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  • Provide Solution for RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 25

Provide Solution for RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 25

Answers (1)

Answer:

Maximum Profit Rs.1800 when Rs.20 were involved in Type A and Rs.40 were involved in Type B.

Hint:

Form Linear Equation and solve graphically.

Given:

A men owns a field of area 1000 m2. He wants to plant fruit trees in it. He has a sum of Rs.1400 to purchase young trees. He has the choice of two types of trees. Type A require 10 m2 of ground per tree and costs Rs.20 per tree and type B requires 20m2 of ground per tree and costs Rs.25 per tree. When fully grown type A produces an average of 20 kg of fruit which can be sold at a profit of Rs.2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of Rs.1.50 per kg.

Solution:

Let the required number of trees of Type A and B be Rs.x and Rs.y respectively.

Number of trees cannot be negative.

x, y \geq 0

To plant tree of Type A requires 10sq.m and Type B requires 20sq.m of ground per tree. And it is given that a man owns a field of are 1000sq.m.

Therefore,

                \begin{aligned} &10 x+20 y \leq 1000 \\ &x+2 y \leq 100 \end{aligned}

Type A costs Rs.20 per tree and type B costs Rs.25 per tree. Therefore, x trees of type A and y trees of type B cost Rs.20x and Rs.25y respectively. A man has a sum of Rs.1400 to purchase young trrs

                 \begin{aligned} &20 x+25 y \leq 1400 \\ &4 x+5 y \leq 280 \end{aligned}

Thus the mathematical formulation of the given LPP is

                Max Z = 40x -20x + 60y - 25y = 20x + 35y

Subject to,
                 x+2 y \leq 100

                \begin{aligned} &4 x+5 y \leq 280 \\ &x, y \geq 0 \end{aligned}

Region 4 x+5 y \leq 280 : the line 4 x+5 y \leq 280  meets axes at A1(70,0) and B1(0,56)respectively.

The region which contains origin represents 4 x+5 y \leq 280 as (0,0) satisfies 4 x+5 y \leq 280 .

Region x+2 y \leq 100 : the line x+2 y \leq 100  meets axes at A2(70,0) and B2(0,56)respectively.

The region which contains origin represents x+2 y \leq 100 as (0,0) satisfies x+2 y \leq 100 .

Region x, y \geq 0 : It represents by  first quadrant.

The maximum value of Z is 1800 which is attained at P(20,40) as

The value of Z at these corner points are as follows

 

Corner Points

Z=20 x+35 y

O

0

A1

1750

P

1800

B2

1400

Thus the, max profit is Rs.1800 obtained when Rs.20 were involved in Type A and Rs.40 were involved in Type B.

Posted by

infoexpert27

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