Problem 67 - Brief Applied Calculus

Question

BUSINESS: Salary An employee's weekly salary is \(\$ 500\) plus \(\$ 15\) per hour of overtime. Find a function \(P(x)\) giving his pay for a week in which he worked \(x\) hours of overtime.

Step-by-Step Solution

Answer

The function is \( P(x) = 500 + 15x \).

Step 1: Understand the Problem

The employee's weekly salary can be broken down into two parts: the base salary and the overtime pay. The problem requires us to create a function that calculates the total weekly pay based on overtime hours worked.

Step 2: Define the Base Salary as a Constant

The base salary is the fixed amount the employee earns each week, which is $500 regardless of the overtime hours. This will be a constant in our function.

Step 3: Define the Overtime Pay Function

Overtime pay is calculated at $15 for every hour worked over the standard hours. If \( x \) is the number of overtime hours, then the overtime pay can be expressed as \( 15x \).

Step 4: Formulate the Pay Function

The total pay function \( P(x) \) is the sum of the base salary and the overtime pay. Thus, the function is defined as: \[ P(x) = 500 + 15x \].

Step 5: Verify the Function

To ensure this function works, consider a test case where the employee works 10 hours of overtime. Substituting \( x = 10 \) in \( P(x) = 500 + 15x \), we find \( P(10) = 500 + 150 = 650 \), which correctly computes the weekly pay.

Key Concepts

Overtime Pay Calculation

When an employee works extra hours beyond their normal schedule, these hours are called "overtime." Overtime is typically paid at a higher rate than regular working hours, to fairly compensate the employee for their additional effort. In this scenario, the rate for each hour of overtime is \(15. This means for every hour worked beyond the standard working hours, the employee earns an extra \)15.

To calculate the overtime pay, we use the formula:

• Identify the number of overtime hours worked, which is represented as \( x \).
• Multiply the number of overtime hours \( x \) by the hourly overtime rate \( 15 \).

This results in the overtime pay, as seen in the expression \( 15x \).

For example, if an employee works 10 hours of overtime, the calculation would be \( 15 \times 10 = 150 \) dollars. This amount is then added to their base salary to calculate their total weekly pay.

Function Formulation

In mathematics, a function represents the relationship between inputs and outputs. In the context of salary calculation, a function can help us understand how overtime hours affect an employee's total weekly pay. The total pay function is a linear equation that combines the base salary with the earnings from overtime hours.

• The input of this function \( x \) is the number of overtime hours.
• The output is the total weekly pay \( P(x) \).
• The relationship is linear, meaning it increases consistently as \( x \) increases.

The function is formulated as \( P(x) = 500 + 15x \). This equation combines the fixed amount of the base salary, \( 500 \), with the variable overtime pay, \( 15x \).

Through this linear function, determining the total pay becomes straightforward. By substituting any value of \( x \), the number of overtime hours, into the function, you can quickly compute the corresponding total weekly pay.

Base Salary

A base salary is a fixed amount of money an employee receives for their work, regardless of any additional factors such as overtime. It serves as a steady foundation, providing security and predictability regardless of how many extra hours an employee may work in a given week.

• In this problem, the base salary is \( \$500 \) per week.
• This amount remains constant every week.

The concept of a base salary simplifies financial planning for both employee and employer. It ensures that regardless of overtime, an employee will receive a guaranteed sum. Adding in the flexibility of overtime pay allows the employee to increase their earnings when feasible, depending upon the overtime hours worked. Ultimately, the base salary and overtime work together in the formulated function \( P(x) = 500 + 15x \), enhancing an employee's potential earnings.