module 20 linear programing

PROGRAM DIDIK CEMERLANG AKADEMIK

SPM

LINEAR PROGRAMMING

ADDITIONAL MATHEMATICSFORM 5

MODULE 16

http://mathsmozac.blogspot.com

http://sahatmozac.blogspot.com

1CHAPTER 16 : LINEAR PROGRAMMING

Contents Page16.1 Concept map 3

16.2 Identify and shade the region that satisfiesseveral inequalities

3-8

16.3 Problem interpretation and the formationof the relevant equations or inequalities

9-11

16.4 To solve problems on linear programmingusing the graphical method

12 15

16.5 SPM Questions 16 13

16.6 Assessment test 18 -19

16.7 Answers 20 25



216.1 Concept Map

16.2 Identify and shade the region that satisfies s

(a) A region is said to satisfy a certain inequalitythe inequality.

(b) When the given inequality is 0 cbyaxthe straight line 0 cbyax must be shad

(c) A solid line ( _________ ) line is used for an ineqor and a dashed line ( _ _ _ _ ) is used for an inequ

LINEAR PROGRAMMIN

Region that satisfies a fewlinear inequalities

To solve problems on lineprogramming using graph

30

2

y

http://mathsmozac.bloeveral inequalities

if each point in the region satisfies

or 0 cbyax the region aboveed.

uality which involves the sign ality which involves the sign > or

3(d) When the given inequality is 0 cbyax or 0 cbyax the region below thestraight line 0 cbyax should be shaded. However, this condition is only truewhen the coefficient of y on the left hand side is positive.

4

2

-2

-4

-6

-8

-5 5

Example 1

Shade the region which satisfies the inequality 632 yx .

Solution

Step 1Draw the straight line 632 yx , determine the x intercept and y intercept.

.x 0 3y 2 0

Step 2Since the inequality is , a full line must be drawn and the region above the straight line

632 yx must be shaded.



4-5 5 10

6

4

2

-2

-4

2x+3y6

2x+3y=6

Example 2


Solution

Step 1Draw the straight line 01535 yx , determine the x intercept and y intercept.

.x 0 - 3Y - 5 0

Step 2Since the inequality is < , a dashed line must be drawn and the region below the straightline 01535 yx must be shaded.

6

4

2

-2

-4

-6

-8

-10 -5 5 x

y

-5

-3A

B



5Example 3


Solution

Step 1

Change the coeffient of y on the left hand side to a positive value.

623623

yxyx

Step 2

Draw the line 623 yx ,determine the x intercept and y intercept.

.x 0 2y - 3 0

Step 3Since the inequality is , a full line is drawn and the region above this line is shaded.

8

6

4

2

-2

-4

-6

-8

-5 5

Change the direction ofthe inequality when thesign is changed



6Exercise 1

1. Draw and shade the region which satisfies the following inequalities:

(a) xy 23 (b) 62 xy

(c) 204 xy (d) xy 2183

2. Draw and shade the region R which satisfies the following inequalities :

(a) 5x , 22 xy and 2054 yx



7(b) 1y , 5 yx and 1025 yx

3. Write down three inequalities which define the shaded region R in the diagrams below(a)

Answer : ..

(b)

Answer : ..

x

R

0 3

y

0 5

3R

(2,7)

(3,6)



816.3 Problem interpretation and the formation of the relevant equations orinequalities

The table below shows the mathematical expressions for the different inequalitiesused.

Mathematical Expressions Inequalitya y greater than x xy b y less than x xy c y not more than x xy d y not less than x xy e The sum of x and y is not more than k kyx f The minimum value of y is k ky g The maximum value of y is k ky h y is at least k times the value of x kxy

Activity 2

1. The cost of a book is 50 cents and that of a pen is RM 1.30. A student wants to buy xbooks and y pens with the following conditions.

I : At least 3 pens must be bought.II: The total number of books and pens bought must not be more than 12.III: The amount of money spent is at most RM10.

Write down the three inequalities other than x 0 and y 0 that satisfy all the aboveconditions.

Answer :

2. A transport company delivers vegetables and flowers using lorries and vans. Eachlorry carries 40 boxes of flowers and 50 boxes of vegetables. Each van carries 20boxes of flowers and 40 boxes of vegetables. In a day, the company uses x lorries andy vans. The following conditions must be satisfied.

I The total number of boxes of flowers must be less or equal to 800.II The total number of boxes of vegetables must be at least 600.III The number of vans must not be more than two times the number of

lorries used.

(a) Write down three inequalities other than x 0 and y 0 that satisfy all the aboveconditions.

