Problem 91 - College Algebra

Question

You will be developing functions that model given conditions. A company that manufactures bicycles has a tixed cost of \(\$ 100,000 .\) It costs \(\$ 100\) to produce each bicycle. The total cost for the company is the sum of its fixed cost and variable costs. Write the total cost, \(C\), as a function of the number of bicycles produced. Then find and interpret

Step-by-Step Solution

Answer

The total cost function for the company is \( C(x) = 100,000 + 100x \). This means, to manufacture \(x\) number of bicycles the company will need to spend $100,000 in fixed costs and $100 for each bicycle manufactured. Therefore, if the company were to manufacture 10 bicycles, the total cost would be $101,000.

Step 1: Create the Cost Function

Given that the fixed cost is $100,000, and the cost to produce each bicycle is $100, the total cost \(C\) for the company, as a function of the number of bicycles \(x\) produced, can be modeled as:\( C(x) = 100,000 + 100x \). This equation represents that for every bicycle made, the cost increases by $100.

Step 2: Interpret the Cost Function

The function \( C(x) = 100,000 + 100x \) shows the total cost at any number of bicycles produced. For example, for ten bicycles, you'd have the cost as \( C(10) = 100,000 + 100*10 = 101,000 \). So, making ten bicycles would cost the company $101,000 in total.

Key Concepts

Understanding Fixed Costs

Fixed costs are expenses that do not change with the number of goods or services produced. These costs remain constant regardless of the company's activity level. In our bicycle company example, the fixed cost is $100,000. This means that whether the company makes one bicycle or thousands, it still incurs $100,000 in costs.
Fixed costs typically include items such as rent, salaries of permanent employees, and insurance.

• They are predictable and consistent.
• Do not change with production levels.

Understanding fixed costs is important because they help businesses determine how much needs to be sold to cover all costs and start making a profit.

Exploring Variable Costs

Variable costs change based on the number of items produced. In the bicycle manufacturer's case, the variable cost is $100 per bicycle. This means that for every additional bicycle made, the cost goes up by $100.
Variable costs can include materials, labor, and other expenses that fluctuate with production.

• They increase as production increases.
• They decrease as production decreases.

This concept allows businesses to estimate how changes in production levels will affect overall costs, playing a crucial role in budgeting and financial forecasting.

Linear Functions in Cost Analysis

Linear functions are mathematical tools that represent relationships with a constant rate of change, often appearing as a straight line when graphed. In the context of cost functions, a linear function helps us understand how costs increase as production increases.
Our example provides the function: \( C(x) = 100,000 + 100x \). This formula indicates that for every bicycle produced, the cost rises by \(100, lining up perfectly with a basic linear function model.

• The slope (100) represents the increase in cost per unit – in this case, per bicycle.
• The y-intercept (\)100,000) is the starting point where no bicycles are produced.

Recognizing how linear functions apply to cost models assists in predicting expenditure and strategically planning for different levels of business activity.