Final answer
Final answer: The linear programming problem determines that 40% of Alloy 3 and 60% of Alloy 5 should be combined to produce the desired alloy at a cost of $2.1 per pound.
Explanation:
To achieve an alloy composition of 30% metal A and 70% metal B, the manufacturer can use a linear programming approach to minimize the cost. This problem involves finding the optimal combination of available alloys while meeting the composition requirements. The objective is to minimize the cost while fulfilling the constraints of the desired alloy composition.
The formulation involves setting up equations based on the percentages of metals A and B in each alloy and their respective prices per pound. By assigning variables for the amounts of alloys to be used, the problem translates into finding the optimal values for these variables. The constraints include the required percentages of metals A and B, leading to a system of equations.
Through linear programming techniques like the simplex method or graphical methods, the calculations determine that 40% of Alloy 3 and 60% of Alloy 5 should be combined. This combination satisfies the composition criteria at the least cost of $2.1 per pound.
By adhering to these proportions, the manufacturer achieves the desired alloy blend economically. This optimal solution minimizes expenses while meeting the specified alloy composition, making it the most cost-effective choice among the available options.