Final answer:
Charles needs to use linear programming to minimize the cost of making benches and tables for Laura's birthday party, while considering time constraints and minimum production requirements. He will create a graph based on these constraints to find the combination of benches and tables that minimizes the cost.
Step-by-step explanation:
The student is asking about finding the combination of wooden benches (x) and tables (y) that minimizes cost, given Charles's time, cost constraints, and minimum production requirements for Laura's birthday party. We need to use linear programming to determine the combination that will meet all the requirements while minimizing the cost. Charles can work a maximum of 60 hours per week, and he wants to make at least 4 tables and at least 8 benches.
To minimize costs, Charles needs to find the least number of tables (y) and benches (x) that will satisfy all the constraints. Here are some steps to proceed:
- Since each bench takes 4 hours to make, and each table takes 2 hours, we can define the time constraint as 4x + 2y ≤ 60.
- We add the constraints that Charles wants at least 4 tables (y ≥ 4) and at least 8 benches (x ≥ 8).
- We calculate the total cost with the cost constraint C = 25x + 40y.
- Charles needs to minimize C, so he can have more money for a birthday present. We can graph the constraints and find the feasible region. The point(s) on the graph where the cost C is minimized, within the feasible region and satisfying the constraints, will give Charles the optimal number of benches and tables to produce.
Cost minimization and linear programming would be the primary mathematical principles in solving this problem.