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  • Explain solution RD Sharma class 12 Chapter 29 Linear Programming exercise 29.4 question 12

Explain solution RD Sharma class 12 Chapter 29 Linear Programming exercise 29.4 question 12

Answers (1)

Answer:

Max \: Profit \Rightarrow Rs. 500 could be obtained when no units of product A and 125 units of product B are manufactured.

Hint:

Form Equation and solve graphically.

Given:

A firm manufactures two products A and B. Each product is processed on two machines. M1  and M2 . Product A requires 4 minutes of processing time on M1  and 8 min on M2  ; Product B requires 4 min on M1  and 4 min on M2

Solution:

Let required production of product A and B be x and y respectively.

Since profit on each product A and B are Rs. 3 and Rs. 4 respectively. So, profits on x number of type A and y number of type B are 3x and 4y respectively.
Let Z denotes total output daily, so,

z=3x+4y

Since, each A and B requires 4 minutes each on machine M1 . So, x of type A and y of type B require 4x and 4y minutes respectively, But total number available on machine M1 is 8 hours 20 minutes is equal to 500 minutes.

So,
\begin{aligned} &4 x+4 y \leq 500 \\ &x+y \leq 125 \end{aligned}                                    { first constraint}

Since each A and B requires 8 minutes and 4 minutes on machine M2  So , 2x of type A and y of type B require 8x and 4y minutes respectively.  But

Total time available on Machine M1 is 10 hours = 600 minutes

So,

\begin{aligned} &8 x+4 y \leq 600 \\ &2 x+y \leq 150 \end{aligned} {Second constraint}
Hence, mathematical formulation of the given L.P.P is,

Max z = 3x + 4y
Subject to constraints,

\begin{aligned} &x+y \leq 125 \\ &2 x+y \leq 150 \end{aligned}

x,y\geq 0 [Since production of chairs and tables cannot be less than 0]

Region: x+4y\leq 125  : Line x+y=125  meets the axes at A(125,0) , B(0,125)   respectively.

Region containing the origin represents x+4y\leq 125   as origin satisfies x+4y\leq 125

Region 2x+5y\leq 150 :  Line  2x+y=150 meets the axes at C(75,0), D(0,150)   respectively.

Region containing the origin represents  2x+5y\leq 150  as origin satisfies 2x+5y\leq 150

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

The corner points are 0(0,0),B(0,125), ?(25,100),C(75,0).

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

Corner Points

z=3x+4y

O

0

B

500

\epsilon

475

C

225

The maximum value of z is 500 which is attained at B(0,125)

Thus, maximum profit is Rs500 obtained when no units of product A and 125 units of product B are manufactured.

 

Posted by

infoexpert27

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