Problem 73 - Precalculus

Question

The Eurostar is a high-speed train that travels between London, Brussels, and Paris. There are 30 cars on each departure. Each train car is designated first-class or second-class. Based on demand for each type of fare, there should always be at least two but no more than four first-class train cars. The management wants to claim that the ratio of first-class to second-class cars never exceeds \(1: 8 .\) If the profit on each first-class train car is twice as much as the profit on each second-class train car, find the number of each class of train car that will generate a maximum profit.

Step-by-Step Solution

Answer

3 first-class cars and 27 second-class cars maximize profit.

Step 1: Define Variables

Let \( x \) represent the number of first-class train cars and \( y \) represent the number of second-class train cars. The total number of train cars is 30, so we have the equation: \( x + y = 30 \).

Step 2: Define Constraints

From the problem, the constraints are as follows: \( 2 \leq x \leq 4 \) for the number of first-class cars and \( \frac{x}{y} \leq \frac{1}{8} \) for the ratio constraint, which simplifies to \( x \leq \frac{y}{8} \).

Step 3: Set Up Objective Function

The profit on each first-class car is twice that of each second-class car. Let the profit on each second-class car be \( P \). Then, the profit on each first-class car is \( 2P \). The objective is to maximize the total profit: \( 2Px + Py \).

Step 4: Express Objective Function

Since \( x + y = 30 \), express \( y \) in terms of \( x \): \( y = 30 - x \). Substitute \( y \) into the profit function to get: \( 2Px + P(30 - x) = 2Px + 30P - Px = Px + 30P \). Thus, the function to maximize is \( Px + 30P \) which simplifies to \( xP + 30P \).

Step 5: Simplify Objective Function to Maximize Profit

The term \( 30P \) is a constant, so we maximize \( Px \). Maximize \( \frac{x}{1} \) subject to the constraints \( 2 \leq x \leq 4 \) and \( x \leq \frac{30-x}{8} \).

Step 6: Solve Constraints and Maximize

Evaluate the constraints: \( x \leq \frac{30-x}{8} \) simplifies to \( 8x \leq 30 - x \), leading to \( 9x \leq 30 \), or simply \( x \leq 3.33 \). Since \( x \) must be an integer, \( x = 3 \) is the maximum integer that satisfies all constraints. Substitute back to find \( y = 30 - x = 27 \).

Step 7: Conclusion

The optimal configuration is 3 first-class cars and 27 second-class cars, leading to maximum profit.

Key Concepts

Optimization

Optimization in linear programming involves finding the best possible solution from a set of feasible solutions, given a list of constraints and an objective function that needs to be maximized or minimized. In the case of the Eurostar train problem, the goal is to maximize the profit earned from a specific combination of first-class and second-class train cars. This involves setting up an equation to represent the total profit, which becomes the focus of the optimization process.

Linear programming techniques are highly efficient for solving such problems. The method consists of setting specific boundaries within which a particular value (in this case, profit) can be optimized. By establishing equations and inequalities that represent these limitations, we can use linear programming to determine the most profitable configuration of train cars that fits within the specified constraints.

Constraints

Constraints in linear programming define the limitations or requirements that a solution must adhere to. In the train problem, constraints include the number of each class of train cars and the ratio of first-class to second-class cars.

These constraints can be summarized as follows:

• The total number of train cars (both classes included) must be exactly 30, leading to the equation: \( x + y = 30 \).
• The number of first-class cars must be between 2 and 4, represented as: \( 2 \leq x \leq 4 \).
• The ratio of first-class cars to second-class cars must not exceed 1:8, which can be translated to: \( x \leq \frac{y}{8} \).

Constraints like these help in establishing a feasible region in which the solution can exist. They effectively narrow down the possibilities, enforcing the conditions that any viable solution must meet, ensuring the train remains efficient and profitable.

Objective Function

The objective function in linear programming tells us what we are trying to optimize. For the Eurostar train problem, the objective function is about maximizing the total profit from the operation of both the first-class and second-class train cars.

This can be framed as: \( 2Px + Py \), where:

• \( x \) is the number of first-class cars,
• \( y \) is the number of second-class cars,
• \( P \) is the profit made on each second-class car.

Since each first-class car makes twice the profit of a second-class car, this formula captures the essence of how the business tries to maximize its earnings. By expressing \( y \) as \( 30-x \) (derived from the constraint of total train cars), the profit function becomes \( Px + 30P \). This simplification lets us focus directly on the numbers of first-class cars in maximizing the company's earnings, given the constraints previously discussed.