Q14 of 97 Page 1376

A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of ₹0.25 and ₹0.50 each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?


Let x and y be number of soaps be manufactured of 1st and 2nd type.


∴According to the question,


2x + 3y 480 , 3x + 5y 480, x , y


Maximize Z = 0.25x + 0.50y


The feasible region determined by 2x + 3y 480 , 3x + 5y 480, x , y is given by



The corner points of feasible region are A(0,96) , B(0,0) , C(160,0).


The value of Z at corner points are



The maximum value of Z is 48 at point (0,96).


Hence, the manufacturer should make 96 soaps of the 2nd type to make maximum profit.


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13

A firm manufactures two types of products, A and B, and sells them at a profit of ₹2 on type A and ₹2 on type B. Each product is processed on two machines, M1 and M2. Type A requires one minute of processing time on M1 and two minutes on M2. Type B requires one minute on M1 and one minute on M2 is available for not more than 6 hours 40 minutes while M2 is available for at most 10 hours a day.

Find how many products of each type the firm should produce each day in order to get maximum profit.


 14

A manufactures produces two types of soap bars using two machines, A and B. A is operated for 2 minutes and B for 3 minutes to manufacture the first type, while it takes 3 minutes on machine A and 5 minutes on machine B to manufacture the second type. Each machine can be operated at the most for 8 hours per day. The two types of soap bars are sold at a profit of ₹0.25 and ₹0.50 each. Assuming that the manufacture can sell all the soap bars he can manufacture, how many bars of soap of each type should be manufactured per day so as to maximize his profit?

 15

A manufacture of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients 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. Furthermore, it takes 3 hours to prepare enough material to fill 1000 bottles of A and it takes 1 hour to prepare enough material to fill 1000 bottles of B, and there are 66 hours available for this operation. The profit is ₹8 per bottle for A and ₹7 per bottle for B.

How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.


 15

A manufacture of a line of patent medicines is preparing a production plan on medicines A and B. There are sufficient ingredients 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. Furthermore, it takes 3 hours to prepare enough material to fill 1000 bottles of A and it takes 1 hour to prepare enough material to fill 1000 bottles of B, and there are 66 hours available for this operation. The profit is ₹8 per bottle for A and ₹7 per bottle for B.

How should the manufacture schedule the production in order to maximize his profit? Also, find the maximum profit.