The maximum value of $Z = 10x + 25y$ subject to $0 \leq x \leq 3$,$0 \leq y \leq 3$,and $x + y \leq 5$ occurs at the point:

The maximum value of $z = 5x + 3y$ subject to constraints $3x + 5y \leq 15, x \geq 0, y \geq 0$ is :

$A$ manufacturer has three machines $I, II$ and $III$ installed in his factory. Machines $I$ and $II$ are capable of being operated for at most $12 \, hours$ whereas machine $III$ must be operated for at least $5 \, hours$ a day. She produces only two items $M$ and $N$ each requiring the use of all the three machines. The number of hours required for producing $1$ unit of each of $M$ and $N$ on the three machines are given in the following table:

Items	Machine $I$	Machine $II$	Machine $III$
$M$	$1$	$2$	$1$
$N$	$2$	$1$	$1.25$

She makes a profit of $Rs. \, 600$ and $Rs. \, 400$ on items $M$ and $N$ respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

$A$ scholarship amount is given by $z = 550x + 300y$ and is to be distributed among $x$ boys and $y$ girls. From the graph given below,the maximum amount of scholarship is . . . . . . .

Reshma wishes to mix two types of food $P$ and $Q$ in such a way that the vitamin contents of the mixture contain at least $8$ $units$ of vitamin $A$ and $11$ $units$ of vitamin $B$. Food $P$ costs Rs $60/kg$ and Food $Q$ costs Rs $80/kg$. Food $P$ contains $3$ $units/kg$ of Vitamin $A$ and $5$ $units/kg$ of Vitamin $B$ while food $Q$ contains $4$ $units/kg$ of Vitamin $A$ and $2$ $units/kg$ of vitamin $B$. Determine the minimum cost of the mixture. Let the mixture contain $x$ kg of food $P$ and $y$ kg of food $Q$. Therefore,$x \geq 0$ and $y \geq 0$. The given information can be compiled in a table as follows:

The minimum value of $Z = 5x + 8y$ subject to the constraints $x + y \geq 5$,$0 \leq x \leq 4$,$y \geq 2$ is: