$A$ diet of a sick person must contain at least $4000$ units of vitamins,$50$ units of proteins,and $1400$ calories. Two foods $A$ and $B$ are available at a cost of ₹ $4$ and ₹ $3$ per unit respectively. If one unit of $A$ contains $200$ units of vitamins,$1$ unit of protein,and $40$ calories,while one unit of food $B$ contains $100$ units of vitamins,$2$ units of protein,and $40$ calories,formulate the problem so that the diet is the cheapest.
- A$200x + 100y \geq 4000, x + 2y \geq 50, 40x + 40y \geq 1400, x \geq 0, y \geq 0, \text{Minimize } z = 4x + 3y$
- B$400x + 200y \geq 100, x + 2y \geq 50, 40x + 40y \geq 1400, x \geq 0, y \geq 0, \text{Minimize } z = 4x + 3y$
- C$100x + 200y \geq 4000, x + 2y \geq 50, 40x + 40y \geq 1400, x \geq 0, y \geq 0, \text{Minimize } z = 4x + 3y$
- DNone of the above
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$A$ manufacturer has three machines $I, II$ and $III$ installed in his factory. Machines $I$ and $II$ are capable of being operated for at most $12 \, hours$ whereas machine $III$ must be operated for at least $5 \, hours$ a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines. The number of hours required for producing $1$ unit of each of $M$ and $N$ on the three machines are given in the following table:
Items Machine $I$ Machine $II$ Machine $III$ $M$ $1$ $2$ $1$ $N$ $2$ $1$ $1.25$
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| Items | Machine $I$ | Machine $II$ | Machine $III$ |
|---|---|---|---|
| $M$ | $1$ | $2$ | $1$ |
| $N$ | $2$ | $1$ | $1.25$ |
She makes a profit of $Rs. \, 600$ and $Rs. \, 400$ on items $M$ and $N$ respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?
Difficult
View SolutionThere 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:
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?
| 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?
Difficult
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