Question

Every gram of wheat provides 0.1 gm of proteins and 0.25 gm of carbohydrates. The corresponding values of rice are 0.05 gm and 0.5 gm respectively. Wheat costs Rs. 4 per kg and rice Rs. 6. The minimum daily requirements of proteins and carbohydrates for an average child are 50 gms and 200 gms respectively. In what quantities should wheat and rice be mixed in the daily diet to provide minimum daily requirements of proteins and carbohydrates at minimum cost.    

Solution

Correct option is

x = 400 & y = 200 and Rs. 2.8.

 

Suppose x gms of wheat and y grams of rice and mixed in the daily diet. Since every gram of wheat provides 0.1 gm of proteins and every gram of rice gives 0.05 gm of proteins. Therefore, x gms of wheat and y grams of rice will provide 0.1x + 0.05y gms of proteins. But the minimum daily requirement of proteins is of 50 gms.   

  

Similarly, x gms of wheat and y gms of rice will provide 0.25x + 0.5y gms of carbohydrates and the minimum daily requirement of carbohydrates is of 200 gms.      

  

Since the quantities of wheat and rice cannot be negative. Therefore,   

                    x ≥ 0, y ≥ 0    

It is given that wheat costs Rs 4 per kg and rice Rs 6 per kg. So, x gms of wheat and y gms of rice will cost Rs   

Hence, the given linear programming problem is   

Minimize   

Subject to the constraints   

                     

                   

  

The solution set of the linear constraints is shaded in fig. The vertices of the shaded region are A2 (800, 0), P (400, 200) and B1 (0, 1000).                                                                

                                                                            

 

The values of the objective function at these points are given in the following table.   

Point (x1x­2)

Value of objective function  

             

A2 (800, 0)

P (400, 200)

B1 (0, 1000)

  

  

Clearly, Z is minimum for x = 400, y = 200 and the minimum value of Z is 2.8.

We observe that the open half plane represented by  does not have points is common with the feasible region. So, Z has minimum value 2.8 at x = 400 and y = 200.

Hence the diet cost in minimum when x = 400 and y = 200. The minimum diet cost is Rs 2.8.

 

SIMILAR QUESTIONS

Q1

 

Solve the following LPP graphically:  

Maximize    Z  = 5x + 3y  

Subject to  

           3x + 5y ≤ 15  

           5x + 2y ≤ 10 

And,    xy ≥ 0.

Q2

 

Solve the following LPP by graphical method: 

Minimize    Z = 20x + 10y  

Subject to   x + 2y ≤ 40

                   3x + y ≥ 30  

                   4x + 3y ≥ 60 

And,           xy ≥ 0

Q3

 

Solve the following LPP graphically:

Minimize and Maximize Z = 5x + 2y  

Subject to –2x – 3y ≤ – 6  

                     x – 2y ≤ 2

                    3x + 2y ≤ 12  

                  –3x + 2y ≤ 3 

                     xy ≥ 0

Q4

 

Solve the following LPP graphically:

Maximize and Minimize   Z = 3x + 5y  

Subject to   3x – 4y + 12 ≥ 0

                       2x – y + 2 ≥ 0 

                   2x + 3y – 12 ≥ 0 

                               0 ≤ x ≤ 4 

                                      y ≥ 2.

Q5

 

Solve the following linear programming problem graphically:

Maximize  Z = 50x + 15y  

Subject to

            5x + y ≤ 100

             x + y ≤ 60

             xy ≥ 0.

Q6

 

Solve the following LPP graphically:  

Maximize   Z = 5x + 7y  

Subject to

              x + y ≤ 4

             3x + 8y ≤ 24  

            10x + 7y ≤ 35

            xy ≥ 0  

Q7

Solve the following LPP graphically:  

Minimize Z = 3x + 5y     

Subject to  

         – 2x + y ≤ 4  

            x + y ≥ 3

           x – 2y ≤ 2   

           xy ≥ 0

Q8

 

A house wife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A,12 units of vitamin B and 8 units of vitamin C.

The vitamin contents of one kg of food is given below:   

 

Vitamin A

Vitamin B

Vitamin C

Food X:

1

2

3

Food Y:

2

2

1

One kg of food X costs Rs 6 and one kg of food Y costs Rs 10. Find the least cost of the mixture which will produce the diet.

Q9

A dietician wishes to mix two types of food in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 10 units of vitamin C. Food ‘I’ contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while food ‘II’ contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 5.00 per kg to purchase food ‘I’ and Rs 7.00 per kg to produce food ‘II’. Determine the minimum cost to such a mixture. formulate the above as a LPP and solve it. 

Q10

A manufacturer produces nuts and bolts for industrial machinery. It takes 1 hour or work on machine A and 3 hours on machine B to produce a package of nuts while it takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs 2.50 per package of nuts and Re 1.00 per package of bolts. How many packages or each should he produce each day so as to maximize hit profit, if he operates his machines for at most 12 hours a day? Formulate this mathematically and then solve it.