Problem 63 - Precalculus

Question

A company produces CD players for a unit cost of \(\$ 15.00\) per CD player. The company has fixed costs of \(\$ 120.00 .\) If each CD player can be sold for \(\$ 30.00\) how many CD players must be sold to break even? Determine the cost equation first. Next, determine the revenue equation. Use the two equations you have found to determine the break-even point.

Step-by-Step Solution

Answer

The company must sell 8 CD players to break even.

Step 1: Identify and Write the Cost Equation

The total cost for producing CD players includes both fixed costs and variable costs depending on the number of units produced. The formula for total cost (C) is given by the sum of fixed costs (F) and variable costs (V), where the variable cost per unit is multiplied by the number of units produced (x). Thus, the cost equation is: \[ C(x) = 15x + 120 \] Here, 15 is the cost per unit and 120 is the fixed cost.

Step 2: Identify and Write the Revenue Equation

The revenue is the amount of money earned by selling a certain number of CD players at a given price per unit. The revenue equation (R) can be expressed as the price per unit sold times the number of units sold (x). Hence, the revenue equation is: \[ R(x) = 30x \] In this case, the price per unit is $30.

Step 3: Set Up the Break-even Equation

The break-even point occurs when total cost equals total revenue. Using the cost and revenue equations from previous steps: \[ 15x + 120 = 30x \] This equation shows that the revenue and cost are equal at the break-even point.

Step 4: Solve for the Break-even Quantity

To find the number of CD players that must be sold to break even, solve the equation: Subtract 15x from both sides:\[ 120 = 15x \]Divide both sides by 15 to solve for x:\[ x = \frac{120}{15} = 8 \]Therefore, the company must sell 8 CD players to break even.

Key Concepts

Cost Equation

A cost equation is essential in understanding how a company calculates its expenses. It helps determine how much it costs to produce a certain number of products. For the given exercise, the company has both fixed and variable costs. Fixed costs remain constant regardless of production levels, while variable costs change according to the number of units produced. Here, the cost for producing each unit, a CD player, is \(15. Adding to this are fixed costs of \)120. Together, the cost equation looks like this:\[ C(x) = 15x + 120 \]- **Fixed Costs (F):** \(120 (costs not dependent on the number of units)- **Variable Costs (V):** \)15 per CD player (cost for each unit)This equation allows you to calculate the total cost for any number of units by plugging in the value of x, the number of CD players produced.

Revenue Equation

The revenue equation is a tool used to measure how much money a company can earn from its sales. It depends on how many units are sold and the price per unit. In this scenario, the company sells each CD player for \(30. To express this in the form of an equation, consider the number of units sold as 'x'. Therefore, the revenue equation is:\[ R(x) = 30x \]- **Price per Unit:** \)30- **Quantity (x):** Number of CD players soldThe revenue equation helps assess potential earnings by varying the number of units sold, which is crucial in reaching the break-even point. This is where total revenue equals total costs.

Fixed and Variable Costs

Understanding fixed and variable costs is key to analyzing a company's financial health. Fixed costs do not change with production levels, while variable costs fluctuate with the amount of output. In this exercise: - **Fixed Costs (F):** These are constant costs, such as rent, salaries, and utilities, that add up to $120. They remain the same no matter how many CD players are manufactured. - **Variable Costs (V):** These depend on the production volume. Here it's $15 for each CD player. To calculate the total cost effectively, the interplay between both types of costs must be recognized. Fixed costs provide predictability, while variable costs allow for scalability of production. Recognizing these costs helps in treasury management and strategic planning, ultimately influencing break-even analysis.