The $L$.$P$.$P$. to maximize $z=x+y$,subject to $x+y \leq 30, x \leq 15, y \leq 20, x+y \geq 15$,and $x, y \geq 0$ has

The maximum value of $z$ in the following equation $z=6xy+y^2$,subject to the constraints $3x+4y \leq 100$,$4x+3y \leq 75$,$x \geq 0$,and $y \geq 0$ is:

Solve the following linear programming problem graphically:
Maximise $Z = 4x + y$......$(1)$
subject to the constraints:
${x + y \leqslant 50}$.......$(2)$
${3x + y \leqslant 90}$......$(3)$
${x \geqslant 0, y \geqslant 0}$......$(4)$

For the feasible region $OCDBO$ shown in the figure,the maximum value of the objective function $z = 3x + 4y$ is:

$A$ merchant plans to sell two types of personal computers - a desktop model and a portable model that will cost $Rs.\,25000$ and $Rs.\,40000$ respectively. He estimates that the total monthly demand of computers will not exceed $250$ units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than $Rs.\,70$ lakhs and if his profit on the desktop model is $Rs.\,4500$ and on portable model is $Rs.\,5000$.

Two godowns $A$ and $B$ have grain capacity of $100$ quintals and $50$ quintals respectively. They supply to $3$ ration shops,$D$,$E$ and $F$ whose requirements are $60, 50$ and $40$ quintals respectively. The cost of transportation per quintal from the godowns to the shops are given in the following table:

Transportation cost per quintal (in $Rs$)
From/To $A$ $B$
$D$ $6$ $4$
$E$ $3$ $2$
$F$ $2.50$ $3$

How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost?