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Linear programming Questions

 

1.  A tailoring business makes two types of garments A and B. Garment A requires 3 metres of  material while garment B requires 2 ½ metres of material. The business uses not more than 600  metres of material daily in making both garments. It must make not more than 100 garments of  type A and not less than 80 of type B each day.

 (a) Write down four inequalities from this information. (3mks)

 (b) Graph these inequalities. (3mks)

(c) If the business makes a profit of shs 80 on garment A and a profit of shs 60 on garment B, how many garments of each type must it make in order to maximize the total profit? (4mks)

2.  A man bakes two types of cakes, queen cakes and marble cakes. Each week he bakes x

queen cakes and y marble cakes. The number of cakes baked are subject to the following

conditions; 30x + 20y ≤ 4800, 30x + 40y ≥ 3600 and 10x >30y

 He makes a profit of shs.10 on each queen cake and shs.12 on each marble cake.

 (i) Draw a graph to represent the above information on the grid provided

  (ii) From the graph, determine how many cakes of each type he should make to

maximize his weekly profit

  (iii) Calculate the maximum profit  

  (iv) If he is to make a weekly profit of at least shs.600, find the least number of marble

cakes he should bake

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3.  A company produces shirts and jerseys using two types of machines. Every shirt made

 requires 2 hours on machine A and 2 hours on machine B. Every Jersey made requires

 3hours on machine A and I hour on machine B. In one day the time limit on machine A

 is 24hours but that on machine B is 12hrs. The number of Jerseys produced must not be

 more than the shirts produced in one day. The company makes a profit of shs.200 on each

 shirt and shs.200 on each Jersey. The company produces x shirts and y jerseys per day

 (a) Write down four inequalities which must be satisfied by x and y and represent these

  inequalities on a grid

 (b) Find the values of x and y which will give the company maximum daily profit and

  also state the maximum profit

 

4.  A trader makes two types of chair, ordinary and special chairs. The cost of each ordinary

chair is shs.300 while each special chair costs shs.700. He is prepared to spend not more than shs.21,000. It is not viable for hi m to make less than 20 chairs. Ordinary chairs must be less

than twice the special chairs but more than 15. By taking the number of ordinary chairs as x

and special chairs as y:

 (a) Write down all the inequalities in x and y  

 (b) Draw the inequalities on the grid provided

(c) He sells a special chair at a profit of shs.140 while ordinary chairs at a profit of shs.120;

Determine the maximum possible profit  

5.  A school has to take 384 people for a tour. There are two types of buses available.

Type X and type Y. Type X can carry 64 passengers and type Y can carry 48 passengers.

They have to use at least 7 buses.

 a) Form all linear inequalities which will represent the above information .

 b) On the grid provided, draw the inequalities and shade the un-wanted region.

 b) The charges for hiring the buses are

Type X: shs.25,000

Type Y: shs.20,000

 Use your graph to determine the number of buses of each type that should be hired to

  minimize the cost.  

 

6. A shoe maker makes two types of shoes A and B. He takes 3 hours to make one pair of type


A and 4 hours to make one pair of type B. He works for a maximum of 120 hours to make x

pairs of type A and y pairs of type B. It costs him Kshs. 400 to make a pair of type A and

Kshs.150 to make a pair of type B. His total cost does not exceed kshs.9000. He must make

at least 8 pairs of type A and 12 pairs of type B.

(a) Write down four inequalities representing the information above  

(b) On the grid provided represent the inequalities and shade the unwanted regions

(c) The shoe maker makes a profit of kshs.40 on each pair of type A and kshs.70 on each pair

 

7.  A theatre has a seating capacity of 250 people. The charges are shs.100 for an ordinary

seat and shs.160 for a special seat. It costs shs.16,000 to stage a show and the threatre

must make a profit. There is never more than 200 ordinary seats and for a show to take

place at least 50 ordinary chairs must be occupied. The number of special seats is always

less than twice the number of ordinary seats.

 a) taking x to be the number of ordinary seats and y the number of special seats ,

write down all the inequalities representing the information above.

 b) On the grid provided, draw the graph to show the inequalities in (a) above

 c) Determine the number of seats of each type that should be booked in order to maximize

the profit.

