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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.

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