Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Cost A company produces a product for which the marginal cost of producing \(x\) units is modeled by \(d C / d x=2 x-12\), and the fixed costs are \(\$ 125\). (a) Find the total cost function and the average cost function. (b) Find the total cost of producing 50 units. (c) In part (b), how much of the total cost is fixed? How much is variable? Give examples of fixed costs associated with the manufacturing of a product. Give examples of variable costs.

Step-by-Step Solution

Answer

The total cost function is \(C(x) = x^2 - 12x + 125\), and the average cost function is \(AC(x) = (x^2 - 12x + 125)/x\). The total cost of producing 50 units is calculated by substituting \(x = 50\) into the total cost function. The fixed costs are \(\$ 125\) and the variable costs can be found by subtracting fixed costs from the total costs for 50 units.

Step 1: Find the Total Cost Function

Integrate the given marginal cost function \(dC/dx = 2x - 12\) with respect to \(x\) to get the cost function \(C(x)\). The antiderivative of \(2x - 12\) is \(x^2 - 12x\). Add the fixed costs to this function to get the total cost function. Write the total cost function as \[C(x) = \int (2x - 12) dx + 125 = x^2 - 12x + 125\]

Step 2: Compute the Average Cost Function

The average cost function is obtained by dividing the total cost function by the number of units produced. Thus, write the average cost function as \[AC(x) = C(x)/x = (x^2 - 12x + 125)/x\]

Step 3: Compute the Total Cost for 50 Units

Substitute \(x = 50\) into the total cost function that was determined in Step 1 to find the total cost of producing 50 units. Write this as \[ C(50) = 50^2 - 12*50 + 125\]

Step 4: Determine the Fixed and Variable Costs

The fixed costs were given as \(\$ 125\). The variable costs can be found by subtracting the fixed costs from the total costs for the 50 units that were computed in Step 3.

Step 5: Give Examples of Fixed and Variable Costs

Fixed costs could include costs such as the cost of a building or production machines, which do not vary with the amount of units produced. Variable costs could include costs such as the cost of raw materials or labor, which do vary with the amount of units produced.

Key Concepts

Marginal Cost

Marginal cost represents the expense of producing an additional unit of a product. In calculus, it's conveyed as the derivative of the total cost function with respect to quantity, denoted as \( \frac{dC}{dx} \). Suppose the marginal cost for manufacturing \(x\) units of an item is expressed by a linear equation like \(2x - 12\). This equation indicates that as production increases, the cost to produce each additional unit changes proportionally to the number of units, in this case, becoming more expensive with higher \(x\).Understanding marginal cost is crucial for businesses, allowing them to determine the optimal level of production where they can maximize profit without escalating costs unsustainably. In the provided exercise, marginal cost aids in establishing the total cost function and informs the company's pricing strategies to cover costs and generate profit.

Total Cost Function

The total cost function is a comprehensive expression that details the entire cost of producing \(x\) units of goods. It combines both fixed and variable costs. To establish this function, one must integrate the marginal cost function and then add any fixed costs.In the context of the exercise, after integrating the given marginal cost function \( 2x - 12 \) with respect to \(x\), and adding the fixed cost of \( \$125 \), the total cost function can be formulated as \( C(x) = x^2 - 12x + 125 \). This equation is fundamental for businesses as it helps to predict total production costs for any number of units made, facilitating budgeting and financial planning.

Average Cost Function

An average cost function expresses the cost per unit of production. It's calculated by dividing the total cost function by the number of units produced (\(x\)). The formula provides insight into cost-efficiency and economies of scale. The resulting value can guide pricing decisions, ensuring that each unit sold covers its production costs and contributes to covering fixed costs.For the problem at hand, the average cost function \( AC(x) \) by dividing the total cost \( C(x) \) by \(x\) to obtain \( AC(x) = (x^2 - 12x + 125) / x \). This calculation reveals how much, on average, each unit costs to make and is instrumental in understanding how producing more or fewer units affects the cost per unit.

Fixed and Variable Costs

Fixed costs are expenses that do not change with the level of goods or services produced within a certain scale. These costs include rent, salaries, insurance, and equipment costs, remaining constant irrespective of the production volume—the exercise lists a fixed cost of \( \$125 \), likely representing such overheads.

Variable Costs

• Raw materials
• Direct labor
• Utilities (depending on usage)
• Commission-based compensations

Variable costs, contrastingly, fluctuate with production levels. Examples include the cost of materials and labor directly tied to the quantity of goods manufactured. These costs increase with each additional unit produced and can significantly impact the total cost function. To manage profitability, companies need to carefully monitor and manage both fixed and variable costs.