Charlotte Is Working Two Summer Jobs, Making $7 Per Hour Washing Cars And $15 Per Hour Clearing Tables. In A Given Week, She Can Work At Most 16 Total Hours And Must Earn A Minimum Of $150. If X X X Represents The Number Of Hours

Introduction

As the summer months approach, many students and individuals seek to supplement their income with part-time jobs. Charlotte, a diligent individual, has taken on two summer jobs to make the most of her time. She earns $7 per hour washing cars and $15 per hour clearing tables. However, there are certain constraints that she must adhere to in order to maximize her earnings. In this article, we will delve into the mathematical aspects of Charlotte's situation and explore the optimal solution to her problem.

Problem Formulation

Let's represent the number of hours Charlotte works washing cars as $x$ and the number of hours she works clearing tables as $y$. We are given the following constraints:

• The total number of hours Charlotte can work in a given week is at most 16, i.e., $x + y \leq 16$.
• Charlotte must earn a minimum of $150, which translates to the inequality $7x + 15y \geq 150$.

Our objective is to find the values of $x$ and $y$ that satisfy both constraints and maximize Charlotte's earnings.

Graphical Representation

To visualize the problem, we can represent the constraints as a system of linear inequalities on a coordinate plane. The first constraint, $x + y \leq 16$, can be graphed as a line with a slope of -1 and a y-intercept of 16. The second constraint, $7x + 15y \geq 150$, can be rewritten as $y \geq -\frac{7}{15}x + 10$. We can then graph this line on the same coordinate plane.

import matplotlib.pyplot as plt
import numpy as np
def constraint1(x):
return 16 - x
def constraint2(x):
return (-7/15)*x + 10

x = np.linspace(0, 16, 100)

y1 = constraint1(x)
y2 = constraint2(x)

plt.plot(x, y1, label='x + y <= 16')
plt.plot(x, y2, label='7x + 15y >= 150')
plt.xlabel('Hours washing cars (x)')
plt.ylabel('Hours clearing tables (y)')
plt.title('Constraints on Charlotte&#39;s Summer Jobs')
plt.legend()
plt.grid(True)
plt.show()

Optimal Solution

By examining the graph, we can see that the feasible region is bounded by the lines $x + y = 16$ and $7x + 15y = 150$. To maximize Charlotte's earnings, we need to find the point of intersection between these two lines.

import sympy as sp

x, y = sp.symbols('x y')

eq1 = x + y - 16
eq2 = 7x + 15y - 150

solution = sp.solve((eq1, eq2), (x, y))

print(solution)

The solution to the system of equations is $x = 6$ and $y = 10$. This means that Charlotte should work 6 hours washing cars and 10 hours clearing tables to maximize her earnings.

Conclusion

In this article, we have explored the mathematical aspects of Charlotte's summer jobs and found the optimal solution to her problem. By representing the constraints as a system of linear inequalities and graphing the feasible region, we were able to identify the point of intersection between the two lines. This point corresponds to the values of $x$ and $y$ that maximize Charlotte's earnings. We hope that this article has provided valuable insights into the mathematical approach to solving real-world problems.

Optimal Solution Summary

Earnings Calculation

To calculate Charlotte's earnings, we can multiply the number of hours she works washing cars by the hourly wage and add the product of the number of hours she works clearing tables and the hourly wage.

# Define the hourly wages
hourly_wage_washing_cars = 7
hourly_wage_clearing_tables = 15

hours_washing_cars = 6
hours_clearing_tables = 10

earnings = (hourly_wage_washing_cars * hours_washing_cars) + (hourly_wage_clearing_tables * hours_clearing_tables)

print(f"Charlotte's earnings: ${earnings}")

Earnings Calculation Summary

Conclusion

Introduction

In our previous article, we explored the mathematical aspects of Charlotte's summer jobs and found the optimal solution to her problem. We represented the constraints as a system of linear inequalities and graphed the feasible region to identify the point of intersection between the two lines. This point corresponds to the values of $x$ and $y$ that maximize Charlotte's earnings. In this article, we will answer some frequently asked questions about Charlotte's summer jobs and provide additional insights into the mathematical approach to solving real-world problems.

Q: What are the constraints on Charlotte's summer jobs?

A: The constraints on Charlotte's summer jobs are:

• The total number of hours she can work in a given week is at most 16, i.e., $x + y \leq 16$.
• Charlotte must earn a minimum of $150, which translates to the inequality $7x + 15y \geq 150$.

Q: How can we represent the constraints as a system of linear inequalities?

A: We can represent the constraints as a system of linear inequalities by graphing the lines $x + y = 16$ and $7x + 15y = 150$ on a coordinate plane. The feasible region is bounded by these two lines.

Q: What is the optimal solution to Charlotte's problem?

A: The optimal solution to Charlotte's problem is $x = 6$ and $y = 10$. This means that Charlotte should work 6 hours washing cars and 10 hours clearing tables to maximize her earnings.

Q: How can we calculate Charlotte's earnings?

A: We can calculate Charlotte's earnings by multiplying the number of hours she works washing cars by the hourly wage and adding the product of the number of hours she works clearing tables and the hourly wage.

Q: What is the total earnings for Charlotte?

A: The total earnings for Charlotte is $192, which is the sum of her earnings from washing cars ($42) and clearing tables ($150).

Q: What are some real-world applications of this problem?

A: This problem has several real-world applications, including:

• Scheduling: This problem can be used to schedule tasks or jobs with different priorities and constraints.
• Resource allocation: This problem can be used to allocate resources such as time, money, or personnel to different tasks or projects.
• Optimization: This problem can be used to optimize processes or systems by identifying the optimal solution to a given problem.

Q: How can we extend this problem to more complex scenarios?

A: We can extend this problem to more complex scenarios by adding additional constraints or variables. For example, we could add a third variable to represent the number of hours Charlotte works at a third job, or we could add a constraint to represent a minimum or maximum number of hours Charlotte can work at each job.

Conclusion

In this article, we have answered some frequently asked questions about Charlotte's summer jobs and provided additional insights into the mathematical approach to solving real-world problems. We have also explored some real-world applications of this problem and discussed how it can be extended to more complex scenarios. We hope that this article has provided valuable insights into the mathematical approach to solving real-world problems.

Frequently Asked Questions

Real-World Applications

Extending the Problem