Problem 65 - 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.) $$C=8650 x+250,000 ; R=9950 x$$

Step-by-Step Solution

Answer

To break even, approximately \(x = [round(250000/1300)]\) units need to be sold.

Step 1: Set up the equation

The first step is to set up the break-even equation. Given that at the break-even point, the cost equals the revenue, we'll set \(C = R\), which leads to: \(8650x + 250000 = 9950x\)

Step 2: Solve the equation

To solve for \(x\), first, rearrange the equation by subtracting \(8650x\) from both sides: \(250000 = 1300x\). Then, divide both sides by 1300 to isolate \(x\): \(x = 250000/1300\).

Step 3: Round the result

After solving the equation, it's essential to round the result to the nearest whole unit since you can't sell a fraction of a unit: \(x \approx [round(250000/1300)]\) units.

Key Concepts

Cost Function

In the realm of business and economics, the **Cost Function** is crucial. It represents the total cost to produce a given number of goods or services. Namely, how much it costs to bring your products to market. A standard cost function is structured as a sum of fixed and variable costs.
Fixed costs remain constant regardless of the production level – think rent or salaries. Meanwhile, variable costs scale with the quantity produced – such as raw material expenses.

In our exercise, the cost function is given by:

• Formula: \(C = 8650x + 250,000\)
• Fixed costs: 250,000 (dollars)
• Variable cost per unit: 8650 (dollars)

Understanding your cost function is vital for managing resources effectively. This knowledge informs pricing strategies and budgeting decisions. Without a clear grasp of cost behavior, a business will struggle to break even or achieve profitability.

Revenue Function

The **Revenue Function** embodies the income a business generates from selling goods or services. It is fundamentally the sale price of the goods times the number of goods sold. This function helps businesses predict how much income will be generated at different sales volumes.
Revenue is a critical figure because it provides insights into sales performance and operational effectiveness. Maximizing revenue is often a primary goal for businesses.

In the exercise, the revenue function is expressed as:

• Formula: \(R = 9950x\)
• Price per unit: 9950 (dollars)

A deep understanding of revenue is essential. It enables businesses to forecast growth, assess the effectiveness of their sales strategies, and make data-driven decisions on future operations.

Algebraic Equation

An **Algebraic Equation** serves as a mathematical sentence, representing relationships where one or more unknowns need solving. In the context of break-even analysis, the equation is set up to find the point where costs equal revenue, making profit or loss zero. At the break-even point, all expenses are covered by income generated.
The exercise involves setting up such an equation by equating the cost and revenue functions, which looks like this:

• Equation: \(8650x + 250,000 = 9950x\)

Solving this equation involves simple algebraic manipulations:

• Subtracting terms: Combine like terms to one side. Here, subtract \(8650x\) from both sides: \(250000 = 1300x\).
• Isolating variables: Divide each side by 1300 to solve for \(x\). This gives \(x = \frac{250000}{1300}\).
• Rounding: Since you can't sell a fraction, round to the nearest whole unit.

These steps emphasize the importance of understanding how algebra translates real-world scenarios into solvable problems. This process enables decision-making grounded in precise calculations.