 

8.  A man sells two types of ice creams in cups and sticks. He can store less than ten packets

in his cooling box. He sells more cups than sticks but less than 3 items as many cups as sticks.

 He also knows that he will sell more than 3 packets of sticks. His profit is shs.3.00 on a packet

 of cups and shs.2.00 on a packet of sticks.

 (a) Form inequalities to represent the above information:

(Let x – packets of cups and

 y – packets of sticks)

(b) On the grid provided graph the inequalities to satisfy the required condition

(c) How many packets of cups and sticks should the man put in his box to give him the highest

profit?

 

9. A shopkeeper bought 50 pangas and 30 jembes :-

(a) From a wholesalers for shs.4,260. He had bought half as many jembes and 5 pangas less,

he would have paid shs.1290 less. Had the shopkeeper bought form wholesaler B, he would

have paid 10% more a panga and 15% less for a jembe. How much would he have Ho saved if he

had bought the 50pangas and 30 jembes from wholesalers B  

(b) The price of a suit if marked at shs.5000. A discount

 

10.  The games master whishes to hire two matatus for a trip. The operators have a Toyota

which carries 10passengers and a Kombi which carries 20 passengers. Altogether 120

people have to travel. The operators have only 20litres of fuel and the Toyota consumes

4 litres on each round trip and the Kombi 1 litre on each round trip. If the Toyota makes x

round trips and the kombi y round trips;

 

 (a) write down four inequalities in x and y which must be satisfied  

 

b) represent the inequalities graphically on the grid provided  

 

 (c) The operators charge shs.100 for each round trip in the Toyota and shs.300 for each

  round trip in the kombi;

(i) determine the number of trips made by each vehicle so as to make the total cost a

Minimum

(ii) find the minimum cost

11.  The velocity of a particle Vm/s moving in a straight line after t seconds is given by

V = 3t2 – 3t – 6. Find the distance covered by the particle between t = 1 and t = 4seconds

 

 

Linear programming Answers

1.

(a) let the No. of garments of type A be x and those of type B be y

(i) 3x + 2 ½ y 600 (material)

(ii) x 1000

y  80

x  0

(b) Lines drawn

3x + 2 ½ y = 600

x = 100

x = 80

x = 0

(c) The object function is

P = 80x + 60y where P = total profit

Either drawn a search line by choosing an appropriate value of P

e.g 12000 = 80x + 60y

or inspect for maximum profit using points further from origin

maximum profit

100 garment og type A

120 garments of type B

 

 

B1

B2

B1

 

 

B1

 

B1

 

B1

 

 

B1

 

 

B1

 

 

B1

B1

 

 

 

all

any two

 

 

lines and shading

 

lines and shading

 

forregion indicated

 

 

objective function

 

 

Use of search line or inspection

 

    

 

 

 

 

 

1.  30x + 20y  4800……..(i)

Image From EcoleBooks.com   30x + 40y  3600……..(ii)

 10x  30 y…………(iii)

  x 0 y  0

 

 objective function 10 x+ 12 y = K

 

Image From EcoleBooks.comImage From EcoleBooks.com3x + 2y = 480 3x + 4y = 360 x = 3y

X

40

60

80

 

X

20

40

60

 

X

30

45

60

y

180

150

120

 

Y

75

60

45

 

Y

10

15

20

 

 (ii) consider (60,40)

 10 (60) + 12(40) = 600 +480

=1080

 10x + 12y = 108 0

Image From EcoleBooks.com  5x + 6y = 540 – search line

Image From EcoleBooks.com

X

20

40

60

y

73

57

40

 

 Maximum profit at ( , 240)

 No queen cake , 240 marble cakes

 

 (iii) 240 X 12 = sh. 2880

 

 (iv) 10x + 12y  600  10x + 12y = 600

 5x + 6y = 300

X

12

60

y

50

40

0

 

