$A$ manufacturing company makes two models $A$ and $B$ of a product. Each piece of Model $A$ requires $9$ labour hours for fabricating and $1$ labour hour for finishing. Each piece of Model $B$ requires $12$ labour hours for fabricating and $3$ labour hours for finishing. For fabricating and finishing,the maximum labour hours available are $180$ and $30$ respectively. The company makes a profit of Rs $8000$ on each piece of model $A$ and Rs $12000$ on each piece of Model $B$. How many pieces of Model $A$ and Model $B$ should be manufactured per week to realise a maximum profit? What is the maximum profit per week?
- A$12$ pieces of $A$,$6$ pieces of $B$; Max Profit = Rs $1,68,000$
- B$6$ pieces of $A$,$12$ pieces of $B$; Max Profit = Rs $1,68,000$
- C$10$ pieces of $A$,$10$ pieces of $B$; Max Profit = Rs $2,00,000$
- D$20$ pieces of $A$,$0$ pieces of $B$; Max Profit = Rs $1,60,000$
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$A$ dietician wishes to mix together two kinds of food $X$ and $Y$ in such a way that the mixture contains at least $10$ units of vitamin $A$,$12$ units of vitamin $B$,and $8$ units of vitamin $C$. The vitamin contents of one $kg$ of food are given below:
Food Vitamin $A$ Vitamin $B$ Vitamin $C$ $X$ $1$ $2$ $3$ $Y$ $2$ $2$ $1$
One $kg$ of food $X$ costs $Rs. 16$ and one $kg$ of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?
| Food | Vitamin $A$ | Vitamin $B$ | Vitamin $C$ |
| $X$ | $1$ | $2$ | $3$ |
| $Y$ | $2$ | $2$ | $1$ |
One $kg$ of food $X$ costs $Rs. 16$ and one $kg$ of food $Y$ costs $Rs. 20$. Find the least cost of the mixture which will produce the required diet?
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View SolutionThe shaded area in the figure below is the solution set for a certain linear programming problem. Then the linear constraints are given by:
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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.
If the grower wants to maximize the amount of nitrogen added to the garden,how many bags of each brand should be added? What is the maximum amount of nitrogen added?
Brand $P$ ($kg$ per bag) Brand $Q$ ($kg$ per bag) Nitrogen $3$ $3.5$ Phosphoric acid $1$ $2$ Potash $3$ $1.5$ Chlorine $1.5$ $2$
If the grower wants to maximize the amount of nitrogen added to the garden,how many bags of each brand should be added? What is the maximum amount of nitrogen added?
| Brand $P$ ($kg$ per bag) | Brand $Q$ ($kg$ per bag) | |
| Nitrogen | $3$ | $3.5$ |
| Phosphoric acid | $1$ | $2$ |
| Potash | $3$ | $1.5$ |
| Chlorine | $1.5$ | $2$ |
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