Answer :



9(b) Using a scale of 2 cm to 5 units on both axes, draw three lines which define theregion satisfying all three inequalities. Shade this region.



10

3. A factory employs skilled and semi-skilled workers. An amount of RM64000 isused to pay the monthly salaries of the workers. The monthly salary of a skilledworker and semi-skilled worker is RM1200 and RM800 respectively. Semi-skilledworkers exceed skilled workers by at least 10 persons. The minimum number of

skilled workers is71 of the number of semi-skilled workers.

(a) Given that x represents the number of skilled workers and y represents the numberof semi-skilled workers, write down three inequalities other than x 0 and y 0 ,that satisfy all of the above conditions.

(b) Using a scale of 2 cm to 5 workers on the x-axis and 2 cm to 10 workers on the y-axis, draw a graph to show these inequalities.Mark and shade the region R which satisfy these conditions.

Answer

(a) ..

(b)



11http://mathsmozac.blogspot.com


12

16.4 To solve problems on linear programming using the graphical method

Step 1 : Interpret the problem and form the equation or inequalities.

Step 2 : Construct the region which satisfies the given inequalities.

Step 3 : Determine the maximum value or minimum value byax from the graph bydrawing the straight line kbyax .

Example 1

A company delivers 900 parcels using x lorries and y vans. Each lorry carries 150parcels while each van carries 60 parcels. The cost of transporting the parcels using alorry is RM 60 while that of a van is RM 40 . The total cost spent on transportation is notmore than RM 480.

(a) Write down two inequalities other than x 0 and y 0 , that satisfy all of the aboveconditions.

(b) Using a scale of 2 cm to 2 units on both axes, draw two lines which define theregion satisfying both inequalities. Shade this region.

(c) Use your graph to find

(i) the maximum number of vans used if 5 lorries are used,(ii) the minimum cost of transportation.

Solution

(a)3025

90060150

yxyx

..

24234804060

yxyx

.

The two inequalities that satisfy the given conditions are :

3025 yx and 2423 yx

1

2



(b) The x -intercept and y - intercept for the straight line 3025 yx

.x 0 6Y 15 0

The x -intercept and y - intercept for the straight line 2423 yx

.x 0 8Y 12 0

(

12

10

8

6

4

2

-2

5 10 15 20

2423 yx

5x

3025 yx

4,5

http://sahatmozac.blogspot.comc) (i) The number of lorFrom the graph, wTherefore the max

(ii) The cost for trans

24234804060

yxyx

Draw a parallel linshaded region andrieh

im

po

ecl

ht 13

s used is 5, so 5x . Draw the line 5x .en 5x , 4y is the maximum value.um number of vans used is 4.

rt, yxk 4060 , assume 480k

to the straight line 2423 yx that passes through theosest to the origin.

tp://mathsmozac.blogspot.com

14

From the graph, the optimum coordinate is (6,0).Therefore the minimum cost of transportation is when 6 lorries are used.

The minimum cost of transportation60 6360

RMRM

Exercise 3

1. A company is appointed by TV12 to make a survey and get feedback on itstelevision programmes. The following conditions must be satisfied.

I At least 500 people surveyed must be from the rural area.II The town people surveyed must not be less than the people from the rural area .III The total number of people surveyed must not be more than 1500.

The company has made a survey on x rural people and y people from the town.(a) Write down three inequalities which satisfy all the above conditions.

(b) Using a scale of 1 cm to 100 people on both axes ,draw three lines which definethe region satisfying all three inequalities. Shade this region.

(c) Use your graph to answer the following questions:

The company is paid RM 2 to each person surveyed. The cost to survey one ruralperson is RM 1.50 and RM 1.00 for a person from the town.Find

(i) the minimum cost to make a survey by the company,(ii) the maximum profit made by the company.

Answer :

(a)

(c)



16

16.5 PAST YEAR SPM QUESTIONS

SPM 2005 Paper 2 Qn.14

An institution offers two computer courses, P and Q. The nmber of participants for courseP is x and for course Q is y .

The enrolment of the participants is based on the following constraints:

I : The total number of participants is not more than 100.

II : The number of participants for course Q is not more than 4 times the number ofparticipants for course P.

III : The number of participants for course Q must exceed the number of participantsfor course P by at least 5.

(a) Write down three inequalities, other than x 0 and y 0 , that satisfy all of the aboveconstraints. 3marks

(b) By using a scale of 2 cm to 10 participants on both axes, construct and shade theregion R that satisfies all the above constraints. 4marks

(c) By using your graph from (b), find

(i) the range of the number of participants for course Q if the number ofparticipants for course P is 30.

