Problem 56 - College Algebra

Question

Solve for the desired quantity. A stuffed animal business has a total cost of production \(C=12 x+30\) and a revenue function \(R=20 x .\) Find the break-even point.

Step-by-Step Solution

Answer

The break-even point is at 3.75 units.

Step 1: Understand the Concept

The break-even point occurs where the total cost equals the total revenue. This is essential for businesses to know when they start making a profit. To find this, we need to set the cost function equal to the revenue function.

Step 2: Set Up the Equation

To find the break-even point, set the total cost function, \( C = 12x + 30 \), equal to the revenue function, \( R = 20x \). This gives us the equation \( 12x + 30 = 20x \).

Step 3: Simplify the Equation

Rearrange the equation \( 12x + 30 = 20x \) by subtracting \( 12x \) from both sides to isolate the variable \( x \) on one side: \( 30 = 20x - 12x \).

Step 4: Solve for x

Simplify \( 30 = 8x \) by dividing both sides by 8 to solve for \( x \): \( x = \frac{30}{8} \).

Step 5: Simplify the Solution

Simplify \( x = \frac{30}{8} \) to \( x = \frac{15}{4} \). This simplifies further to \( x = 3.75 \).

Step 6: Interpret the Result

At the break-even point, the stuffed animal business needs to produce and sell 3.75 units. Since you can't sell a fraction of a physical product, they would need to produce at least 4 units to realistically break even.

Key Concepts

Cost Function

The cost function is a mathematical model used to represent the total cost incurred by a business in the production of a certain number of goods. It typically includes two components:

• Fixed Costs: These are costs that do not change with the level of production. They can include rents, salaries, and other overheads that must be paid regardless of how much is produced. In our cost function, this is represented by the constant term, 30.
• Variable Costs: These change with production volume and can include materials and labor directly associated with production. In our cost equation, variable costs are represented by the term 12x, where x is the number of units produced.

Understanding the cost function is vital for businesses to predict expenses and make informed pricing and production decisions.

Revenue Function

The revenue function is another crucial component in business mathematics. It describes how much money a business earns based on the quantity of goods sold. Mathematically, it is often a simple equation where price per unit is multiplied by the number of units sold.

In the example we have, the revenue function is given by:

• R = 20x

Here, the number 20 represents the price at which each stuffed animal is sold, and x is the number of animals sold. For any business, understanding the revenue function helps to estimate potential income and assess financial viability.

Solving Equations

When comparing cost and revenue functions to find the break-even point, you set them equal to solve the equation:

• 12x + 30 = 20x

The goal here is to find the value of x that satisfies the equation, representing the production level where costs equal revenues. Here's how it's done:

• Reorganize the equation by eliminating like terms, ideally placing all terms with x on one side. Subtraction can achieve this, like subtracting 12x from both sides.
• This simplifies to 30 = 8x.
• To solve for x, perform division, 30 divided by 8 results in x = 3.75.

This step-by-step approach is invaluable in breaking down complex equations in business mathematics.

Business Mathematics

These calculations sit at the heart of business mathematics, crucial for strategizing and planning financials in any enterprise. By knowing how to calculate the break-even point, a business can anticipate when it will start making profits rather than losses.

The break-even analysis encompasses:

• Determining how production affects financial standing.
• Informing pricing strategies based on cost recovery.
• Anticipating future revenue streams and profitability.

For the stuffed animal business, the break-even point calculation indicated they must sell 4 full units (rounding up from 3.75) to cover all production costs. Essential knowledge for guiding production decisions and ensuring sustainability in the marketplace.