To determine the optimal number of standard-mix and deluxe-mix packages to maximize revenue, we need to formulate and solve a linear programming problem. Let's summarize and define the problem in a structured manner:
### Objective Function:
Our goal is to maximize the revenue [tex]\( R \)[/tex] given by:
[tex]\[ R(x, y) = 1.95x + 2.25y \][/tex]
where:
- [tex]\( x \)[/tex] is the number of standard-mix packages,
- [tex]\( y \)[/tex] is the number of deluxe-mix packages.
### Constraints:
1. Cashews Constraint: There are 15,000 grams of cashews available. Each standard-mix uses 100 grams and each deluxe-mix uses 150 grams:
[tex]\[ 100x + 150y \leq 15000 \][/tex]
Simplifying:
[tex]\[ 2x + 3y \leq 300 \][/tex]
2. Peanuts Constraint: There are 20,000 grams of peanuts available. Each standard-mix uses 200 grams and each deluxe-mix uses 50 grams:
[tex]\[ 200x + 50y \leq 20000 \][/tex]
Simplifying:
[tex]\[ 4x + y \leq 400 \][/tex]
3. Standard-Deluxe Ratio Constraint: The number of standard-mix packages must be at least as great as the number of deluxe-mix packages:
[tex]\[ x \geq y \][/tex]
Or equivalently:
[tex]\[ y \leq x \][/tex]
4. Non-Negativity Constraints: The number of packages cannot be negative:
[tex]\[ x \geq 0 \][/tex]
[tex]\[ y \geq 0 \][/tex]
### Solution:
Given the constraints and the objective function, the problem can be solved using linear programming methods. The results are:
- Number of standard-mix packages ([tex]\( x \)[/tex]): 90
- Number of deluxe-mix packages ([tex]\( y \)[/tex]): 40
The maximum revenue achieved by this combination is:
[tex]\[ \text{Maximum Revenue} = 1.95 \cdot 90 + 2.25 \cdot 40 \][/tex]
[tex]\[ \text{Maximum Revenue} = 175.5 + 90 \][/tex]
[tex]\[ \text{Maximum Revenue} = 265.5 \][/tex]
### Conclusion:
The confectioner should package:
- 90 standard-mix packages
- 40 deluxe-mix packages
This combination will maximize the revenue, resulting in a total revenue of \$265.50.
