Problem 67 - College Algebra and Calculus: An Applied Approach

Question

Break-Even Analysis, find the sales necessary to break even \((R=C)\) for the cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round your answer to the nearest whole unit.) $$C=2.65 x+350,000 ; R=4.15 x$$

Step-by-Step Solution

Answer

The number of units needed to break even is \(x = 233,334\), rounded to the nearest whole unit.

Step 1: Equalize Revenue and Cost

Set the revenue function equal to the cost function. The resulting equation will be \(4.15x = 2.65x + 350,000\).

Step 2: Simplify the Equation

Solve for \(x\) to find the quantity necessary to break-even. Start by getting terms with \(x\) on the one side: \(4.15x - 2.65x = 350,000\) which simplifies to \(1.5x = 350,000\).

Step 3: Solve for \(x\)

To solve for x, divide both sides by 1.5, so \(x = 350,000 / 1.5\).

Step 4: Calculate and Round

Once you perform the division, don't forget to round your answer to the nearest whole unit as the instructions request.

Key Concepts

Understanding the Revenue Function

In break-even analysis, the revenue function is crucial because it represents the income generated from selling a certain number of units, which we have labeled as \(R\). Revenue can be simply thought of as the product of the price at which each unit is sold and the number of units sold. In this exercise, the revenue function is given as \(R = 4.15x\). Here, \(4.15\) is the selling price per unit, and \(x\) is the quantity of units sold.

The goal is to align revenue with costs to find the break-even point, where no profit or loss occurs. When using this function, remember that any change in the selling price or the number of units will directly impact revenue.

Exploring the Cost Function

The cost function in break-even analysis denotes the total expense involved in producing a certain amount of goods. It's represented as \(C\) in the equation, and it combines both fixed and variable costs. In our given problem, the cost function is expressed as \(C = 2.65x + 350,000\).

Let's break it down:

• \(2.65x\) - This represents the variable cost per unit, meaning each unit produced adds \(2.65\) to the total cost.
• \(350,000\) - This is the fixed cost, which does not change regardless of the amount of units produced.

The fixed cost remains constant as it covers expenses like rent, salaries, and equipment that don't fluctuate with production volume. Understanding these components helps in accurately setting and managing production budgets.

Solving for x

The most critical step in determining the break-even point is solving for \(x\). This means finding the number of units \(x\) necessary to ensure that revenue matches the cost. Follow these easy steps:

• Set the revenue function equal to the cost function: \(4.15x = 2.65x + 350,000\).
• Simplify the equation by moving all terms involving \(x\) to one side: \(4.15x - 2.65x = 350,000\), resulting in \(1.5x = 350,000\).
• To isolate \(x\), divide both sides of the equation by \(1.5\): \(x = \frac{350,000}{1.5}\).
• Calculate the division and round the resulting number to the nearest whole unit.

Completing these steps reveals the exact quantity needed to "break-even," meaning the business neither profits nor incurs a loss at this volume of product sold. Accurate calculation and rounding are essential for practical application in business decisions.