 

 

2.  Machine A Machine B

  Shirts Jerseys Shirts Jerseys

No. x y x y

Hrs. @2hrs @3hrs @2hrs @1hr

 

(i) 2x + 3y  24

(ii) 2x + y  12

(iii) y > x

(iv) x > 0

  y > 0

Max pt(3,6)

Image From EcoleBooks.comMax profit = 22 x 3 + 200 x 6

= 600 + 1200

= Shs. 1800

 

 

 

 

 

 

 

 

Image From EcoleBooks.com

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.  (a)   3x + 7y  210

x + y 20

x < 2y

x > 15

(b) refer

(c) 120x + 140y = 120 x 130 + 140 x 10

Profit = shs.5960

x = 31

y = 16

4.  Passengers

64x + 48y ≥ 384 i.e. 8x + 6y ≥ 48

x > 0

y > 0

x + y ≥ 7

 

Cost equation

Total cost = 2500x + 20000y

(3,4)

3 type x

4 type y

 

Image From EcoleBooks.com

 

 

 

 

 

Image From EcoleBooks.com

 

 

 

 

Image From EcoleBooks.com

Image From EcoleBooks.com

Image From EcoleBooks.comImage From EcoleBooks.comImage From EcoleBooks.com

 

 

 

 

 

 

5.   3x + 4y ≤120

400x + 150y ≥ 9000

  x ≥ 8 y >12

(b)(i) 3x + 4y 20

(ii) 40x + 15y 900

(iii) x 8

(iv) y 12

 

Points

Objective function 40x + 70y

Profit

(i) (8,24)

320 + 1680

2000

(ii) (24, 12)

960 + 840

1800

(iii) (8, 12)

320 + 840

1160

Image From EcoleBooks.com

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c)  12 type x

12 type y

Profit = 40(24) + 70(12)

  = 1800

6.  100 x = 160y = 16000 5 x 200 + 8 x 50

=100 x 200 + 160 x 50 1000 + 4000

  20000 + 8000 10 x 200 + 16 x 50=

 28000/= 10x + 16y = 1600

5x + 8y = 800

5 x 20 + 100

8y = 800 – 100

y = 700

8


800/5 = 160

 

a) y<2x, 50≤x≤200 x >100

  y >0, x+y ≤250, 100x + 160y ≥ 16000

 

b) See graph

 

 

Image From EcoleBooks.comImage From EcoleBooks.com

 

 

 

 

 

 

 

 

 

 

 

 

Image From EcoleBooks.com

profit?

 

 

7.  x + y  10

y  3x

y > 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  1. Obejctive function 3x = 2y = I or use of serach line

5 packets of cups and 4packets of stucks

x

y

Profit

2

2

3

3

3

4

4

5

4

5

4

5

6

4

5

4

14

16

17

19

21

20

22

23

 

8. Panga – P, Jembe J

(a) 50P + 30J = 4260

50P + 15J = 1290

50P + 30J = 4260

10P + 30J = 1290

40P = 1680

 P = 168 = 42

4

50(42) + 30J = 4260

  1. + 30J = 4260

30J = 2160

J = (2160)

  30

J = 72

 

Wholesaler

110 x 42 = shs.46.50 = pangas

100

85 x 72 = shs 60 = jembes

100

 

For B

50 x 46.50 + 30 x 61.2

2310 + 1836 = 4146

Saving =  4260

4116

144

 

(b) Discount 5000 – 3500 = 1500

% discount = 1500 x 100

 5000

  = 30%

9.  a) X ≥ 0, y = ≥ 0

10x + 20y ≥ 120

4x + y ≥ 20

 

b) On the graph.

 

c) i) (4,4)

4 x 100 + 4 x 300

400 + 1200 = 1600

 

 

10.  Distance Covered = (3t2 – 3t -6)dt

  = t33/2 t2 – 6t 4 4


1

433 (4)2 – 6(4) – 13 – 3(1)2 – 6(1)

2

16 – -13/2=


 




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