The maximum value of $z=3x+4y$,subject to the constraints $x+y \leq 40$,$x+2y \leq 60$ and $x, y \geq 0$ is

$A$ man rides his motorcycle at the speed of $50 \, km/h$. He has to spend $Rs. \, 2$ per $km$ on petrol. If he rides it at a faster speed of $80 \, km/h$,the petrol cost increases to $Rs. \, 3$ per $km$. He has at most $Rs. \, 120$ to spend on petrol and one hour's time. He wishes to find the maximum distance that he can travel. Express this problem as a linear programming problem.

The solution set of the constraints $x+2y \leq 2000$,$x+y \leq 1500$,$y \leq 600$ and $x \geq 0$ does not include the point

For the $LPP$,minimize $z = x_{1} + x_{2}$ subject to the constraints $5x_{1} + 10x_{2} \geq 0$,$x_{1} + x_{2} \leq 1$,$x_{2} \leq 4$ and $x_{1}, x_{2} \geq 0$.

$A$ company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type $A$ require $5 \text{ minutes}$ each for cutting and $10 \text{ minutes}$ each for assembling. Souvenirs of type $B$ require $8 \text{ minutes}$ each for cutting and $8 \text{ minutes}$ each for assembling. There are $3 \text{ hours } 20 \text{ minutes}$ available for cutting and $4 \text{ hours}$ for assembling. The profit is $Rs. 5$ each for type $A$ and $Rs. 6$ each for type $B$ souvenirs. How many souvenirs of each type should the company manufacture in order to maximise the profit?

$A$ cottage industry manufactures pedestal lamps and wooden shades,each requiring the use of a grinding/cutting machine and a sprayer. It takes $2 \text{ hours}$ on the grinding/cutting machine and $3 \text{ hours}$ on the sprayer to manufacture a pedestal lamp. It takes $1 \text{ hour}$ on the grinding/cutting machine and $2 \text{ hours}$ on the sprayer to manufacture a shade. On any day,the sprayer is available for at most $20 \text{ hours}$ and the grinding/cutting machine for at most $12 \text{ hours}$. The profit from the sale of a lamp is $Rs. 5$ and that from a shade is $Rs. 3$. Assuming that the manufacturer can sell all the lamps and shades that he produces,how should he schedule his daily production in order to maximise his profit?