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  • Need solution for RD Sharma Maths Class 12 Chapter 29 Linear Programmig Excercise 29.4 Question 17

Need solution for RD Sharma Maths Class 12 Chapter 29 Linear Programmig Excercise 29.4 Question 17

Answers (1)

Answer:

Min\: cost = Rs 254 when 14 units of compound A and 33 unit compound B are produced.

Hint:

Form Linear Equation and solve graphically.

Given:

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 requirement of C and D and costs per kg of A and B

 

compound

Minimum

 

A

B

requirement

Ingredient C

1

2

80

Ingredient D

3

1

75

Cost(inE) perkg

4

6

 

Solution:

Let required quantity of compound A and B are x and y kg. Since, cost of 1 kg of compound A and B are Rs.4 and Rs.6 per kg. So, Cost of x kg compound A and y kg of compound B are Rs.4x and Rs.6 respectively.

Let z be the total cost of compounds.

So,z= 4x + 6y

Since compound A and B contain 1 and 2 units of ingredient C per kg respectively , So x kg of Compound A and y kg of Compound B contain x and 2y units of ingredient C respectively but minimum requirement of ingredient C is 80 units.

So ,
x+2y\geq 80                                    {first constraint}

Since, compound A and B contain 3 and 1 units of ingredient D per kg respectively.

So x kg of compound A and y kg of compound B contain 3x and y units of ingredient D respectively but minimum requirement of ingredient C is 75 units

So ,

3x+y\geq 75                                 {Second constraint}
Hence mathematical formulation of the given L.P.P is,

Min  z = 4x + 6 y
Subject to constraints,

\begin{aligned} &x+2 y \geq 80 \\ &5 x+y \geq 75 \\ & \end{aligned}

x, y \geq 0 [Since production cannot be less than 0]

Region represented by x+2y\geq 80  : The line x+2y=80  meets the axes at A(80,0) , B(0,40) respectively.

Region not containing represents x+2y=80  as (0,0) does not satisfy satisfies  x+2y=80

Region  3x-1y\geq 75   : line  3x+y=75  meets axes at C(25,0),D(0,7)  respectively.
Region not containing origin represents  3x-1y\geq 75  as (0,0) does not satisfy 3x-1y\geq 75 .
Region  x,y\geq 0:   it represents first quadrant.

The corner points are D(0,75),\epsilon (14,33),A(80,0)  .

The value of z at these corner points are as follows.

Corner Points

z=4x+6y

D

450

ε

254

A

320

The minimum value of z is 254 which is attained at \epsilon (14,33)
Thus, the minimum cost is Rs. 254 obtained when 14 units of compound A and 33 units compound B are produced.

Posted by

infoexpert27

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