish at A and wishes to minimise his total cost.

Use Prim's algorithm, starting at A , to find a minimum spanning tree for this network. You must list the arcs that form the tree in the order in which you select them.
State the weight of the minimum spanning tree.
Use your answer to (b) to help you calculate an initial upper bound for the total cost of Srinjoy's route.
Show that there are two nearest neighbour routes that start from A. You must make the routes and their corresponding costs clear.
State the best upper bound that can be obtained by using your answers to (c) and (d).
Starting by deleting A and all of its arcs, find a lower bound for the total cost of Srinjoy's route. You must make your method and working clear.
Use your results to write down the smallest interval that must contain the optimal cost of Srinjoy's route.

Edexcel D1 2021 October Q4

11 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-06_757_1163_226_459} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} The network in Figure 2 shows the activities that need to be carried out by a company to complete a project. Each activity is represented by an arc, and the duration, in days, is shown in brackets. Each activity requires one worker. The early event times and the late event times are shown at each vertex.

Complete the precedence table in the answer book.
(2) A cascade chart for this project is shown on Grid 1. \includegraphics[max width=\textwidth, alt={}, center]{d409aaae-811d-4eca-b118-efc927885f97-07_885_1358_276_356} \section*{Grid 1}
Use Figure 2 and Grid 1 to find the values of $v , w , x , y$ and $z$. The project is to be completed in the minimum time using as few workers as possible.
Calculate a lower bound for the minimum number of workers required. You must show your working.
On Grid 2 in your answer book, construct a scheduling diagram for this project. Before the project begins it is found that activity F will require an additional 5 hours to complete. The durations of all other activities are unchanged. The project is still to be completed in the shortest possible time using as few workers as possible.
State the new minimum project completion time and state the new critical path.

Edexcel D1 2021 October Q5

10 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-08_588_1428_230_322} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} [The total weight of the network is 166] Figure 3 models a network of cycle lanes that must be inspected. The number on each arc represents the length, in km, of the corresponding cycle lane. Lance needs to cycle along each lane at least once and wishes to minimise the length of his inspection route. He must start and finish at A.

Use an appropriate algorithm to find the length of the route. State the cycle lanes that Lance will need to traverse twice. You should make your method and working clear.
(6)
State the number of times that vertex C appears in Lance's route.
(1) It is now decided that the inspection route may finish at any vertex. Lance will still start at A and must cycle along each lane at least once.
Determine the finishing point so that the length of the route is minimised. You must give reasons for your answer and state the length of this new minimum route.
(3)

Edexcel D1 2021 October Q6

13 marks Standard +0.3

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: $\quad 3 y \geqslant x$ $$\begin{aligned} x + 2 y & \leqslant 130 \\ 4 x + y & \geqslant 100 \\ 4 x + 3 y & \leqslant 300 \end{aligned}$$

Add lines and shading to Diagram 1 in the answer book to represent these constraints. Hence determine the feasible region and label it $R$.
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 $R$,
determine the range of possible values for $k$. You must make your method and working clear.

Edexcel D1 2021 October Q7

10 marks Moderate -0.8

7. The numbers listed below are to be packed into bins of size $n$, where $n$ is a positive integer.
14
20
23
17
15
22
19
25

Edexcel D1 2021 October Q20

Easy -1.2

20
23
17
15
22
19
25
13
28
32 A lower bound for the number of bins required is 4

Determine the range of possible values of $n$. You must make your method clear.
(3)
Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly.
(4) When the first-fit bin packing algorithm is applied to the original list of numbers, the following allocation is achieved. \end{table}
\includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-14_1193_1586_1270_185}
Shortest path from A to J: $\_\_\_\_$ Length of shortest path from A to J: $\_\_\_\_$ \section*{2. $\_\_\_\_$} \section*{3.}
A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-22_755_1157_246_404} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
1. Activity
  Immediately
  preceding
  activities
  A
  B
  C
  D
  E
  
  Activity
  Immediately
  preceding
  activities
  F
  G
  H
  I
  J
  
  Activity
  Immediately
  preceding
  activities
  K
  L
  M
  $$v = \ldots \quad x = \ldots \quad y = \ldots$$ \includegraphics[max width=\textwidth, alt={}, center]{d409aaae-811d-4eca-b118-efc927885f97-23_2255_56_315_37}
  \includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-23_1153_1338_303_310}
  \section*{Grid 2} 5. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-24_591_1433_255_260} \captionsetup{labelformat=empty} \caption{Figure 3
  [0pt] [The total weight of the network is 166]}
  \end{figure} 6.
  \includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-26_1287_1645_301_162}
  \section*{Diagram 1} 7. $\begin{array} { l l l l l l l l l l l } 14 & 20 & 23 & 17 & 15 & 22 & 19 & 25 & 13 & 28 & 32 \end{array}$

Edexcel D1 2013 Specimen Q1

The table shows the names of nine people.

