91Ó°ÊÓ

Understanding the total-cost function is essential in any business calculation. The total-cost function, denoted as \(C(x)\), comprises both fixed and variable costs. In the exercise given, the total-cost function is \(C(x) = 15x + 3100\). Here, the term \(15x\) represents the variable cost per unit produced, while the constant term \(3100\) stands for the fixed costs.
Fixed costs include expenses that remain constant, regardless of the number of units produced, such as rent, salaries, and insurance. Variable costs, on the other hand, change in direct proportion to the number of units produced, like raw materials and labor.
Calculating total cost accurately helps businesses to understand how their costs scale with production and is critical for determining profitability.

The total-revenue function, denoted as \(R(x)\), is another fundamental concept in analyzing business operations. It represents the total income generated from selling a certain number of goods or services. In our example, the total-revenue function is \(R(x) = 40x\). This implies that each unit sold generates $40 in revenue.
To calculate total revenue, simply multiply the price per unit by the number of units sold. For instance, if 10 units are sold, the total revenue would be \(40 \times 10 = 400\).
Understanding the total-revenue function helps businesses to predict income based on sales volumes, which is crucial for planning and strategy purposes.

The break-even point is a critical milestone in business, indicating when total revenues match total costs, resulting in neither profit nor loss. This point is determined by setting the total-profit function equal to zero. The formula used is \(P(x) = R(x) - C(x)\).
From our example, you first determine the total-profit function: \(P(x) = 40x - (15x + 3100) = 25x - 3100\). To find the break-even point, set \(P(x) = 0\) and solve for \(x\):

• Set the profit function to 0: \(0 = 25x - 3100\)
• Add 3100 to both sides: \(3100 = 25x\)
• Divide by 25: \(x = \frac{3100}{25} = 124\)

Thus, the break-even point is at 124 units. Reaching this point means the business covers all costs, with any further production resulting in profit.