The maximum value of $z=50x+15y$ subject to the constraints $x+y \leq 60$,$5x+y \leq 100$,$x \geq 0$,$y \geq 0$ is at the point:

$A$ 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?

The shaded area in the given figure is a solution set for some system of inequations. The maximum value of the function $z=10x+25y$ subject to the linear constraints given by the system is

One kind of cake requires $200 \,g$ of flour and $25 \,g$ of fat,and another kind of cake requires $100 \,g$ of flour and $50 \,g$ of fat. Find the maximum number of cakes which can be made from $5 \,kg$ of flour and $1 \,kg$ of fat,assuming that there is no shortage of the other ingredients used in making the cakes.

In the following figure, the shaded region represents the system of constraints:

$A$ farmer mixes two brands $P$ and $Q$ of cattle feed. Brand $P$,costing $Rs. 250$ per bag,contains $3$ units of nutritional element $A$,$2.5$ units of element $B$ and $2$ units of element $C$. Brand $Q$ costing $Rs. 200$ per bag contains $1.5$ units of nutritional element $A$,$11.25$ units of element $B$,and $3$ units of element $C$. The minimum requirements of nutrients $A$,$B$ and $C$ are $18$ units,$45$ units and $24$ units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?