Problem 75 - Precalculus

Question

A manufacturer of skis produces two models: a regular ski and a slalom ski. A set of regular skis produces a \(\$ 25.00\) profit and a set of slalom skis produces a profit of \(\$ 50.00\) The manufacturer expects a customer demand of at least 200 pairs of regular skis and at least 80 pair of slalom skis. The maximum number of pairs of skis that can be produced by this company is 400. How many of each model of skis should be produced to maximize profits?

Step-by-Step Solution

Answer

Produce 200 regular skis and 200 slalom skis for maximum profit of $15,000.

Step 1: Define the Variables

Let \( x \) represent the number of regular skis produced and \( y \) represent the number of slalom skis produced.

Step 2: Set Up the Objective Function

The objective function, which represents the total profit \( P \), is given by:\[ P = 25x + 50y \] where 25 is the profit per regular ski and 50 is the profit per slalom ski.

Step 3: Write the Constraints

The problem gives several constraints:1. Customer demand: - At least 200 pairs of regular skis: \( x \geq 200 \) - At least 80 pairs of slalom skis: \( y \geq 80 \) 2. Production capacity: - Total skis must not exceed 400 pairs: \( x + y \leq 400 \)

Step 4: Identify the Feasible Region

The feasible region is determined by the intersection of planes defined by the constraints. We have to look for values of \( x \) and \( y \) that satisfy:- \( 200 \leq x \leq 320 \)- \( 80 \leq y \leq 200 \)- \( x + y \leq 400 \)

Step 5: Evaluate Profit at Corner Points

Vertex (corner) points of the feasible region are:1. (200, 200)2. (200, 80)3. (320, 80)Calculate profit at each corner point:- At (200, 200): \( P = 25(200) + 50(200) = 15000 \)- At (200, 80): \( P = 25(200) + 50(80) = 9000 \)- At (320, 80): \( P = 25(320) + 50(80) = 13000 \)

Step 6: Identify Maximum Profit

Comparing profits at each vertex, the maximum profit of \( 15000 \) occurs at \( (200, 200) \).

Key Concepts

Objective Function

In linear programming, setting up the objective function is crucial. It's a mathematical expression that defines what you're trying to achieve. In the context of the ski manufacturing problem, the objective function represents the total profit. For our scenario, it is expressed as the equation \( P = 25x + 50y \). Here,

• \( x \) represents the number of regular skis produced.
• \( y \) is the number of slalom skis produced.
• *P* stands for the total profit.

To maximize profits, you need to find values for \( x \) and \( y \) that maximize this equation, considering the production constraints.
The coefficients 25 and 50 are the profit contributions per unit, showing how each type of ski affects overall profitability.

Constraints

Constraints in linear programming are the conditions that must be met. They play a vital role in determining the possible solutions. In our ski exercise, we have specific constraints based on demand and production capacity:

• **Demand Constraints**
  • At least 200 pairs of regular skis: \( x \geq 200 \)
  • At least 80 pairs of slalom skis: \( y \geq 80 \)


• **Production Constraint**
  • The total number of skis produced cannot exceed 400: \( x + y \leq 400 \)

These constraints form the boundaries within which we can operate. They ensure that production levels meet minimum customer demands while not overextending the manufacturing capacity.

Feasible Region

The feasible region is the set of all possible solutions that simultaneously satisfy all constraints. It is typically represented as a shaded area on a graph. In the ski problem, the feasible region is determined by intersecting the planes defined by our constraints:

• \( 200 \leq x \leq 320 \)
• \( 80 \leq y \leq 200 \)
• \( x + y \leq 400 \)

Any combination of \( x \) and \( y \) within this region represents a valid production plan. The boundaries of this region are crucial because the optimal solution is typically found at one of these boundaries or vertices, where all conditions are just met but not exceeded.

Corner Points

In linear programming, the most crucial points to examine within the feasible region are the corner points or vertices. They are where the boundary lines of the constraints intersect. For our ski production problem, the corner points are:

• (200, 200)
• (200, 80)
• (320, 80)

Each corner point is evaluated to find the potential maximum profit. By calculating the profit at these points:

• At (200, 200), the profit is \( P = 15000 \).
• At (200, 80), the profit is \( P = 9000 \).
• At (320, 80), the profit is \( P = 13000 \).

From this, we see that the maximum profit occurs at the point (200, 200). This means that to achieve the highest profit, producing and selling 200 regular skis and 200 slalom skis is the optimal strategy.