Problem 68 - Brief Applied Calculus

Question

BUSINESS: Salary A sales clerk's weekly salary is \(\$ 300\) plus \(2 \%\) of her total week's sales. Find a function \(P(x)\) for her pay for a week in which she sold \(x\) dollars of merchandise.

Step-by-Step Solution

Answer

The function for her pay is \(P(x) = 300 + 0.02x\).

Step 1: Understanding the Components of Salary

The sales clerk's weekly salary is composed of two parts: a fixed salary of \(\$300\) and a variable component that depends on her total week's sales. Specifically, she earns an additional \(2\%\) of her total sales in a week.

Step 2: Expressing the Variable Component

The variable component of her salary is calculated as \(2\%\) of her total sales, or \(0.02x\), where \(x\) represents the dollar amount of sales made in a week.

Step 3: Forming the Pay Function

Add the fixed salary and the variable component together to express the total pay function. The function for her weekly pay, given \(x\) dollars of merchandise sold, can be written as: \(P(x) = 300 + 0.02x\).

Key Concepts

Pay Function

Understanding a pay function is an essential aspect of salary calculation. It serves as a mathematical expression that helps determine how much someone earns over a certain period. Here, you can think of the pay function as a recipe—combining different parts to form the "salary meal." A pay function considers both the fixed and variable components of salary. For a sales clerk, it means combining a base salary with a performance-based addition. One way to express this clearly is by using an algebraic equation. Let's look at the sales clerk example:

The pay function is represented as:

• \[ P(x) = 300 + 0.02x \]

This equation holds two parts: a constant fixed salary of \(300\) and a variable component of \(0.02x\), based on sales. Understanding this formula helps you see how earnings adjust with sales.

Every additional dollar in sales boosts the total pay by 2%. Therefore, the essence of a pay function lies in identifying and adapting different salary components based on set conditions.

Fixed Salary

A fixed salary is a stable component of an individual's earnings. It does not sway with performance or output. Think of the fixed salary as a guaranteed minimum income, which provides security and financial predictability. In our sales clerk scenario, the fixed salary is \(\\(300\). Regardless of how many sales she makes, this amount remains constant throughout the week.

This concept is crucial in salary calculation. It ensures that no matter how variable the circumstances, there is still a base payment. Fixed salaries are common in many jobs where regular, consistent work is expected. It forms a safety net for employees who can rely on this amount even in weeks with fewer sales. This component illustrates stability in a pay function, where the clerk doesn’t have to worry about earning below \(\\)300\).

• Stable and predictable
• Unchanged by performance
• A critical part of financial security

Variable Component

The variable component of a salary is what makes pay calculations dynamic and performance-based. It reflects how earnings increase with successful output, in this case, sales. For the sales clerk, the variable component is set at 2% of total sales. This means that for every dollar in sales, her earnings increase by two cents.

In a pay function, the variable component showcases the link between effort and reward. When she sells more, her total pay grows, thus motivating higher performance. Let's break this down further:

• The variable part of the pay is shown as \(0.02x\).
• This part introduces flexibility and rewards growth.
• It provides an incentive to increase sales efforts.

Thus, the variable component is crucial in roles where performance can vary greatly from week to week, offering employees a share in the success they help create.