A retailer sells file cabinets for 80-x dollars each, where...

A retailer sells file cabinets for $80-x$ dollars each, where $x$ is the number of cabinets she receives from the supplier each week. She pays $\$ 10$ for each file cabinet and has fixed costs of $\$ 600$ per week. How many file cabinets should she order from the supplier each week to guarantee that she makes a profit?

Key Concepts

Revenue Function

The revenue function represents the amount of money a business earns from selling its product. It is typically formulated as the product of the price per unit and the number of units sold. In scenarios where the price depends on the quantity sold, this function becomes a composite that captures the relationship between price and demand.

Cost Function

The cost function includes all expenses incurred in producing or obtaining a product. It typically consists of fixed costs, which remain constant regardless of output level, and variable costs, which change with the number of units produced or ordered. Understanding this function is critical for determining the overall expense structure in business operations.

Profit Function

The profit function is calculated by subtracting the total costs from the total revenue. This function is used to determine financial gains or losses and is central for decision-making. In many business problems, the profit function has to be analyzed to determine the conditions under which a business is profitable.

Break-Even Analysis

Break-even analysis is a financial calculation used to determine the point at which total revenues equal total costs, meaning no profit or loss is incurred. This concept is key in identifying the minimum sales volume or order quantity required to avoid losses and begin generating profit.

Quadratic Inequalities

Quadratic inequalities involve solving expressions in which a quadratic function is set greater than or less than zero. They are commonly encountered in profit analysis when the revenue function or profit function is quadratic, and the solutions help determine the range of values for which profitability conditions are met.

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