Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Production Limit \(\ln\) Exercises 75 and 76, use the following information. Variable costs depend on the number of units produced. Fixed costs are the same regardless of how many units are produced. Find the greatest number of units the company can produce each month. The company has fixed monthly costs of \(\$ 15,000\) and variable monthly costs of \(\$ 8.75\) per unit. The company has \(\$ 90,000\) available each month to cover costs.

Step-by-Step Solution

Answer

The greatest number of units the company can produce each month is 8571.

Step 1: Formulate the cost equation

The total monthly costs of production for the company is the sum of the fixed costs and the variable costs. Variable costs are the product of the number of units produced and the cost per unit. Thus, the total monthly cost equation is \(Total Cost = Fixed Costs + Variable Cost * Units Produced\). In terms of the given variables, the equation becomes \(90,000 = 15,000 + 8.75 * X\).

Step 2: Solve for X (units produced)

Solving for units produced (X), we isolate X on one side of the equation. We subtract \(15,000\) from both sides to isolate the term containing X, \(90,000 - 15,000 = 15,000 + 8.75 * X - 15,000\). Simplifying, the equation becomes \(75,000 = 8.75 * X\). Then, we divide both sides by the cost per unit (8.75) to find the units produced, \(75,000 / 8.75 = X\).

Step 3: Compute and round down

After performing the division, we find that X is approximately \(8571.43\). Because it is not possible to produce a fraction of a unit, we round this number down to the nearest whole number. Thus, the maximum number of units the company can produce is 8571 units.

Key Concepts

Cost Equation

Understanding the cost equation is critical for businesses in managing their operations. This equation represents the total production cost of goods or services, combining both variable and fixed expenses. Specifically, the cost equation is typically formulated as Total Cost = Fixed Costs + (Variable Cost per Unit × Units Produced).

In practice, if a company has fixed monthly costs and variable costs that depend on the number of units produced, this equation helps determine the total cost for different production levels. The exercise provided illustrates how a company can use this cost equation to plan financially, ensuring that it does not exceed the available budget. By inserting the relevant figures into this formula, one can easily calculate the maximum production capacity within the financial constraint.

Variable Costs

Variable costs vary directly with the level of production. These costs rise as production increases and fall when production decreases. Examples include raw materials, packaging, and direct labor. In our exercise, the variable cost is given as \(8.75 per unit produced, which signifies that each additional unit brings an extra cost of \)8.75 to the company.

This poses a contrast to fixed costs, as variable costs fluctuate with production volume, thus significantly impacting the cost equation. Clear comprehension of variable costs is essential for setting the right price for products and making decisions on scaling production.

Fixed Costs

Fixed costs remain unchanged regardless of the number of units a company produces. These can include rent, salaries, insurance, and other overheads, which do not vary with the level of production. Understanding fixed costs is crucial because they comprise a significant portion of the total costs for many businesses.

In the exercise, the fixed monthly costs are $15,000, which must be paid regardless of production volume. It is essential to cover these costs to keep the business operational, and they are a key component of the cost equation. Accurately projecting fixed costs is imperative for budgeting and financial planning.

Units Produced

The term 'units produced' refers to the number of complete units of a product that a company manufactures. In the context of the cost equation, the units produced will determine the total variable costs since variable costs are incurred for each unit. The balance between the units produced and the costs associated with production is critical for achieving maximum profitability.

In the given exercise, calculating the number of units produced involves considering the maximum budget available for costs and solving the cost equation. The goal is to determine the greatest number of units that can be produced without exceeding the budget. In this case, the company can produce up to 8571 units before reaching the financial limit. Understanding production capacity is vital for operational planning and meeting market demand.