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 \(\$ 10,000\) and variable monthly costs of \(\$ 9.30\) per unit. The company has \(\$ 85,000\) available each month to cover costs.

Short answer

The maximum number of units the company can produce each month is approximately 8064.

Step-by-step solution

- Understanding the problem

The company’s total monthly cost consists of a fixed part and a variable part, depending on the quantity, denoted by \( q \), of the units produced. The formula then for the total cost is given by: \[ \text{TOTAL\_COST} = \text{FIXED\_COST} + \text{UNIT\_VARIABLE\_COST} \times q \]In this exercise, the total cost must not exceed the budget of \(\$ 85,000\) and we want to solve for \( q \), the quantity.

- Creating the cost equation

Plug the given terms into the equation where the fixed costs are \(\$ 10,000\) and variable cost per unit is \(\$ 9.30\).\[ \$ 85,000 = \$ 10,000 + \$ 9.30 \times q \]

- Solve for q

To solve for \( q \), we isolate \( q \) by subtracting \(\$ 10,000\) from both sides of the equation, and then dividing both sides by \(\$ 9.30\).\[ q = \frac{\$ 85,000 - \$ 10,000}{\$ 9.30} \]

Key concepts

Fixed Costs

Fixed costs are expenses that remain constant, no matter how many units a company produces. For example, rents, insurance, and salaries of permanent staff often make up a company's fixed costs. In our exercise, the company has a fixed monthly cost of $10,000. This amount does not change regardless of the production quantity. Such costs provide stability in financial planning, as businesses can expect to incur these expenses regularly. Fixed costs need to be managed efficiently to ensure that they do not put unnecessary financial strain on a company. By understanding fixed costs, companies can more effectively calculate how many units need to be sold to cover these expenses.

Variable Costs

Variable costs change according to the production volume. The more units produced, the higher the variable costs. In the given exercise, the variable cost is $9.30 per unit. This means for every additional unit produced, the company's expenses increase by $9.30.
This type of cost is often associated with the materials and labor directly used in production. Unlike fixed costs, variable costs can fluctuate and are more controllable by adjusting the production rate. By accurately predicting variable costs, businesses can plan their pricing strategies and production accordingly to maximize profit.

Budget Constraint

A budget constraint refers to the limit on the spending capacity of a business. In simple terms, it's the maximum amount a company can spend based on its financial resources. For the company in our exercise, this budget constraint is $85,000 per month. This value includes both fixed and variable costs.
Operating within a budget constraint ensures that a business doesn’t overspend which could lead to financial difficulties. Companies must make strategic decisions on how they allocate their budget between fixed and variable costs to efficiently run operations and remain profitable.

Systems of Equations

Systems of equations are pivotal in solving economic and financial problems, such as determining production limits while considering costs and budget constraints. In the exercise, the system equation was set up to determine how many units could be produced without exceeding the budget of $85,000.
The equation is structured as: \[ \text{Total Cost} = \text{Fixed Cost} + \text{Variable Cost per Unit} \times q \]By implementing algebraic methods, you solve for \( q \), giving the highest possible number of units that can be produced each month. Solving these types of problems helps businesses optimize production schedules in line with financial limits.