$A$ farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$,costing $Rs. 250$ per bag,contains $3$ units of nutritional element $A$,$2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing $Rs. 200$ per bag contains $1.5$ units of nutritional element $A$,$11.25$ units of element $B$,and $3$ units of element $C$. The minimum requirements of nutrients $A$,$B$ and $C$ are $18$ units,$45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?
- A$3$ bags of $P$ and $6$ bags of $Q$,Minimum cost $= Rs. 1950$
- B$6$ bags of $P$ and $3$ bags of $Q$,Minimum cost $= Rs. 2100$
- C$4$ bags of $P$ and $5$ bags of $Q$,Minimum cost $= Rs. 2000$
- D$2$ bags of $P$ and $7$ bags of $Q$,Minimum cost $= Rs. 1900$
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There are two factories located at place $P$ and place $Q$. From these locations,a certain commodity is to be delivered to each of the three depots situated at $A, B$ and $C$. The weekly requirements of the depots are $5, 5$ and $4$ units respectively,while the production capacities of the factories at $P$ and $Q$ are $8$ and $6$ units respectively. The cost of transportation per unit is given below:
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How many units should be transported from each factory to each depot in order that the transportation cost is minimum? What will be the minimum transportation cost?
| From/To | $A$ | $B$ | $C$ |
| $P$ | $160$ | $100$ | $150$ |
| $Q$ | $100$ | $120$ | $100$ |
How many units should be transported from each factory to each depot in order that the transportation cost is minimum? What will be the minimum transportation cost?
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View Solution$A$ factory makes tennis rackets and cricket bats. $A$ tennis racket takes $1.5\, \text{hours}$ of machine time and $3\, \text{hours}$ of craftsman's time in its making, while a cricket bat takes $3\, \text{hours}$ of machine time and $1\, \text{hour}$ of craftsman's time. In a day, the factory has the availability of not more than $42\, \text{hours}$ of machine time and $24\, \text{hours}$ of craftsman's time. What number of rackets and bats must be made if the factory is to work at full capacity?
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View SolutionThe shaded area in the figure below is the solution set for a certain linear programming problem. Then the linear constraints are given by:
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