The maximum value of $Z=3x+5y$,subject to the constraints $x+4y \leq 24$,$y \leq 4$,$x \geq 0$,$y \geq 0$ is:

$A$ 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 $A$?

If $Z = 7x + y$ subject to $5x + y \geq 5$,$x + y \geq 3$,$x \geq 0$,$y \geq 0$,then the minimum value of $Z$ is

The objective function $z = 4x + 5y$ subject to the constraints $2x + y \geq 7$,$2x + 3y \leq 15$,$y \leq 3$,$x \geq 0$,and $y \geq 0$ has a minimum value at which point?

$A$ manufacturer produces nuts and bolts. It takes $1\, hour$ of work on machine $A$ and $3\, hours$ on machine $B$ to produce a package of nuts. It takes $3\, hours$ on machine $A$ and $1\, hour$ on machine $B$ to produce a package of bolts. He earns a profit of $Rs.\,17.50$ per package on nuts and $Rs.\,7$ per package on bolts. How many packages of each should be produced each day so as to maximise his profit,if he operates his machines for at the most $12\, hours$ a day?

The difference between the maximum and minimum values of the objective function $Z = 3x + 5y$,subject to the constraints $x + 3y \leqslant 60$,$x + y \geqslant 10$,$x - y \leqslant 0$,$x \geqslant 0$,$y \geqslant 0$ is