A firm manufactures two products A and B . Each product is processed on two machines M 1 and M 2 . Product A requires 4 minutes of processing time on M 1 and 8 m i n on M 2 . product B requires 4 minutes on M 1 and 4 min on M 2 . The machine M 1 is available for not more than 8 h r s 20 m i n , while machine M 2 is available for 10 h r s during any working day. The products A and B are sold at a profit of R s 3 and R s 4 respectively.Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

Let x products of type A and y products of type B are manufactured. Number of products cannot be negative. Therefore, x , y ≥ 0 The given information can be tabulated as f Product M 1 (minutes) M 2 (minutes) A ( x ) 4 8 B ( y ) 4 4 Availability 500 600 Therefore, the constraints are as follows: 4 x + 4 y ≤ 500 8 x + 4 y ≤ 600 The products A and B are sold at a profit of R s 3 and R s 4 respectively.Therefore, Profit gained from x products of type A and y products of type B is R s 3 x and R s 4 y respectively. Total profit = Z = 3 x + 4 y which is to be maximized. Thus, the mathematical formulation of the given linear programming problem is Maximize Z = 3 x + 4 y subject to the constraints 4 x + 4 y ≤ 500 8 x + 4 y ≤ 600 , x ≥ 0 , y ≥ 0 The above inequalities form a feasible region depicted below. Notice that the feasible region is O B E C O . The corner points are O ( 0 , 0 ) , B ( 0 , 125 ) , E ( 25 , 100 ) and C ( 75 , 0 ) . The values of Z at these corner points are as follows Corner point Z = 3 x + 4 y O 0 B 500 E 475 C 225 The maximum value of Z is 500 which is attained at B ( 0 , 125 ) . Thus, the maximum profit is R s 500 obtained when no units of product A and 125 units of product B were manufactured.

EASY

12th CBSE

IMPORTANT

Earn 100

A manufacturer of Furniture makes two products : chairs and tables. Processing of these products is done on two machines $A$ and $B$ . A chair requires $2 hrs$ on machine $A$ and $6 hrs$ on machine $B$ . A table requires $4 hrs$ on machine $A$ and $2 hrs$ on machine $B$ . There are $16$ hrs of time per day available on machine $A$ and $30 hrs$ on machine $B$ . Profit gained by the manufacturer from a chair and a table is $Rs 3$ and $Rs 5$ respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

Important Questions on Linear Programming

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 11

A furniture manufacturing company plans to make two products : chairs and tables. From its available resources which consists of

400

square feet to teak wood and

590

man hours. It is known that to make a chair requires

5

square feet of wood and

10

man-hours and yields a profit of

Rs 45

, while each table uses

20

square feet of wood and

25

man-hours and yields a profit of

Rs 80

. How many items of each product should be produced by the company so that the profit is maximum?

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 12

A firm manufactures two products

A

and

B

. Each product is processed on two machines

M 1

and

M 2

. Product $A$ requires

4

minutes of processing time on

M 1

and

8 m i n

M 2

. product

B

requires

4

minutes on

M 1

and

4

min on

M 2

. The machine

M 1

is available for not more than

\frac{9}{5}

, while machine

M 2

is available for

10 h r s

during any working day. The products

A

and

B

are sold at a profit of

R s 3

and

R s 4

respectively.Formulate the problem as a linear programming problem and find how many products of each type should be produced by the firm each day in order to get maximum profit.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 13

A firm manufacturing two types of electric items,

A

and ,

B

can make a profit of

R s 20

per unit of

A

and

R s 30

per unit of

B

. Each unit of

A

requires

3

motors and

4

transformers and each unit of

B

requires

2

motors and

4

transformers. The total supply of these per month is restricted to

210

motors and

300

transformers. Type

B

is an export model requiring a voltage stabilizer which has a supply restricted to

65

units per month. Formulate the linear programing problem for maximum profit and solve it graphically.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 14

A factory uses three different resources for the manufacture of two different products,

20

units of the resources

A,

12

units of

B

and

16

units ofCbeing available.

1

unit of the first product requires

2, 2

and

4

units of the respective resources and unit of the second product requires

4, 2

and

0

units of respective resources. It is known that the first product gives a profit of

2

monetary units per unit and the second

3

. Formulate the linear programming problem. How many units of each product should be manufactured for maximizing the profit? Solve it graphically.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 15

A publisher sells a hard cover edition of a text book for

R s 72.00

and paperback edition of the same ext for

R s 40.00

. Costs to the publisher are

R s 56.00

and

R s 28.00

per book respectively in addition to weekly costs of

R s 9600.00 .

Both types require

5

minutes of printing time, although hardcover requires

10

minutes binding time and the paperback requires only

2

minutes. Both the printing and binding operations have

4, 800

minutes available each week. How many of each type of book should be produced in order to maximize profit?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 16

A firm manufactures headache pills in two sizes

A

and

B

. SizeAcontains

2

grains of aspirin,

5

grains of bicarbonate and

1

grain of codeine; size

B

contains

1

grain of aspirin,

8

grains of bicarbonate and

66

grains of codeine. It has been found by users that it requires at least

12

grains of aspirin,

7.4

grains of bicarbonate and

24

grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief. Determine also the quantity of codeine consumed by patient.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 17

A chemical company produces two compounds,

A

and

B

. The following table gives the units of ingredients,

C

and

D

per kg of compounds

A

and

B

as well as minimum requirements of

C

and

D

and costs per kg of

A

and

B

. Find the quantities of

A

and

B

which would give a supply of

C

and

D

at a minimum cost.

	Compound	Minimum requirement
$A$	$B$
Ingredient $C$ Ingredient $D$	$1$ $3$	$2 1$	$80 75$
Cost (in Rs) per kg	$4$	$6$

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 11 - Linear Programming>EXERCISE>Q 18

A company manufactures two types of novelty Souvenirs made of plywood. Souvenirs of type A require

5

minutes each for cutting and

10

minutes each for assembling. Souvenirs of type

B

require

8

minutes each for cutting and

8

minutes each for assembling. There are

3

hours

20

minutes available for cutting and

4

hours available for assembling. The profit is

50

paise each for typeAand 60 paise each for type

B

souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

Important Questions on Linear Programming

Learn and Prepare for any exam you want !