10. linear programming

7/29/2019 10. Linear Programming

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CHAPTER 10 LINEAR PROGRAMMING

1. Understand and use the concept of graphs of linear inequalities.

1.1 Identify and shade the region on the graph that satisfies a linear inequality.

Shade the region that satisfies each of the following inequalities

(a) 2x (b) 5x

(e) 3y x

(f) 3x y (j) 2 20x y+ (l) 3 1x y

10. Linear Programming 2


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1.2 Find the linear inequality that defines a shaded region

.Write the linear inequality that defines the shaded region other than 0 0x and y

(a)

Linear inequality :

(b)

Linear inequality :

(c)

Linear inequality :

(d)

Linear inequality :

1.3 Shade region on the graph that satisfies several linear inequalities.

(a)

10, 3 , 0y y x y

(b)

2 5 10, 5, 0x y x x+ > <

Answers for 1.2 (a) y


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(c)

3 2y x + 2 20, 0, 0x y y x+ < (d)

1x y > , 4y > , 8x

(e)

2 5, 2 5, 5y x y x x + + (f)

2 4, , 0x y y x x+ <

(g)

2 4, 3 , 3, 0y x y x x y +

(h)

2 12, 6, 2y x x y y< + + >



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1.4 Find linear inequalities that define a shaded region

Write the linear inequalities which define the shaded region.

(a)

Linear inequalities :

(b)


(c)


[ 7, 1y x< ]

(d)


[ 2 4, 7, 1y x y x + < ]

(e)


[ 2 4, 0, 0, 2 8y x x y y x + < + ]

(f)


[ 2 4, 2 8, 4y x y x y x + < + + ]



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2. Understand and use the concept of linear programming.

2.1 Solve problems related to linear programming by:

A) writing linear inequalities and equations describing a situation.

No Constraints Inequality

(1) The value of x must be at least 5

(2) The value of x is at most 5

(3) The value of x must be more than 5

(4) The minimum value of x is 5

(5) The value of x must exceed y by at least 5

(6) The value of x must be more than twice of y

(7) The value of x is at most three times of y

(8) The value of x is more than the value of y by at least 5

(9) The value of x is not less than twice the value of y

(10) The value of x is at least three time the value of y

(11) The sum of x and y is 45

(12) The sum of x and y is not less than 15

(13) The maximum value of x + y is 15

(14) The sum of x and y is more than 15

(15) The value of 2x+y is not more than 15

(16) The value of 2x+y is less than 10

(17) The ratio of x to y is at least 2

(18) The ratio of y to x is at most 1: 5

Answer: (1) 5,x (2) 5;(3) 5, (4) 5,(5) 5,(6) 2 , (7) 3x x x x y x y x y > > ,(8) 5, (9) 2 ,(10) 3x y x y x y

(11) 45,(12) 15,(13) 15(14) 15, (15) 2 15x y x y x y x y x y+ = + + + > + ,

1(16) 2 10,(17) 2 (18)

5

x yx y

y x+ <



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B) writing linear inequalities and equations describing a situation and shading the region of

feasible solutions.

( Use graph paper to answer these questions)

(1) Shankar has an allocation of RM 225 to buy x kg of prawns and y kg of fish. The total mass

of the commodities is not less than 15 kg. The mass of prawn is at most three times that of

fish. The price of 1 kg of prawns is RM9 and the price of 1 kg of fish is RM5.i. Write down three inequalities, other than 0x and 0y , that satisfy all the above

constraints.

ii Hence, using a scale of 2cm to 5 kg for both axes, construct and shade the region R thatsatisfy all the above constraints.

(2) A district education office intends to organize a course on the teaching of Mathematics and

Science in English.

The course will be attended by x Mathematics participants and y Science participants.

The selection of participants is based on the following constraints:I : The total number of participants is at least 40

II : The number of Science participants is at most twice that of Mathematics.

