vitamin | Litre of milk | Kg of beaf | Doze of eggs | Minimum daily requirements |
A | 1 | 1 | 10 | 1 mg |
B | 100 | 10 | 10 | 50 mg |
C | 10 | 100 | 10 | 10 mg |
Cost | Rs. 1.00 | Rs. 1.10 | Rs. 0.50 |
Answer:
Let the daily diet consists of x litres of milk, y kgs of beaf and z dozens of eggs. Then, Total cost per day \[=\text{ }Rs.\text{ (}x+1.10y+0.50z\text{)}\] Let Z denotes the total cost in Rs. Then, \[Z=x+1.10y+0.50z\] Total amount of vitamin A in the daily diet is \[(x+y+10z)mg.\] But the minimum requirement is 1 mg of vitamin A. \[\therefore \] \[x+y+10z\ge 1\] Similarly, total amounts of vitamins B and C in the daily diet are \[(100x+10y+10z)\] mg and \[(10x+100y+10z)\] mg respectively and their minimum requirements are of 50 mg and 10 mg respectively \[\therefore \] \[100x+10y+10z\ge 50\] and \[10x+100y+10z\ge 10\] Finally, the quantity of milk, kgs of beaf and dozens of eggs cannot assume negative values. \[\therefore \] \[x\ge 0,\] \[y\ge 0,\] \[z\ge 0\] Hence, the mathematical formulation of the given LPP is Minimize \[Z=x+1.10y+0.50z\] Subject to constraints \[x+y+10z\ge 1\] \[100x+10y+10z\ge 50\] \[10x+100y+10z\ge 10\] and \[x\ge 0,\] \[y\ge 0,\] \[z\ge 0\]
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