Variables and Formulas

Product $29$: Thuringer Rostbratwurst, costing $$58$ each.

Product $59$: Raclette Courdavault, costing $$30$ each.

Product $38$: Cote de Blaye, costing $$100$ each.

Total budget: $$250,000.$

Minimum stock for Product $59$: $1000$ units.

Maximum stock for Product $38$: $2000$ units.

Concepts

Linear Programming: A mathematical optimization technique used to find the best possible solution for a problem with multiple constraints.

Objective Function: A mathematical expression that represents the goal of the problem, in this case, the total profit.

Constraints: Restrictions on the variables, such as the budget and stock requirements.

Main Body

Let's define the variables and set up the linear programming problem:

Step $1$: Define the Variables

Let x$1$ be the number of units of Product $29,$ x$2$ be the number of units of Product $59,$ and x$3$ be the number of units of Product $38.$

Step $2$: Formulate the Objective Function

The objective is to maximize the total profit, which is given by:

Profit $=58$x$1+30$x$2+100$x$3$

Step $3$: Define the Constraints

Budget constraint: $58$x$1+30$x$2+100$x$3<=250,000$

Stock constraint for Product $59$: x$2>=1000$

Stock constraint for Product $38$: x$3<=2000$

Step $4$: Solve the Linear Programming Problem

To find the optimal solution, we can use a linear programming solver or software. The solution will provide the optimal values for x$1,$ x$2,$ and x$3,$ which represent the number of units for each product.

Final Answer

To ensure maximum profit, the optimal quantity for each product is determined by solving the linear programming problem. The solution will provide the number of units for each product $($x$1,$ x$2,$ and x$3).$ The potential revenue amount can be calculated by substituting the optimal values into the objective function: Profit $=58$x$1+30$x$2+100$x$3.$

Please note that the actual values will depend on the specific linear programming solver or software used.

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