(ii) the maximum total fees that can be collected if the fees per month forcourses P and Q are RM50 and RM60 respectively.

4marksAnswer :

(a)

(c)



17

16.6 Assesment Test



18

16.6 AssesmentEncik Ahmad sells two types of fried noodles. A pack of plain noodles uses 120 g ofprawns and 300 g of beef, while the special fried noodles uses 240 g of prawns and 200 gof beef. Encik Ahmad has 8.4 kg of prawns and 12 kg beef to make x packs of plainnodles and y packs of special noodles.The number of packs for plain noodles cannot be more than two times the number ofspecial noodles.

(a) Write down 3 inequalities other than x 0 and y 0 , that satisfy all of the aboveconstraints.

2marks(b) Using a scale of 2 cm to 10 units for x axis and 2 cm for 5 units for the y axis ,

draw all the three lines for the inequalities. , construct and shade the region R thatsatisfies all the above constraints. 3marks

(c) Use the graph in (b) to answer the following questions :

(i) If Encik Ahmad fries the special noodles 10 packs more than the plain noodles,state the maximum packs of plain noodles and maximum number of sspecialnoodles cooked.

(ii) What is the maximum profit obtained if a pack of plain noodles and a pack ofspecial noodles cost RM5 and RM7 each.

5marksAnswer

(a) ..

(c)



16.7 Answers

Exercise 1

1 (a)

(b)

6

4

2

-2_

8

_

6

_

4

_

2

_

-2

_

-4

_

-6

_

-8

_

-10

_

-5

_

5

_

10

3 2y x

2 6y x

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(c)

_ 8_

6

_

4

_

2

_

-2

_

-4

_

-6

_

-8

_

-20

_

-10

_

10

_

20

4 20y x

(d)

8

6

4

2

-15

2 (a) x

-

-5

3 18 2y x

http://sahatmozac.blogspot.com-10

5,2y

6

4

2

2-5

x

5

x

4x

R21

-2

-4

-6

-8

5 10 15 20

2, 4 5 20x y (b) 1, 5,5 2 10y x y x y

10

5

2 2y x

5 20y


64

2

-2

5

3. (a) 3, 0, 2x y y x (b) 2 3,3 5 15,7 3 35y x x y x y

Exercise 2

1. 3, 12,5 13 100x x y x y

2. 40 20 800,50 40 600, 2x y x y y x

Activity 3

1.(a) 500,x y x , 1500x y

(c) (i) Cost 1.5 1.0x y . Draw the line 1.5 150x y . ( 150 = 1.5 X 1.0 X 100 )From the graph, the minimum cost occurs when 500, 500x y

Therefore the minimum cost 1.5(500) 1.0(500)1250RM

(ii)Area Payment

receivedCost Profit

Rural ( x person) RM 2.00 RM 1.50 RM 0.50Town ( y person) RM 2.00 RM 1.00 RM 1.00

Profit 0.5 1.0x y , draw the

From the graph the maximum

Therefore the maximum profi

1y

5 2 10x y

5x y

R

http

http://sahatmozac.blogspot.comline 0.5 500x y .

profit occurs when 500, 1000x y .

t 0.5(500) 1.0(1000)1250RM

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23

1600

1400

1200

1000

800

600

400

200

150

0

100 200 400 600 800 1000

R

500x

y x

max.(500,1000)

min.(500,500)

0.5 500x y


http://sahatmozac.blogspot.com1200 140

1500x y

24

Answer SPM 2005

(a) 100x y , 4y x , 5y x

(b) 100x y 4y x 5y x

(c) (i) When 30,35 70x y

(ii) Total Fees paid, 50 60k x y Let 3000,50 60 3000k x y

max , 20, 80k x y

Therefore total fees paid 50(20) 60(80)5800Rm

Assesment Test

(a) 120x + 240y 8400 or x + 2y 70300x + 200y 12000 or 3x + 2y 120

x 2y or y 21

x

(b) Rujuk Lampiran B

(c)(i) y = x + 10

x maximum = 16 and y maximum = 26

Plain noodles = 16 packs

Special noodles = 26 packs

(c)(ii) (24, 23) and K maximum = 5x + 7y

Maximum profit = 5(24) + 7(23)= RM 281

.x 0 100y 100 0

.x 0 10y 0 40

.x 0 15y 5 20



25

1(b)

0 10 20 30 40 50 60 70 80x

5

10

15

20

25

30

35

40

y

(24, 23)

x + 2y = 70

y = x + 10

y = 31 x

R

45

50

55

60

3x + 2y = 120



module 20 linear programing

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