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  • Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 40 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 40 Maths Textbook Solution.

Answers (1)

Answer: The factory makes 4 tennis racket and 12 cricket bats. Maximum profit is 200.

Hint:

Let the number of tennis rackets and cricket bats be x and y.

Given:

A tennis racket takes 1.5 hours if machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day factory has availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. If the profit on a rackets and bat is 20Rs and 10Rs.

Solution:

Let the number of tennis racket and cricket bats manufactured by factory be x and y.

Hence, Profit is the objective function Z.

Z = 20x + 10y                                                      … (i)

We have to maximize Z subject to the constraints.

\begin{aligned} &1.5 x+3 y \leq 42 \\ & \end{aligned}                                                   … (ii)      [Constraints for machine hour]

3 x+y \leq 24 \\                                                         … (iii)     [Constraints for craftman’s hour]

x \geq 0 \& y \geq 0

Graph of x = 0 and y = 0 is the y-axis and x-axis.

Graph of  x \geq 0 \& y \geq 0  is the 1st quadrant.

Graph of 1.5 x+3 y=42

x

0

28

y

14

0

 

Graph for 1.5 x+3 y \leq 42 is the part of 1st quadrant which contains the origin.

Graph for \begin{aligned} &3 x+y \leq 24 \\ & \end{aligned}

               3 x+y=24

x

0

8

y

24

0

 

Graph for 3 x+y \leq 24 is the part of st quadrant in which origin lies.

Hence, shaded area OACB is the feasible region for coordinate of C equation

\begin{aligned} &1.5 x+3 y=42 \\ & \end{aligned}                                                                  … (iv)

3 x+y=24 \\                                                                         … (v)

\begin{aligned} 2 \times(\mathrm{iv})-(\mathrm{v}) \Rightarrow 3 \mathrm{x}+6 \mathrm{y}=84 \\ & \end{aligned}

                                  3 \mathrm{x}+\mathrm{y}=24 \\

                                           \overline{5 y=60}\; \Rightarrow y=12

         X = 4 (Substituting y = 12 in (iv))

Now the value of objective function Z at each corner of feasible region is

 

Corner Points

Z=20x+10y

O(0,0)

Z=0+0=0

A(8,0)

20 \times 8+10 \times 0=160

B(0,14)

20 \times 0+10 \times 14=140

C(4,12)

20 \times 4+10 \times 12=200

 

Therefore, maximum profit is Rs.200 when factory makes 4 tennis rackets and 12 cricket bats.

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