Final answer:
The manufacturer can determine the optimal number of downhill and cross-country skis to produce for maximum weekly profit by using linear programming, setting up a set of linear inequalities based on manufacturing and finishing time constraints, and optimizing a profit function.
Step-by-step explanation:
The manufacturer needs to determine the number of each type of ski to produce to maximize weekly profit given the available manufacturing and finishing hours, along with the profit for each ski type. To solve this, one should use linear programming: a method to achieve the best outcome (such as maximum profit) in a mathematical model whose requirements are represented by linear relationships.
Let x be the number of downhill skis and y be the number of cross-country skis produced in a week. The manufacturing time constraints can be described by the inequalities 5x + 2y ≤ 18 (manufacturing hours) and 4x + y ≤ 12 (finishing hours). Additionally, the number of skis cannot be negative, so x ≥ 0 and y ≥ 0.
The objective is to maximize the profit function P = 51x + 75y. A feasible set of solutions satisfying all constraints needs to be identified, and then the objective function is evaluated at each vertex of this feasible set to find the optimal solution.
To solve the set of linear inequalities graphically, plot the constraints on a coordinate plane and find the feasible region. Then, identify the vertex or vertices where the objective function attains its maximum value. This point or points will give the number of each type of ski to produce to maximize the weekly profit.