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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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