III : The maximum allocation for the course is RM7200. The expenditure for a

Mathematics participant is RM 120, and for a science participant isRM80.

i Write down three inequalities, other than x 0 and y 0, which satisfy the above

constraints.

ii. Hence, by using a scale of 2 cm to 10 participants on both axes, construct and shade

the region R which satisfies all the above constraints.

(3) An institution offers two computer courses, P and Q. The number of participants for the

course P 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 of participants for course P

III : The number of participants for course Q must exceed the number of

Participants for course P by at least 5

i Write down three inequalities, other than x 0 and y 0, which satisfy all the aboveconstraints.

ii By using a scale of 2 cm to 10 participants on both axes, construct the shade the

region R that satisfy all the above constraints.

(4) A workshop produces two types of rack, P and Q. The production of each type of rack

involves two processes, making and painting. Table below shows the time taken to make

and paint a rack of type P and a rack of type Q.

Rack Time taken (minutes)

Making Painting

P 60 30

Q 20 40

The workshop produces x racks of type P and y racks of type Q per day. The

production of racks per day is based on the following constraints:



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I : The maximum total time for making both racks is 720 minutes.

II : The total time for painting both racks is at least 360 minutes.III : The ratio of the number of racks of type P to the number of racks of type Q

is at least 1:3

i Write three inequalities, other than x 0 and y 0, which satisfy all the above

constraints.ii Using a scale of 2 cm to 2 racks on both axes, construct and shaded the region R are

satisfies all the above constraints.

(5) A factory produces two components, P and Q. In a particular day, the factory

produced x pieces of component P and y pieces of component Q. The profit fromthe sales of a piece of component P is RM15 and a piece of component Q is RM12.

The production of the components per day is based on the following constraints:

I : The total number of components produced is at most 500.II : The number of component P produced is not more than three times the

number of component Q

III : The minimum total profit for both components is RM4200i Write three inequalities, other than x 0 and y 0, which satisfy all the

constraints.

ii. Using a scale of 2 cm to 50 components on both axes, construct and shade the

region R which satisfy all the above constraints.

(6) The members of a youth association plan to organize a picnic. They agree to rent x buses

and y vans. The rental of a bus is RM 800 and the rental of a van is RM300. The rental of

the vehicles for the picnics is based on the following constraints:

I : The total number of vehicles to be rented is not more than 8

II: the number of buses is at most twice the number of vans

III: the maximum allocation for the rental of the vehicles is RM4000

(a) Write three inequalities, other than x 0 and y 0, which satisfy all the above

constraints(b) Using a scale of 2 cm to 1 vehicle on both axes, construct and shade the region

R which satisfy all the above constraints.

C) Determining and drawing the objective function ax + by = kwhere a, b and kare

constants.

Determining graphically the optimum value of the objective function.

( Use graph paper to answer these questions)

(1) SPM 2003 P2 Q 14Shanker has an allocation of RM 225 to buy x kg of prawns and y kg of fish. The total mass

of the commodities is not less than 15 kg. The mass of prawns is at most three times that offish. The price of 1 kg of prawns is RM9 and the price of 1 kg of fish is RM5.

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

(b) Hence, using a scale of 2 cm to 5 kg for both axes, construct and shade the region R

that satisfies all the above conditions. [4 marks]

(c) If Shanker buys 10 kg of fish, what is the maximum amount of money that could

remain from his allocation? [3marks]

answers: 15, 3 ,9 5 225, 130x y x y x y RM+ +



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(2) SPM 2004 P2 Q14

A district education office intends to organize a course on the teaching of Mathematics and

Science in English.

The course will be attended by x Mathematics participants and y Science participants.

The selection of participants is based on the following constraints:

I : The total number of participants is at least 40II : The number of Science participants is at most twice that of Mathematics.

III : The maximum allocation for the course is RM7200. The expenditure for a

Mathematics participant is RM 120, and for a science participant is

RM80.

