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CBSE Class 12-science Answered

A dealer wishes to purchase a number of fans and sewing machines.  He has only Rs. 5760.00  to invest and has space for at most 20 items. A fan costs him Rs.360.00 and a sewing machine Rs. 240.00.  His expectation is that he can sell a fan at a profit of Rs. 22.00 and sewing machine at a profit of Rs. 80.00. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize his profit? Translate this problem mathematically and then solve it.
Asked by Topperlearning User | 18 Sep, 2014, 03:34: PM
answered-by-expert Expert Answer

Suppose the dealer buys x fans and y sewing machines. Since the dealer has space for at most 20 items.  Therefore,

                  

          A fan costs Rs. 360 and a sewing machine costs Rs. 240.  Therefore, total cost of x fans and y sewing machine is Rs. (360 x  + 240y).  But the dealer has only Rs.5760 to invest.

          Therefore,

                  

          Since the dealer can sell all the items that he can buy and the profit on a fan is Rs.22 and on a sewing machine the profit is of Rs. 18.  Therefore, total profit on selling x fans and y sewing is of Rs. (22 x + 18 y).

          Let Z denote the total profit.  Then, Z = 22x + 18y.

          Clearly, .

          Thus, the mathematical formulation of the given problem is

                   Maximize

          Subject        to

                  

                  

          and

          To solve this LPP graphically, we first convert the inequations into equations and draw the corresponding lines. The feasible region of the LPP is shaded in Figure. The corner points of the feasible region OA2 PB1 are (0, 0), A2 (16, 0), P(8, 12) and B1 (0, 20).

 

 

          These points have been obtained by solving the corresponding intersecting lines, simultaneously.

          The values of the objective function Z at corner – points of the feasible region are given in the following table.

         

Point (x, y)

Value of the objective function

Z = 22 x+ 18y

O(0 0)

A2 (16, 0)

P(8, 12)

B1 (0, 20)

 

          Clearly, Z is maximum at x = 8 and y = 12.  The maximum value of Z is 392.

          Hence, the dealer should purchase 8 fans and 12 sewing machines to obtain the maximum  profit under given conditions.

Answered by | 18 Sep, 2014, 05:34: PM

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