Given that and , find the smallest possible value of .
Two packages of food supplies are available; the first gives 4 calories and has 6 units of vitamin C, and the second gives 3 calories and has 4 units of vitamin C. We need at least 37 calories and 22 units of vitamin C. The first costs 6 LE per package, and the second costs 8 LE per package. Using to represent the amount of packages of the first type and to represent the number of packages of the second type, state the objective function used to determine the minimum cost of buying packages to fulfil the nutrient needs.
In a workshop, two workers produce two types of iron desks: type A and type B. One worker builds the desks and the other sprays them. It takes the first worker 4 hours to build one desk of type A and 3 hours to build one desk of type B. It takes the second worker 3 hours to spray one desk of type A and 4 hours to spray one desk of type B. The first person works at least 5 hours a day, and the other works a maximum of 7 hours a day. If the workshop earns a profit of 60 LE from each desk (of either type), determine the objective function and inequalities required for calculating the number of desks of each type to be produced every day to maximise the profit .