(a) Write down three inequalities, other than x 0 and y 0, which satisfy theabove constraints. [3marks]

(b) Hence, by using a scale of 2 cm to 10 participants on both axes, construct and

shade the region R which satisfies all the above constraints. [3marks]

(c) Using your graph from (b) , find

(i) the maximum and minimum number of Mathematics participants when the

number of Science participants is 10,

(ii) the minimum cost to run the course. [4marks]Answer : 720080120,2,40 ++ yxxyyx ,30, 53 (13,27)

RM3720

(3) SPM 2005 P2 Q 14

An institution offers two computer courses, P and Q. The number of participants for the

course P 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 of participants for course PIII : The number of participants for course Q must exceed the number of

Participants for course P by at least 5

(a) Write down three inequalities, other than x 0 and y 0, which satisfy all theabove constraints. [3 marks]

(b) By using a scale of 2 cm to 10 participants on both axes, construct the shade

the region R that satisfy all the above constraints. [3marks]

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

(i) the range of the number of participants for course Q if the number

of participants for course P is 30.

(ii) The maximum total fees per month that can be collected if the feesper month for courses P and Q are RM50 and RM 60 respectively.

[4marks]

Answers 5800,7035,5,4,100 RMyxyxyyx +



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(4) SPM 2006 P2 Q14A workshop produces two types of rack, P and Q. The production of each type of rack

involves two processes, making and painting. Table below shows the time taken to make

and paint a rack of type P and a rack of type Q.

Rack Time taken (minutes)Making Painting

P 60 30

Q 20 40

The workshop produces x racks of type P and y racks of type Q per day. The

production of racks per day is based on the following constraints:

I : The maximum total time for making both racks is 720 minutes.

II : The total time for painting both racks is at least 360 minutes.

III : The ratio of the number of racks of type P to the number of racks of type Q

is at least 1:3(a) Write three inequalities, other than x 0 and y 0, which satisfy all the above

constraints.

(b) Using a scale of 2 cm to 2 racks on both axes, construct and shaded the region Rare satisfies all the above constraints.

(c) By using your graph from part (b), find(i) the minimum number of racks of type Q if 7 racks of type P are

produced per day.

(ii) The maximum total profit per day if the profit from one rack of type Pis RM24 and from one rack of type Q is RM32.00

Answers

4,3

1

,3604030,7202060++

y

x

yxyx ,RM720

(5) SPM 2007 P2 Q14

A factory produces two components,Pand Q. In a particular day, the factory producedx

pieces of componentPandy pieces of component Q. The profit from the sales of a piece of

componentPis RM15 and a piece of component Q is RM12.

The production of the components per day is based on the following constraints:

I : The total number of components produced is at most 500.

II : The number of componentPproduced is not more than three times the

number of component Q

III : The minimum total profit for both components is RM4200

a) Write three inequalities, other than x 0 and y 0, which satisfy all the

constraints. [3marks]b) Using a scale of 2 cm to 50 components on both axes, construct and shade the

region R which satisfy all the above constraints. [3marks]



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c) Use your graph in part (b) to find

(i) the minimum number of pieces of component Q if the number of pieces of

componentPproduced on a particular day is 100

(ii) the maximum total profit per day [4marks]

[ ]7125,225,42001215,3,500 RMyxyxyx ++

(6) SPM 2008 P2 Q15

The members of a youth association plan to organize a picnic. They agree to rent x buses

and y vans. The rental of a bus is RM 800 and the rental of a van is RM300. The rental of

the vehicles for the picnics is based on the following constraints:

I : The total number of vehicles to be rented is not more than 8

II: the number of buses is at most twice the number of vans

III: the maximum allocation for the rental of the vehicles is RM4000

(a) Write three inequalities, other than x 0 and y 0, which satisfy all the above

constraints [ 3 marks]

(b) Using a scale of 2 cm to 1 vehicle on both axes, construct and shade the region R which

satisfy all the above constraints. [ 3 marks](c) Use the graph constructed in part (a) , to find

(i) the minimum number of vans rented if 3 buses are rented

(ii) the maximum number of members that can be accommodated into the rented vehicles ifa bus can accommodate 48 passengers and a van can accommodate 12 passengers.

[4 marks]

Answers: ,4000300800,2,8 ++ yxyxyx 2, (4,2) 216]

(7) An institution offers two types of Mathematics courses, Calculus and Statistics .

The number of students taking Calculus course is x and the number of students

taking Statistics course isy. The number of students taking the Mathematics

courses is based on the following constraints:

I : The ratio of the number of students taking Calculus course to the

number of students taking Statistic course is not more than 80 : 20.

II : The total number of students taking Mathematics courses is less than

or equal to 80.

III : The number of students taking Statistics course is at least 10

IV: The number of students taking Calculus course is more than 20.

(a) Write four inequalities, other than x 0 and y 0,which satisfy all the

above constraints. [4 marks]

(b) Using a scale of 2 cm to 10 students on both axes, construct and shadethe regionR which satisfies all of the above constraints. [3 marks]

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

(i) the range of the number of students for Calculus course if the

number of students for Statistics course is 20.

(ii) the maximum examination fee that can be collected if the examination

fees for Calculus and Statistics courses are RM 200 and RM 400

respectively. [3 marks]



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Answer:80

, 80, 10, 2020

xx y y x

y + > ,

6021 x ,k=200x+400y, (21,59), RM27,800(8) A factory produces two components ,A andB. In a particular day, the factory produced x

pieces of componentA and y pieces of componentB. The production of the two

components is based on the following constraints.:

I : The total numbers of component is not more than 500.

II : The number of componentB produced is at most three times the

number of componentA,

III : The minimum number of component B is 200.

a) Write three inequalities, other than 0x and 0y , which satisfy all theabove constraints. [ 3 marks]

b) Using a scale of 2 cm to 50 components on both axes, construct and shade the

regionR which satisfies all the above constraints. [ 3 marks]

c) Use your graph in part a), to findi. the maximum number of componentA if the number of componentB

produced on a particular day is 300. [ 1 mark]

ii.The maximum total profit per day if RM 25 and RM 20 are the profit from the

sales of component A and B respectively. [3 marks]

Answer: 500+ yx , 3y x , 200y , 200, max point (300,200) ,

25(300)+20(200)=11500

(9) xparticipants andy participants took part in eventA and eventB of a fund raising

telematch. Each participant of eventA and eventB donated RM40 and RM50 respectively.

The following conditions are imposed.

(i) Not more than 40 participants for eventA

(ii) At least 20 participants for eventB

(iii)The number of participants of eventB is not more than twice the number of

participants of eventA.

(iv) The total number of participants for both events is not more than 70.

(a) Write an inequality for each condition above. [3 marks]

(b) By using a scale of 2 cm to 10 units on both axes, construct and shade the

regionR that satisfies all the inequalities above. [4 marks]

(c) Based on your graph, determine the maximum donation collected. [3 marks]

Answer: 70,2,20,40 + yxxyyx Donation = 40x +50y, (23, 46), RM3220

(10) Table below shows the time needed for machine I and machine II to produce two types of



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electrical componentsPand Q in a certain factory.

Electrical component Machine I Machine II

P 4 minutes 10 minutes

Q 8 minutes 4 minutes

The factory producesx component P andy component Q based on the following

constraints:

I Machine I can be operated for not more than 640 minutes in a day.

II Machine II must be operated for at least 400 minutes in a day.

III The number of componentsPproduced must not be more than two times the

number of component Q produced.

(a) State three inequalities, other thanx 0 andy 0., which satisfy all of the above

constraints [3 marks]

(b) By using scales of 2 cm to 20 units on the x-axis and 2 cm to 10 units on the y-axis,draw and shade the region R which satisfy all the above constraints. [3 marks]

(c) The profits made from selling each componentPand component Q is RM 30 and

RM20 respectively.(i) Given that 60 components Q have been produced, find the maximum number of

componentPthat can be produced. [1 mark]

(ii) calculate the maximum profit that can be made by the factory in a day.[3 marks]

Answers [ ,2,400410,64084 yxyxyx ++ , 50, 3200]


10. linear programming

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