Problem 68 - 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 pay function is \(P(x) = 300 + 0.02x\).

Step 1: Identify Components of the Salary

The sales clerk's salary consists of two parts: a fixed salary of $300, and a commission of 2% on her total weekly sales. We need to combine these to express the total weekly pay as a function of the week's sales.

Step 2: Express Commission as a Mathematical Expression

The commission is 2% of the total sales, which can be written as \(0.02x\), where \(x\) is the total dollar amount of merchandise sold in a week.

Step 3: Formulate the Total Weekly Pay Function

The total weekly pay, \(P(x)\), is the sum of the fixed salary and the commission. Therefore, the function can be represented as \(P(x) = 300 + 0.02x\).

Key Concepts

Salary Calculation

Calculating a salary often begins with understanding the basic components that make up a paycheck. In this case, the sales clerk's weekly earnings include a base salary of $300. This is the guaranteed amount she receives regardless of her sales performance. Salary components typically involve a fixed aspect and possibly variable components like overtime pay or bonuses. Here, the fixed part is the $300. Understanding this helps in easily determining the minimum earnings a worker can expect each week. In scenarios where there's sales or performance-based income like in many sales jobs, you combine these with stable earnings to calculate total pay. Let’s move on to how this coincides with commission.

Commission Calculation

Commission is a form of variable income based on performance, especially prevalent in sales jobs. It acts as an incentive for better job performance. Here, the commission rate is 2%, which means the clerk earns 2% of her total weekly sales. Commission can be calculated by the formula: - Commission Amount = Commission Rate * Total Sales. In mathematical terms, 2% can be expressed as 0.02. Thus, if the sales clerk sells goods worth $x, her commission is computed as 0.02x. This motivates employees to sell more, as their earnings directly relate to their performance, allowing them to increase their weekly pay according to their sales efforts.

Weekly Pay Function

The weekly pay function is a mathematical expression representing the total earnings a sales clerk receives. To build this function, we sum both the fixed salary and the commission. The function is given as:- Total Weekly Pay, \( P(x) = 300 + 0.02x \) Here

• \(300\) is the fixed salary,
• \(0.02x\) is the result of the commission calculation.

This formula allows you to compute total pay for any given week by plugging in the value of sales \(x\). As sales increase, the total pay increases, reflecting the influence of commission on the overall salary. This formula helps both employees and employers to plan and predict earnings, calculate budgets, and understand financial growth prospects based on sales activities.