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  • Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 19 Maths Textbook Solution.

Please Solve RD Sharma Class 12 Chapter 29 Linear Programming Exercise 29.4 Question 19 Maths Textbook Solution.

Answers (1)

Answer:

Max Profit = Rs1,440 when 48 units of product A and 16 units of product B are manufactured

Hint:

Form Linear Equation and solve graphically.

Given:

A manufacturer makes two products A and B. Product A sells at Rs.200 each and takes 2hrs to make. Product B sells at rs.300 each and rakes 1 hr. to make. There is a permanent order for 14 of product A and 16 of product B. A working week consists of 40 hrs. of production and weekly turn over must not be less that Rs. 10000. If the profit on product A is 20Rs and on Product B is Rs.300

Solution:

Let x be units of product A and y be units of product B are manufactured.

Number of units cannot be negative

Therefore,  \text { x, } y \geq 0

According to question, the given information can be tabulated as:

               

 

Selling Price (Rs)

Manufacturing time (hrs.)

Product A(x)

200

0.5

Product B(y)

300

1

 

Also, the availability of time is 40 hrs and the revenue should be at least Rs.10000

Further, it is given that there is a permanent order for 14 units of Product A and 16 units of Product B

Therefore, the constraints are,

                \begin{aligned} &200 x+300 y \geq 10000 \\ &0.5 x+y \leq 40 \\ &x \geq 14 \& y \geq 16 \end{aligned}

If the profit on each of product A is Rs,20 and on product B is Rs,30. Therefore, profit gained on x units of product A and y units of Product B is Rs. 20x and Rs.30y resepectively.

Total Profit=20x+30y which is to be maximized.

Thus, the mathematical formulation of the given LPP is,

Max: z = 20x +30y

Subject to constraints,

\begin{aligned} &200 x+300 y \geq 10000 \\ &0.5 x+y \leq 40 \\ &x \geq 14 \& y \geq 16 \\ &x, y \geq 0 \end{aligned}
Region 200 x+300 y \geq 10000: \text { Line } 200 x+300 y=10000  meets the areas at A(50,0) , \mathrm{A}(50,0), \mathrm{B}\left(0, \frac{100}{3}\right)  respectively.

Region not containing origin represents 200 x+300 y \geq 10000  as (0,0) does not satisfy  200 x+300 y \geq 10000

Region 0.5 x+y \leq 40: \text { line } 0.5 x+y=40  meets the area at C(80,0) D(0,40) respectively.

Region containing origin represents 0.5 x+y \leq 40 \text { as }(0,0) \text { satisfies } 0.5 x+y \leq 40

Region represented by  x\geq 14

x=14 is the line passes through (14,0) and is parallel to the X-axis.

The region to the right of the line y=14 will satisfy the in equation

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

The corner points of the feasible region are E(26,16), F(48,16), G(14,33) and H(14,24)

The value of Z at the corner points are as follows:

Corner Points

z=20x+30y

E

1000

F

1440

G

1270

H

1000

The maximum of Z is 1440 which is attained at F(48,16)

Thus, the maximum profit is Rs.1440 obtained when 48 units of Product A and 16 units of Product B are manufactured.

 

Posted by

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