An Employee Is Paid $ \$50 $ Per Week Plus $ \$20 $ For Every Hour Worked. They Are Not Allowed To Work Overtime Or Over 40 Hours Per Week. Their Weekly Income Can Be Modeled By The Equation $ Y = 20x + 50 $, Where $ X
Introduction
In this article, we will explore a mathematical model that represents an employee's weekly income. The model takes into account the employee's hourly wage and the maximum number of hours they are allowed to work per week. We will analyze the equation that represents the employee's weekly income and discuss its implications.
The Mathematical Model
The employee's weekly income can be modeled by the equation $ y = 20x + 50 $, where $ x $ represents the number of hours worked per week. The equation is a linear function, which means that the relationship between the number of hours worked and the weekly income is directly proportional.
Understanding the Equation
Let's break down the equation and understand its components. The equation is in the form of $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. In this case, the slope $ m $ is 20, which represents the hourly wage. The y-intercept $ b $ is 50, which represents the weekly income when no hours are worked.
Interpreting the Equation
To interpret the equation, let's consider a few scenarios:
- If the employee works 0 hours per week, their weekly income will be $ 50 $.
- If the employee works 1 hour per week, their weekly income will be $ 20(1) + 50 = 70 $.
- If the employee works 2 hours per week, their weekly income will be $ 20(2) + 50 = 90 $.
As we can see, the equation represents a linear relationship between the number of hours worked and the weekly income.
Graphing the Equation
To visualize the equation, let's graph it on a coordinate plane. The x-axis represents the number of hours worked per week, and the y-axis represents the weekly income.
import matplotlib.pyplot as plt
import numpy as np
# Define the x-values
x = np.linspace(0, 40, 100)
# Define the y-values
y = 20*x + 50
# Create the plot
plt.plot(x, y)
plt.xlabel('Hours Worked per Week')
plt.ylabel('Weekly Income')
plt.title('Employee\'s Weekly Income')
plt.grid(True)
plt.show()
The graph shows a straight line with a positive slope, indicating a direct proportional relationship between the number of hours worked and the weekly income.
Implications of the Equation
The equation has several implications for the employee and the employer:
- The employee's weekly income is directly proportional to the number of hours worked.
- The employee's weekly income is capped at $ 20(40) + 50 = 900 $, since they are not allowed to work overtime or over 40 hours per week.
- The employer can use the equation to determine the employee's weekly income based on the number of hours worked.
Conclusion
In conclusion, the equation $ y = 20x + 50 $ represents a mathematical model that describes an employee's weekly income. The equation is a linear function that takes into account the employee's hourly wage and the maximum number of hours they are allowed to work per week. The equation has several implications for the employee and the employer, and it can be used to determine the employee's weekly income based on the number of hours worked.
References
Q: What is the equation that represents an employee's weekly income?
A: The equation that represents an employee's weekly income is $ y = 20x + 50 $, where $ x $ represents the number of hours worked per week.
Q: What is the hourly wage represented by the equation?
A: The hourly wage represented by the equation is $ 20 $ per hour.
Q: What is the weekly income when no hours are worked?
A: The weekly income when no hours are worked is $ 50 $.
Q: What is the maximum number of hours an employee can work per week?
A: The maximum number of hours an employee can work per week is 40 hours.
Q: What is the maximum weekly income an employee can earn?
A: The maximum weekly income an employee can earn is $ 20(40) + 50 = 900 $.
Q: Can an employee work overtime or over 40 hours per week?
A: No, an employee is not allowed to work overtime or over 40 hours per week.
Q: How can an employer determine an employee's weekly income based on the number of hours worked?
A: An employer can use the equation $ y = 20x + 50 $ to determine an employee's weekly income based on the number of hours worked.
Q: What is the relationship between the number of hours worked and the weekly income?
A: The relationship between the number of hours worked and the weekly income is directly proportional.
Q: Can the equation be used to model an employee's income for any number of hours worked?
A: Yes, the equation can be used to model an employee's income for any number of hours worked.
Q: What are the implications of the equation for the employee and the employer?
A: The equation has several implications for the employee and the employer, including the employee's weekly income being directly proportional to the number of hours worked, the employee's weekly income being capped at $ 20(40) + 50 = 900 $, and the employer being able to use the equation to determine the employee's weekly income based on the number of hours worked.
Q: Can the equation be graphed on a coordinate plane?
A: Yes, the equation can be graphed on a coordinate plane, with the x-axis representing the number of hours worked per week and the y-axis representing the weekly income.
Q: What is the slope of the equation?
A: The slope of the equation is 20, which represents the hourly wage.
Q: What is the y-intercept of the equation?
A: The y-intercept of the equation is 50, which represents the weekly income when no hours are worked.
Q: Can the equation be used to model an employee's income for different scenarios?
A: Yes, the equation can be used to model an employee's income for different scenarios, such as when the employee works 0 hours per week, 1 hour per week, or 2 hours per week.
Q: What are the benefits of using the equation to model an employee's income?
A: The benefits of using the equation to model an employee's income include being able to determine the employee's weekly income based on the number of hours worked, being able to model the employee's income for different scenarios, and being able to understand the relationship between the number of hours worked and the weekly income.