A Company Has A $$ 150$ Budget To Provide Lunch For Its 20 Employees. The Options Are To Provide Either Roast Beef Sandwiches, Which Cost $$ 5$ Apiece, Or Tuna Sandwiches, Which Also Cost $$ 5$

by ADMIN 203 views

Introduction

In today's fast-paced business world, companies often face difficult decisions when it comes to allocating resources. One such decision is how to manage a limited budget to provide lunch for employees. In this article, we will explore a classic problem in mathematics, where a company has a budget of $150 to provide lunch for its 20 employees. The options are to provide either roast beef sandwiches or tuna sandwiches, both of which cost $5 apiece. We will analyze this problem using mathematical concepts and provide a solution to help the company make an informed decision.

The Problem

A company has a budget of $150 to provide lunch for its 20 employees. The options are to provide either roast beef sandwiches or tuna sandwiches, both of which cost $5 apiece. The company wants to know how many sandwiches it can buy with its budget and which option is more cost-effective.

Mathematical Modeling

Let's denote the number of roast beef sandwiches as x and the number of tuna sandwiches as y. Since the company has a budget of $150, we can set up the following equation:

5x + 5y ≤ 150

This equation represents the constraint that the total cost of the sandwiches cannot exceed the budget. We also know that the company has 20 employees, so the total number of sandwiches must be equal to or less than 20:

x + y ≤ 20

Solving the Problem

To solve this problem, we can use a graphical approach or algebraic methods. Let's use a graphical approach to visualize the problem.

Graphical Approach

We can graph the two constraints on a coordinate plane, where the x-axis represents the number of roast beef sandwiches and the y-axis represents the number of tuna sandwiches.

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(0, 20, 100)
y = 20 - x



plt.plot(x, y, label='x + y ≤ 20')
plt.plot(x, 30 - x, label='5x + 5y ≤ 150')
plt.xlabel('Number of Roast Beef Sandwiches')
plt.ylabel('Number of Tuna Sandwiches')
plt.title('Constraints on the Number of Sandwiches')
plt.legend()
plt.show()

The graph shows the two constraints as lines on the coordinate plane. The shaded region represents the feasible region, where the company can buy sandwiches within its budget.

Algebraic Approach

We can also solve this problem using algebraic methods. Let's solve the first equation for x:

x ≤ (150 - 5y) / 5

Simplifying the equation, we get:

x ≤ 30 - y

This equation represents the constraint that the number of roast beef sandwiches cannot exceed 30 minus the number of tuna sandwiches.

Optimal Solution

To find the optimal solution, we need to maximize the number of sandwiches while satisfying the constraints. Let's analyze the feasible region and find the point that maximizes the number of sandwiches.

From the graph, we can see that the feasible region is a triangle with vertices at (0, 20), (0, 30), and (20, 0). The optimal solution is the point that maximizes the number of sandwiches within the feasible region.

Conclusion

In conclusion, the company can buy a maximum of 18 roast beef sandwiches and 2 tuna sandwiches within its budget of $150. This solution maximizes the number of sandwiches while satisfying the constraints.

Recommendation

Based on the analysis, we recommend that the company buy 18 roast beef sandwiches and 2 tuna sandwiches. This solution provides the maximum number of sandwiches within the budget and is the most cost-effective option.

Future Research Directions

This problem can be extended to include other variables, such as the cost of each sandwich, the number of employees, and the budget. Future research directions can include:

  • Analyzing the impact of different budget scenarios on the optimal solution
  • Investigating the effect of varying the cost of each sandwich on the optimal solution
  • Developing a more general model that can handle multiple variables and constraints

Introduction

In our previous article, we explored a classic problem in mathematics, where a company has a budget of $150 to provide lunch for its 20 employees. The options are to provide either roast beef sandwiches or tuna sandwiches, both of which cost $5 apiece. We analyzed this problem using mathematical concepts and provided a solution to help the company make an informed decision.

Q&A Session

In this article, we will address some of the most frequently asked questions related to the problem.

Q: What is the optimal solution for the company?

A: The optimal solution is to buy 18 roast beef sandwiches and 2 tuna sandwiches. This solution maximizes the number of sandwiches within the budget of $150.

Q: How did you determine the optimal solution?

A: We used a graphical approach to visualize the problem and find the point that maximizes the number of sandwiches within the feasible region.

Q: What are the constraints of the problem?

A: The constraints are:

  • The total cost of the sandwiches cannot exceed the budget of $150.
  • The total number of sandwiches must be equal to or less than 20.

Q: How can the company adjust the budget to accommodate more employees?

A: If the company wants to accommodate more employees, it can increase the budget accordingly. However, the optimal solution will depend on the new budget and the number of employees.

Q: What if the cost of each sandwich changes?

A: If the cost of each sandwich changes, the optimal solution will also change. The company will need to recalculate the optimal solution based on the new cost of each sandwich.

Q: Can the company buy a mix of roast beef and tuna sandwiches?

A: Yes, the company can buy a mix of roast beef and tuna sandwiches. However, the optimal solution will depend on the specific mix of sandwiches and the budget.

Q: How can the company use this problem as a case study for future budgeting decisions?

A: The company can use this problem as a case study to develop a more general model that can handle multiple variables and constraints. This will help the company make more informed decisions when faced with similar dilemmas in the future.

Q: What are some potential extensions of this problem?

A: Some potential extensions of this problem include:

  • Analyzing the impact of different budget scenarios on the optimal solution
  • Investigating the effect of varying the cost of each sandwich on the optimal solution
  • Developing a more general model that can handle multiple variables and constraints

Conclusion

In conclusion, the company can buy a maximum of 18 roast beef sandwiches and 2 tuna sandwiches within its budget of $150. This solution maximizes the number of sandwiches while satisfying the constraints. We hope this Q&A session has provided valuable insights into the problem and its solution.

Recommendation

Based on the analysis, we recommend that the company buy 18 roast beef sandwiches and 2 tuna sandwiches. This solution provides the maximum number of sandwiches within the budget and is the most cost-effective option.

Future Research Directions

This problem can be extended to include other variables, such as the cost of each sandwich, the number of employees, and the budget. Future research directions can include:

  • Analyzing the impact of different budget scenarios on the optimal solution
  • Investigating the effect of varying the cost of each sandwich on the optimal solution
  • Developing a more general model that can handle multiple variables and constraints

By exploring these research directions, we can gain a deeper understanding of the problem and develop more effective solutions for companies facing similar dilemmas.