The problem involves optimizing the production of two types of coffee blends, each using fixed amounts of A grade and B grade coffee, subject to the total available supply of these two ingredients. To maximize profit, we will set up a linear programming problem.

Given:

• Total amount of A grade coffee: 1728 ounces
• Total amount of B grade coffee: 1760 ounces

Coffee Blends:

• Economy blend:

  • Uses 2 ounces of A grade coffee and 10 ounces of B grade coffee per package.
  • Profit per economy package = $3.

• Superior blend:

  • Uses 9 ounces of A grade coffee and 2 ounces of B grade coffee per package.
  • Profit per superior package = $4.

Variables:

Let:

• $x_1$ = number of economy blend packages.
• $x_2$ = number of superior blend packages.

Constraints:

1. Total amount of A grade coffee used: $2x_1 + 9x_2 \leq 1728$

2. Total amount of B grade coffee used: $10x_1 + 2x_2 \leq 1760$

3. Non-negativity constraint: $x_1 \geq 0, \quad x_2 \geq 0$

Objective function (Profit):

The profit is given by: $P = 3x_1 + 4x_2$ We want to maximize this profit function subject to the constraints above.

Next Steps:

1. Solve the linear programming problem using either graphical methods or a solver to find the values of $x_1$ and $x_2$ that maximize the profit.

Would you like me to proceed with solving this system to find the optimal number of packages for each blend?

Follow-up questions:

1. How do we graphically solve linear programming problems with two variables?
2. What methods can be used to solve linear programming problems?
3. How can slack variables be used in constraints for optimization problems?
4. How would the problem change if profits or coffee amounts were different?
5. What would happen if the constraints had no upper bounds?

Tip:

Linear programming problems can be solved efficiently using algorithms like the Simplex Method or software tools like Excel Solver or Python's SciPy module.