Problem 56 - College Algebra with Corequisite Support

Question

For the following exercises, 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 stuffed animals.

Step 1: Understand the Break-even Concept

The break-even point is where total costs equal total revenue. This means the company is making neither a profit nor a loss.

Step 2: Set Cost Equal to Revenue

To find the break-even point, set the cost equation equal to the revenue equation: \( C = R \).

Step 3: Substitute Provided Equations

The cost function is \( C = 12x + 30 \) and the revenue function is \( R = 20x \). Substitute these into the equation: \( 12x + 30 = 20x \).

Step 4: Isolate the Variable

Rearrange the equation to solve for \( x \) by subtracting \( 12x \) from both sides: \( 30 = 20x - 12x \).

Step 5: Simplify the Equation

Simplify the equation to get \( 30 = 8x \).

Step 6: Solve for x

Divide both sides by 8 to isolate \( x \): \( x = \frac{30}{8} \).

Step 7: Calculate the Value

Calculate \( x = \frac{30}{8} = 3.75 \).

Key Concepts

Cost Function

When we talk about a cost function in business mathematics, we're referring to a mathematical way of expressing how the cost of production changes with varying levels of output. Imagine it as a formula that tells a business owner how much they're spending to produce their products.

In the original exercise, the cost function for the stuffed animal business is given by the equation \( C = 12x + 30 \). Here:

• \( C \) represents the total cost.
• \( x \) is the number of stuffed animals produced.
• \( 12x \) indicates the variable cost, which is $12 per animal.
• \( 30 \) is the fixed cost, which is the constant costs of running the business regardless of production level.

This function helps businesses to plan how costs change as production increases. Understanding their cost function allows businesses to make better budgeting and pricing decisions.

Revenue Function

A revenue function is another critical concept in business mathematics. It helps in calculating the total income from selling goods or services. Just like the cost function, it uses an equation to represent revenues depending on output levels.

For the stuffed animal business, the revenue function is given by \( R = 20x \). In this case:

• \( R \) stands for the total revenue.
• \( x \) is the number of units sold, i.e., stuffed animals.
• \( 20x \) signifies that each stuffed animal is sold at $20.

By understanding their revenue function, businesses can forecast how much money they will make based on their sales volumes, which guides pricing and marketing strategies.

Algebraic Equations

Algebraic equations are fundamental tools in solving business-related problems, like finding the break-even point. They involve using algebra to manipulate equations so you can find unknown values.

In the exercise, to find the break-even point, we used the following equation: \( 12x + 30 = 20x \). This shows when the cost equals revenue, indicating neither profit nor loss.

To solve these kinds of problems:

• Set the cost function equal to the revenue function.
• Combine like terms and rearrange to isolate the variable \( x \).
• Solve for \( x \) to find the quantity required for break-even.

Through these steps, Algebra becomes a powerful tool in analyzing and planning business finances.

Business Mathematics

Business mathematics encompasses all mathematical concepts used in commercial enterprise. It's about applying mathematical techniques to solve real business problems, focusing on functions like cost and revenue.

Understanding how to utilize these mathematical functions allows businesses to make informed decisions. For instance:

• Identifying break-even points to know when a business's costs and revenue are balanced.
• Budgeting effectively by understanding fixed and variable costs.
• Maximizing profits through optimal pricing and cost reduction strategies.

With these skills, professionals can better navigate the financial aspects of their businesses, leading to more successful outcomes.