$A$ dietician has to develop a special diet using two foods $P$ and $Q$. Each packet (containing $30 \, g$) of food $P$ contains $12$ units of calcium,$4$ units of iron,$6$ units of cholesterol,and $6$ units of vitamin $A$. Each packet of the same quantity of food $Q$ contains $3$ units of calcium,$20$ units of iron,$4$ units of cholesterol,and $3$ units of vitamin $A$. The diet requires at least $240$ units of calcium,at least $460$ units of iron,and at most $300$ units of cholesterol. How many packets of each food should be used to maximize the amount of vitamin $A$ in the diet? What is the maximum amount of vitamin $A$ in the diet?
- A$40$ packets of $P$ and $15$ packets of $Q$; Maximum vitamin $A = 285$ units
- B$15$ packets of $P$ and $40$ packets of $Q$; Maximum vitamin $A = 210$ units
- C$20$ packets of $P$ and $40$ packets of $Q$; Maximum vitamin $A = 240$ units
- D$10$ packets of $P$ and $50$ packets of $Q$; Maximum vitamin $A = 210$ units
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The shaded area in the figure below is the solution set for a certain linear programming problem. Then the linear constraints are given by:
$A$ 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 $III$ $M$ $1$ $2$ $1$ $N$ $2$ $1$ $1.25$
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?
| 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?
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View SolutionMinimize the objective function $Z = 3x + 2y$ subject to the constraints: $x + y \geq 8$,$x + y \leq 5$,$x \geq 0$,$y \geq 0$.
The maximum value of $Z = 4x + 5y$, subject to the constraints $3x + y \leq 15$, $3x + 4y \leq 24$, $x \geq 0$, $y \geq 0$ is
DifficultMHT CET 2026
View SolutionThe shaded part of the given figure indicates the feasible region. Then the constraints are
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