$A$ 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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$A$ fruit grower can use two types of fertilizer in his garden,brand $P$ and brand $Q$. The amounts (in $kg$) of nitrogen,phosphoric acid,potash,and chlorine in a bag of each brand are given in the table. Tests indicate that the garden needs at least $240 \, kg$ of phosphoric acid,at least $270 \, kg$ of potash,and at most $310 \, kg$ of chlorine.
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| Nitrogen | $3$ | $3.5$ |
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| Potash | $3$ | $1.5$ |
| Chlorine | $1.5$ | $2$ |
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