91Ó°ÊÓ

Understanding the cost function is paramount when businesses evaluate their financial health. In essence, a cost function represents the total cost of production as a function of the number of units produced, typically formulated as

\( C(x) = Vx + F \)

where \( V \) is the variable cost per unit, \( F \) is the total fixed cost, and \( x \) is the number of units produced. In the given exercise, the cost function \( C(x) = 150x + 20,000 \) encapsulates both the variable costs (\( 150 \) per unit) and the fixed cost (\( 20,000 \)), which does not change with the level of output.

All businesses need to manage their fixed costs such as rent and salaries, and their variable costs like raw materials, as they impact profitability.

The revenue function is another core concept that outlines the earnings a firm receives from selling its products or services. It's computed by multiplying the price per unit by the number of units sold, notated as \( R(x) = Px \), where \( P \) signifies the price per unit and \( x \) is the number of units.

In our problem, the revenue function is given by \( R(x) = 270x \). Here, each unit is sold for \( 270 \) dollars. A company's aim is to maximize revenue while controlling costs, as the interplay between these functions determines profitability and sustainability.

The phrase 'solve for \( x \)' is commonly used in algebra and refers to finding the value of the variable \( x \) that makes an equation true. In the context of our exercise, solving for \( x \) means determining the number of units that must be produced and sold for the cost and revenue to be equal.

This is often achieved by isolating \( x \) on one side of the equation, using algebraic operations such as addition, subtraction, multiplication, or division. In the solution steps you have encountered, \( x \) was isolated by subtracting and then dividing, which are the typical steps in solving linear equations.

Break-even quantity is a crucial metric in business and economics, referring to the number of units that need to be sold for a company's total costs and total revenues to be exactly equal. This point is significant because it represents the threshold at which a business starts to turn a profit.

At the break-even quantity, represented mathematically as \( x \) in our earlier steps, there is no net loss or gain. Using the formula and solving for \( x \), as shown in the step-by-step solution, the firm's break-even quantity is 167 units. This means after selling 167 units, every additional sale contributes to the profit of the firm.

Understanding how to calculate and interpret the break-even quantity can give valuable insights into the pricing strategy, cost structure, and the financial feasibility of new projects.