Problem 66 - Brief Applied Calculus

Question

BUSINESS: Cost Functions A company manufactures bicycles at a cost of \(\$ 55\) each. If the company's fixed costs are \(\$ 900\), express the company's costs as a linear function of \(x\), the number of bicycles produced.

Step-by-Step Solution

Answer

The cost function is \( C(x) = 55x + 900 \).

Step 1: Identify Variables and Constants

Let's identify the variables and constants in the problem. The variable here is the number of bicycles produced, which is denoted by \(x\). The cost per bicycle is a constant value of \(\\(55\) and the fixed costs are \(\\)900\).

Step 2: Understand the Linear Cost Function

A linear cost function can generally be written as \( C(x) = mx + b \), where \(m\) is the variable cost per unit, \(x\) is the number of units, and \(b\) is the fixed cost. Here, \(m = 55\) and \(b = 900\).

Step 3: Formulate the Cost Function

Using the linear cost function formula \( C(x) = mx + b \), substitute the values for \(m\) and \(b\). Thus, the cost function becomes \( C(x) = 55x + 900 \).

Step 4: Interpret the Cost Function

The cost function \( C(x) = 55x + 900 \) indicates that for every bicycle produced, the cost increases by \\(55, and there is a constant cost of \\)900 irrespective of the number of bicycles produced.

Key Concepts

Linear Function

A linear function is a mathematical expression that creates a straight line when plotted on a graph. It is represented in the form \( y = mx + b \), where:

• \(y\) is the dependent variable or the output.
• \(x\) is the independent variable or input.
• \(m\) is the slope of the line, indicating how much \(y\) changes for a unit change in \(x\).
• \(b\) is the y-intercept, the value of \(y\) when \(x=0\).

For a business like bicycle manufacturing, the linear function helps model costs that increase in a predictable way with each additional unit produced. In the given exercise, the linear cost function helps to calculate the total cost of producing a certain number of bicycles by applying the variable cost per bicycle and adding fixed costs. This approach simplifies the problem of linking production volume to total cost using straightforward algebra.

Fixed Costs

Fixed costs are business expenses that remain constant regardless of production volume. These costs do not change, whether a company produces one unit or a thousand units. Examples include rent, salaries, and machinery costs.In the context of our exercise with bicycle manufacturing, the fixed cost is \(\$900\). This is the cost the company incurs even if no bicycles are produced. In the cost function \(C(x) = 55x + 900\), the number \(900\) represents the fixed costs. These costs are important because they ensure the business can operate smoothly and cover basic operational expenses, regardless of production levels.

Variable Costs

Variable costs fluctuate with production volume, meaning they increase as more units are produced and decrease as fewer units are made. This is because they are directly tied to the output level.For example, in our bicycle manufacturing scenario, producing each bicycle incurs a cost of \(\\(55\). Therefore, if the company makes 5 bicycles, the variable cost is \(5 \times 55 = \\)275\). In the cost function \(C(x) = 55x + 900\), \(55\) is the variable cost per bicycle. Understanding variable costs is crucial for businesses as it helps them determine the marginal cost of production, set prices appropriately, and analyze profitability as production scales.