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  • Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 21 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 21 Maths Textbook Solution.

Answers (1)

Answer:

Max Profit = Rs.3, 25,500 when 10500 bottles of A and 34500 bottles of B are manufactured.

Hint:

Form Linear Equation and solve graphically.

Given:

A manufacturer of patent medicines is preparing a production plan on medicines A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B. But there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation.

Solution:

Let production of each bottle of A and B are x and y respectively.

Since profits on each bottle of A and B are Rs.8 and Rs.7 per bottle respectively. So, profit on x bottles of A and y bottles of B are 8x and 7y respectively. Let Z be total profit on bottles so,

                Z = 8x + 7y

Since, it takes 3 hours and 1 hour to prepare enough material to fill 1000 bottles of Type A and Type B respectively, so x bottles of A and y bottles of B are preparing is \frac{3 x}{1000}  hours and  \frac{y}{1000}  hours respectively, about only 66 hours are available, so,

                \begin{aligned} &\frac{3 x}{1000}+\frac{y}{1000} \leq 66 \\ &3 x+y \leq 66000 \end{aligned}

Since raw materials available to make 2000 bottles of A and 4000 bottles of B but there are 45000 bottles in which either of these medicines can be put so,

                \begin{aligned} &x \leq 20000 \\ &y \leq 40000 \\ &x+y \leq 45000 \\ &x, y \geq 0 \end{aligned}

[Since production of bottles cannot be negative]

Hence mathematical formulation of the given LPP is,

Max Z = 8x + 7y

Subject to constraints,

                \begin{aligned} &3 x+y \leq 66000 \\ &x \leq 20000 \\ &y \leq 40000 \\ &x+y \leq 45000 \\ &x, y \geq 0 \end{aligned}

Region 3 x+y \leq 66000 : line   meets the axes at A (22000,0), B(0,66000) respectively.

Region containing origin represents 3 x+y \leq 66000  as (0,0) satisfy 3 x+y \leq 66000

Region x+y \leq 45000 : line x+y=45000  meets the axes at C (45000,0) and D(0,45000) respectively.

Region towards the origin will satisfy the in equation as (0,0) satisfies the in equation.

Region represented by x \leq 20000

x= 200000  is the line passes through (20000,0) and is parallel to the y-axis. The region towards the origin will satisfy the in equation.

Region represented by  y \leq 40000,

Y=40000 is the line passes through (0, 40000) and is parallel to the x- axis. The region towards the origin will satisfy the in equation.

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

                                                                                                                                   

                                                                                                                                                         Scale: On y-axis, 1 Big division=20000 units

                                                                                                                                                                    On x-axis, 1 Big division=10000 units

              

The corner points are O(0,0), B(0,40000), G(10500,34500), H(20000,6000), A(20000,0)

The value of Z at these corner points are,

Corner Points

z=8x+7y

O

0

B

280000

G

325500

H

188000

A

160000

The maximum value of Z is 325500 which is attained at G (10500, 34500)

Thus the maximum profit is Rs.325500 obtained when 10500 bottles of A and 34500 bottles of B are manufactured.

Posted by

infoexpert27

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