Problem 49 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

BREAK-EVEN ANALYSIS A manufacturer can sell a certain product for \(\$ 80\) per unit. Total cost consists of a fixed overhead of \(\$ 4,500\) plus production costs of \(\$ 50\) per unit. a. How many units must the manufacturer sell to break even? b. What is the manufacturer's profit or loss if 200 units are sold? c. How many units must the manufacturer sell to realize a profit of \(\$ 900\) ?

Step-by-Step Solution

Answer

a. 150 units. b. Profit of \( \$1500 \). c. 180 units.

Step 1 - Define variables

Let the number of units be denoted by \( x \). The selling price per unit is \( \$80 \). The fixed overhead costs are \( \$4500 \), and the production cost per unit is \( \$50 \).

Step 2 - Break-even point formula

To find the break-even point, set total revenue equal to total cost. The total revenue can be calculated as: \[ \text{Total Revenue} = 80x \] The total cost can be calculated as: \[ \text{Total Cost} = 4500 + 50x \]

Step 3 - Solve for the break-even point

Set the total revenue equal to the total cost: \[ 80x = 4500 + 50x \] Subtract \(50x\) from both sides: \[ 30x = 4500 \] Divide both sides by 30: \[ x = 150 \] Therefore, the manufacturer must sell 150 units to break even.

Step 4 - Calculate profit or loss for 200 units

Calculate the total revenue for 200 units: \[ \text{Total Revenue} = 80 \times 200 = 16000 \] Calculate the total cost for 200 units: \[ \text{Total Cost} = 4500 + 50 \times 200 = 14500 \] Calculate the profit or loss: \[ \text{Profit or Loss} = 16000 - 14500 = 1500 \] Therefore, the manufacturer's profit is \( \$1500 \) if 200 units are sold.

Step 5 - Calculate units for a profit of $900

Set up the equation for a profit of \( \$900 \): \[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} \] Therefore: \[ 900 = 80x - (4500 + 50x) \] Simplify the equation: \[ 900 = 30x - 4500 \] Add 4500 to both sides: \[ 5400 = 30x \] Divide both sides by 30: \[ x = 180 \] Therefore, the manufacturer must sell 180 units to realize a profit of \( \$900 \).

Key Concepts

Fixed Costs

Understanding fixed costs is crucial for break-even analysis. Fixed costs are expenses that do not change with the level of production or sales. In this exercise, the fixed overhead cost is \( \$4500 \). This means that regardless of how many units are produced or sold, the manufacturer will always incur this \( \$4500 \) cost. Examples of fixed costs include rent, salaries, and insurance. These costs remain constant and must be covered before the company can start making a profit. Hence, knowing fixed costs helps in determining the break-even point where total revenue equals total cost.

Variable Costs

Variable costs change with the level of production. For our problem, the variable cost is \( \$50 \) per unit produced. This means for each additional unit produced, the costs associated with materials, labor, and other such expenses increase by \( \$50 \). Variable costs are also referred to as direct or unit-level costs. They are important because they directly impact the total cost, and understanding them helps in planning production levels and pricing strategies.

Total Revenue

Total revenue is the total amount of money a company earns from selling its products. It is calculated by multiplying the selling price per unit by the number of units sold. In our exercise, the selling price is \( \$80 \) per unit. Thus, the total revenue formula would be: \[ \text{Total Revenue} = 80x \] where \( x \) is the number of units sold. Total revenue is essential for calculating profits and understanding how sales impact the financial health of a business.

Total Cost

Total cost combines both fixed and variable costs. In the given exercise, the total cost is calculated by adding fixed costs to the product of the variable cost per unit and the number of units produced: \[ \text{Total Cost} = 4500 + 50x \] Here, \( 4500 \) is the fixed cost, and \( 50x \) represents the variable costs. Calculating total cost is important because it helps determine the break-even point and assess profitability. Lowering total cost can improve profit margins and overall business performance.

Profit Calculation

Calculating profit involves subtracting the total cost from the total revenue. The basic formula for profit is: \[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} \] For example, if 200 units are sold: \[ \text{Total Revenue} = 80 \times 200 = 16000 \] and \[ \text{Total Cost} = 4500 + 50 \times 200 = 14500 \] then the profit is: \[ \text{Profit} = 16000 - 14500 = 1500 \] Thus, the manufacturer's profit when selling 200 units would be \( \$1500 \). Understanding this calculation helps in setting sales targets and managing resources effectively.