Use a quick sort to produce the list of names in ascending alphabetical order. You must make your pivots clear.
Use the binary search algorithm on your list to locate the name Paul.

Edexcel D1 2013 Specimen Q2

9 marks Easy -1.2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-03_719_1161_223_452} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 represents the distances, in metres, between eight vertices, $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G }$ and H , in a network.

Use Kruskal's algorithm to find a minimum spanning tree for the network. You should list the arcs in the order in which you consider them. In each case, state whether you are adding the arc to your minimum spanning tree.
Complete Matrix 1 in your answer book, to represent the network.
Starting at A, use Prim's algorithm to determine a minimum spanning tree. You must clearly state the order in which you considered the vertices and the order in which you included the arcs.
State the weight of the minimum spanning tree.

Edexcel D1 2013 Specimen Q3

9 marks Easy -1.8

3. $$\begin{array} { l l l l l l l } 41 & 28 & 42 & 31 & 36 & 32 & 29 \end{array}$$ The numbers in the list represent the weights, in kilograms, of seven statues. They are to be transported in crates that will each hold a maximum weight of 60 kilograms.

Calculate a lower bound for the number of crates that will be needed to transport the statues.
Use the first-fit bin packing algorithm to allocate the statues to the crates.
Use the full bin algorithm to allocate the statues to the crates.
Explain why it is not possible to transport the statues using fewer crates than the number needed for part (c).

Edexcel D1 2013 Specimen Q4

10 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-05_879_1068_248_497} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} [The total weight of the network is 73.3 km ]
Figure 2 models a network of tunnels that have to be inspected. The number on each arc represents the length, in km , of that tunnel.
Malcolm needs to travel through each tunnel at least once and wishes to minimise the length of his inspection route.
He must start and finish at A .

Use the route inspection algorithm to find the tunnels that will need to be traversed twice. You should make your method and working clear.
Find a route of minimum length, starting and finishing at A . State the length of your route. A new tunnel, CG, is under construction. It will be 10 km long.
Malcolm will have to include the new tunnel in his inspection route.
What effect will the new tunnel have on the total length of his route? Justify your answer.

Edexcel D1 2013 Specimen Q5

8 marks Moderate -0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-06_730_597_269_315} \captionsetup{labelformat=empty} \caption{Figure 3}

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

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-06_722_583_274_1160} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 3 shows the possible allocations of six people, Amelia, Charlie, Ellie, Gemma, Jimmy and Saskia, to six tasks, 1, 2, 3, 4, 5 and 6.
Figure 4 shows an initial matching.

Use the maximum matching algorithm once to find an improved matching. You must state the alternating path used and your improved matching.
Explain why a complete matching is not possible. After training, Jimmy can be assigned to tasks 4 or 5 and Ellie to tasks 2, 3, 5 or 6.
Starting with your current maximal matching, use the maximum matching algorithm to obtain a complete matching. You must state the alternating path used and your final matching.

Edexcel D1 2013 Specimen Q6

9 marks Easy -1.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-07_602_1182_244_440} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a network of cycle tracks within a national park. The number on each arc represents the time taken, in minutes, to cycle along the corresponding track.

Use Dijkstra's algorithm to find the quickest route from S to T. State your quickest route and the time it takes.
(6)
Explain how you determined your quickest route from your labelled diagram.
Write down the quickest route from E to T .

Edexcel D1 2013 Specimen Q7

11 marks Easy -1.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-08_1372_1769_278_189} \captionsetup{labelformat=empty} \caption{Figure 6}

\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.

Explain why $y = 6$ is shown as a dotted line. Two further constraints are $$\begin{aligned} & 3 x \geqslant 2 y \\ \text { and } \quad 5 x + 4 y & \geqslant 80 \end{aligned}$$
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.
Write down the objective function, C , in terms of $x$ and $y$.
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.

Edexcel D1 2013 Specimen Q8

11 marks Moderate -0.8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5fa867a2-0a3d-4f0b-9f9c-15584f2be5c0-10_705_1207_248_427} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure} A project is modelled by the activity network shown in Figure 7. The activities are represented by the arcs. The number in brackets on each arc gives the time, in days, to complete the activity. Each activity requires one worker. The project is to be completed in the shortest possible time.

Complete Diagram 2 in the answer book to show the early and late event times.
State the critical activities.
On Grid 1 in the answer book, draw a cascade (Gantt) chart for this project.
Use your cascade chart to determine a lower bound for the number of workers needed. You must justify your answer. \section*{END}

Edexcel D1 2008 January Q1

9 marks Easy -1.2

(a) Define the terms
1. alternating path,
2. matching.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-2_568_621_689_344} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-2_554_540_701_1171} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} At a school fair, five teachers, $A , B , C , D$ and $E$, are to supervise five stalls, $1,2,3,4$ and 5 .
A bipartite graph showing their possible allocations is given in Figure 1. An initial matching is given in Figure 2.
(b) Use the maximum matching algorithm twice to obtain a complete matching. List clearly the alternating paths you use.

Edexcel D1 2008 January Q2

10 marks Easy -1.2

2. (a) $\begin{array} { l l l l l l l l l l l } 18 & 20 & 11 & 7 & 17 & 15 & 14 & 21 & 23 & 16 & 9 \end{array}$ The list of numbers shown above is to be sorted into ascending order. Apply quick sort to obtain the sorted list. You must make your pivots clear. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-3_839_1275_614_395} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 represents a network of paths in a park. The number on each arc represents the length of the path in metres.
(b) Using your answer to part (a) and Kruskal's algorithm, find a minimum spanning tree for the network in Figure 3. You should list the arcs in the order in which you consider them and state whether you are adding it to your minimum spanning tree.
(c) Find the total weight of the minimum spanning tree.

Edexcel D1 2008 January Q3

9 marks Moderate -0.8

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-4_755_1132_239_468} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 models a network of roads in a housing estate. The number on each arc represents the length, in km , of the road. The total weight of the network is 11 km .
A council worker needs to travel along each road once to inspect the road surface. He will start and finish at A and wishes to minimise the length of his route.

Use an appropriate algorithm to find a route for the council worker. You should make your method and working clear. State your route and its length.
(6) A postal worker needs to walk along each road twice, once on each side of the road. She must start and finish at A . The length of her route is to be minimised. You should ignore the width of the road.
1. Explain how this differs from the standard route inspection problem.
  (1)
2. Find the length of the shortest route for the postal worker.
  (2)

Edexcel D1 2008 January Q4

11 marks Moderate -0.5

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-5_1079_1392_267_338} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} A project is modelled by the activity network shown in Figure 5. The activities are represented by the arcs. The number in brackets on each arc gives the time, in hours, to complete the activity. Some of the early and late times for each event are shown.

Calculate the missing early and late times and hence complete Diagram 1 in your answer book.
Calculate the total float on activities D, G and I. You must make your calculations clear.
List the critical activities. Each activity requires one worker.
Calculate a lower bound for the number of workers needed to complete the project in the minimum time.
(2)

Edexcel D1 2008 January Q5

7 marks Moderate -0.3

5. (a) Draw the activity network described in this precedence table, using activity on arc and exactly two dummies.
(4)

Activity	Immediately preceding activities
A	-
B	-
C	A
D	B
E	B, C
F	B, C

(b) Explain why each of the two dummies is necessary.
(3)

Edexcel D1 2008 January Q6

13 marks Moderate -0.8

6. (a) Define the term 'cut' as it applies to a directed network.
(2) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-7_659_1367_376_349} \captionsetup{labelformat=empty} \caption{Figure 6}

\end{figure} Figure 6 shows a capacitated, directed network. The number on each arc represents the capacity of that arc. The numbers in circles represent an initial flow.
(b) Complete the labelling procedure on Diagram 2 in your answer book by entering values along CE, EG, HT and GT.
(c) Find the maximum flow through the network. You must list each flow-augmenting route you use together with its flow.
(d) Show a maximal flow pattern on Diagram 3 in your answer book.
(e) State the value of the maximum flow through the network.
(f) Prove that your flow is maximal.

Edexcel D1 2008 January Q7

16 marks Moderate -0.8

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7396d930-0143-4876-b019-a4d73e09b172-8_2158_1803_239_137} \captionsetup{labelformat=empty} \caption{Figure 7}

\end{figure}

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.

Write down the inequality represented by the fourth constraint. Two further constraints are: $$\begin{aligned} & x + 2 y \leqslant 160 \\ & \text { and } y > 60 \end{aligned}$$
Add two lines and shading to Diagram 4 in your answer book to represent these inequalities.
Hence determine and label the feasible region, R .
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.
Write down the objective function.
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.
Find Phil's maximum daily profit.

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

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	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-