Problem 105 - College Algebra Essentials

Question

In Exercises \(105-108,\) you will be developing functions that model given conditions. A company that manufactures bicycles has a fixed 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, \(x .\) Then find and interpret \(C(90)\)

Step-by-Step Solution

Answer

The total cost of producing 90 bicycles is $109,000.

Step 1: Writing the Function

Start by formulating the total cost function. The company's fixed cost is $100,000 and the cost to produce each bike is $100. Given that the total cost, \(C\), is the sum of fixed and variable costs, the function can be represented as \(C(x) = 100000 + 100x\). Where \(x\) is the number of bicycles made.

Step 2: Evaluate the Function

After formulating the function, proceed by substituting \(x = 90\) into the function to evaluate \(C(90)\). Thus, \(C(90) = 100000 + 100*90\).

Step 3: Calculate C(90)

Perform the calculation in the function to find the total cost when 90 bicycles are produced. So, \(C(90) = 100000 + 9000 = 109000\).

Key Concepts

Linear Functions

Linear functions are fundamental in algebra and play a vital role in various real-world scenarios, like calculating expenses in a business. A linear function is expressed in the form of the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope represents the rate of change between the independent variable \(x\) and dependent variable \(y\), while the y-intercept is the point where the line crosses the y-axis.

When we apply this to a business context, the linear function allows us to model relationships between different quantities, such as the total cost and the number of units produced. In our example exercise, the cost to produce each bicycle represents the slope (\(m = 100\)), and the fixed cost denotes the y-intercept (\(b = 100000\)). The linear nature of this function implies a constant increase in total cost with each additional bicycle produced - a perfect embodiment of a direct, proportional relationship.

Fixed and Variable Costs

Understanding fixed and variable costs is crucial in the analysis of a company's overall expenses. Fixed costs, such as rent or salaries, do not change with the level of goods produced. Contrastingly, variable costs are dependent on production output; the more you produce, the higher these costs are.

In our scenario, the company's fixed cost is depicted by \(\$100,000\), a value that remains constant regardless of the number of bicycles manufactured. Variable costs, on the other hand, depend on the number of bicycles produced (\(x\)) and the cost to produce each one (\$100). By dissecting the costs in this manner, managers can make informed decisions about pricing, budgeting, and even profitability thresholds. Moreover, a good grasp of these concepts can help students like you to solve problems relating to cost analysis intuitively and accurately.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. This concept becomes particularly practical when calculating real-world quantities, such as total costs in a business operation. To evaluate a function, you replace the independent variable with a given value and perform the necessary calculations to determine the dependent variable.

In the textbook exercise, evaluating the cost function \(C(x)\) at \(x = 90\) involves substituting 90 for \(x\) and following through with the arithmetic operation: \(C(90) = 100000 + 100 \cdot 90 = 109000\). This yields a total cost of \$109,000 for the production of 90 bicycles. Grasping this evaluation process is essential as it extends beyond the classroom; for instance, it is used to forecast financial scenarios and to analyze the impact of scaling production on a company's expenses.