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A dietician has to develop a special diet using two foods $\mathbf{P}$ and Q. Each packet (containing $30 \mathrm{~g}$ ) 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 atleast 240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. How many packets of each food should be used to maximise the amount of vitamin $A$ in the diet? What is the maximum amount of vitamin A in the diet?
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Let $x$ and $y$ be the number of packets of food. $P$ and $Q$
Objective function is to maximize $\mathrm{Z}=6 \mathrm{x}+3 \mathrm{y}$, subject to constraints are $12 x+3 y \geq 240$,
$$
4 x+20 y \geq 460,6 x+4 y \leq 300, x, y \geq 0
$$
The objective function is $Z=6 x+3 y$
At $\mathrm{P}(2,72), \quad \mathrm{Z}=12+3 \times 72=228$
At $\mathrm{Q}(15,20), \mathrm{Z}=90+60=150$
At $\mathrm{R}(40,15), Z=240+45=285 \mathrm{max}^{\mathrm{m}}$
40 packets of food $P$ and 15 packets of food $Q$ are required to get maximum amount of vitamin A of 285 units.
Objective function is to maximize $\mathrm{Z}=6 \mathrm{x}+3 \mathrm{y}$, subject to constraints are $12 x+3 y \geq 240$,
$$
4 x+20 y \geq 460,6 x+4 y \leq 300, x, y \geq 0
$$
The objective function is $Z=6 x+3 y$
At $\mathrm{P}(2,72), \quad \mathrm{Z}=12+3 \times 72=228$
At $\mathrm{Q}(15,20), \mathrm{Z}=90+60=150$
At $\mathrm{R}(40,15), Z=240+45=285 \mathrm{max}^{\mathrm{m}}$
40 packets of food $P$ and 15 packets of food $Q$ are required to get maximum amount of vitamin A of 285 units.
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