Problem 57 - Precalculus with Limits

Question

Break-Even Analysis In Exercises 57 and 58 , find the sales necessary to break even \((R=C)\) for the total cost \(C\) of producing \(x\) units and the revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) $$C=8650 x+250,000, \quad R=9950 x$$

Step-by-Step Solution

Answer

Approximately 192 units need to be sold to break even.

Step 1: Set up the break-even equation

Break-even point is where total revenue \(R\) equals total cost \(C\). Set \(C=R\): \n\[8650x + 250,000 = 9950x \]

Step 2: Rearrange the equation

Convert the equation by moving all terms involving \(x\) to one side and the remaining terms to the other side: \n\[ 9950x - 8650x = 250,000\]

Step 3: Solve for \(x\)

Combine similar terms, then divide both sides by the coefficient of \(x\) to isolate \(x\): \n\[ x = \frac{250,000}{1300} \] This calculates the break-even volume.

Step 4: Evaluate and round the result

Evaluate the expression to get the exact value, then round it to the nearest whole unit to get the break-even volume. This step involves numerical computation.

Key Concepts

Break-Even Point

The break-even point is a fundamental concept in economics and business, representing the sales level at which total revenue equals total cost. At this point, a business is not making a profit, but it is also not incurring any losses. It’s the moment where expenses and income are perfectly balanced.

Understanding the break-even point helps businesses determine the minimum amount of product they need to sell to cover their costs. This critical piece of information is essential for pricing strategies, budget planning, and financial forecasting.

Using the provided exercise as an example, the sales necessary to break even can be found by setting up an equation where the total cost of producing x units (\(C\text{ for cost}\)) is equivalent to the revenue obtained by selling x units (\(R\text{ for revenue}\)).

Total Revenue

Total revenue is the entire amount of money generated from the sale of goods or services before any expenses are deducted. For a business selling multiple products, it would be the sum of the revenues from each product.

In an algebraic sense, total revenue is often represented as the product of the number of units sold (\(x\text{ for units}\)) and the price per unit. In the given exercise, total revenue is expressed as \(R=9950x\), where \(9950\) is the price per unit, and \(x\) is the number of units sold.

Being aware of total revenue helps businesses to understand their sales performance and is an essential variable in break-even analysis.

Total Cost

Total cost refers to the sum of all expenses incurred by a business in producing a certain number of goods or services. This includes both fixed costs (which do not change with the amount of product produced) and variable costs (which vary with production volume).

In the context of our example exercise, the total cost formula is given by \(C = 8650x + 250,000\). Here, \(8650\) represents the variable cost per unit, \(x\) is the number of units, and \(250,000\) stands for the fixed costs. Such an equation helps a business determine how their costs will change as production levels scale.

Algebraic Equations

Algebraic equations are mathematical statements that involve letters (representing variables) and numbers, along with arithmetic operations. They set two expressions equal to one another, such as in the break-even analysis equation where \(C=R\).

To find the break-even point algebraically, one rearranges the equation to get the variable of interest, in this case, \(x\), on one side. The goal is to isolate the variable and solve for its value, as seen in Step 3 of our example, resulting in \(x = \frac{250,000}{1300}\). Algebraic manipulation includes combining like terms and using arithmetic operations to simplify and solve for variables.