(A) Let $x$ and $y$ be the number of packets of food $P$ and $Q$ respectively. The objective is to minimise $Z = 6x + 3y$ (vitamin $A$).Constraints ar

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$15$ packets of $P$,$20$ packets of $Q$; Minimum vitamin $A = 150$ units

(A) Let $x$ and $y$ be the number of packets of food $P$ and $Q$ respectively. The objective is to minimise $Z = 6x + 3y$ (vitamin $A$).Constraints are:$12x + 3y \geq 240 \implies 4x + y \geq 80$$4x + 20y \geq 460 \implies x + 5y \geq 115$$6x + 4y \leq 300 \implies 3x + 2y \leq 150$$x, y \geq 0$The feasible region is bounded by the corner points $L(2, 72)$,$M(15, 20)$,and $N(40, 15)$.Corner Point$Z = 6x + 3y$$L(2, 72)$$6(2) + 3(72) = 12 + 216 = 228$$M(15, 20)$$6(15) + 3(20) = 90 + 60 = 150$$N(40, 15)$$6(40) + 3(15) = 240 + 45 = 285$The minimum value of $Z$ is $150$ at point $(15, 20)$. Thus,$15$ packets of food $P$ and $20$ packets of food $Q$ should be used.

Class 12 Mathematics Linear Programming Word problem of Linear programming

Difficult

$A$ 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 minimise the amount of vitamin $A$ in the diet? What is the minimum amount of vitamin $A$?

A
$15$ packets of $P$,$20$ packets of $Q$; Minimum vitamin $A = 150$ units
B
$20$ packets of $P$,$15$ packets of $Q$; Minimum vitamin $A = 165$ units
C
$10$ packets of $P$,$25$ packets of $Q$; Minimum vitamin $A = 135$ units
D
$25$ packets of $P$,$10$ packets of $Q$; Minimum vitamin $A = 180$ units

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Linear Programming

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Word problem of Linear programming

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The difference between the maximum value and minimum value of the objective function $z = 3x + 5y$ of a linear programming problem subject to constraints $5x + 10y \leq 50$, $x + y \geq 1$, $y \leq 4$, $x \geq 0$, $y \geq 0$ is $3\lambda$. Then the value of $\lambda$ is

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