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  • Provide Solution for RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 14

Provide Solution for RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 14

Answers (1)

Answer:

Max\: Profit \Rightarrow Rs. 16 when two units of first product and 4 units of second product were manufactured.

Hint:

Form Linear Equation and solve graphically.

Given:

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 of C. 1 unit of the first Product requires 2, 2 and 4 units of the respective resources and 1 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.

Solution:

Let number of product I and product II are x and y respectively.

Since profit on each product I and II requires 2 an 4 units of resources A:50 , x units of product I and y units of product II requires 2x and 4y minutes respectively. But maximum available quantity of resources A is 20 units.

So,

\begin{aligned} &2 x+4 y \leq 20 \\ &x+2 y \leq 10 \end{aligned}                                  { first constraint}

Since each I and II requires 2 and 2 units of resources B . So, x units of product I and y units of product II requires 2x and 2y minutes respectively. But maximum available quantity of resources A is 12 units

So ,

\begin{aligned} &2 x+2 y \leq 12 \\ &x+y \leq 6 \end{aligned}                                      {Second constraint}

Since each units of product I requires 4 units of resources C . It is not required product II. So x units of product I require 4x units of resource C . But maximum available quantity of resource C is 16 units.

So ,

\begin{aligned} &4 x \leq 16 \\ &x \leq 4 \end{aligned}                        {Third constraint}

Hence mathematical formulation of the given L.P.P is,

Max  z = 2x + 3y
Subject to constraints,

\begin{aligned} &x+2 y \leq 10 \\ &x+y \leq 6 \\ &x \leq 4 \end{aligned}                            

 

x,y\geq 0 [Since production of I AND II cannot be less than 0]

 

Region represented by x+2y\leq 10  The line  x+2y=10  meets the axes at A(10,0) , B(0,5) respectively.

Region containing the origin represents x+2y\leq 10 as origin satisfies  x+2y\leq 10

Region represented by  x+y\leq 6 :   Line  x+y=6   meets the axes at C(6,0), D(0,6) respectively.

Region containing the origin represents  x+y\leq 6   as origin satisfies  x+y\leq 6

Region  x,y\geq 0:  it represents the first quadrant.

The corner points are 0(0,0),B(0,5), G(2,4),?(4,0).

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

Corner Points

z=3x+4y

O

0

B

15

G

16

F

14

?

8

The maximum value of z is 16 which is attained at G(12,4)

Thus, maximum profit is 16 monetary units obtained when 2 units of the first product and 4 units of the second product were manufactured.

 

Posted by

infoexpert27

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