Problem 66 - 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 \(C(x) = 900 + 55x\).

Step 1: Understanding Fixed and Variable Costs

The fixed costs are those expenses that do not change with the number of items produced. In this problem, the fixed cost is given as $900. The variable cost depends on the number of bicycles produced, which is $55 per bicycle.

Step 2: Setting Up the Cost Function

A cost function is generally expressed as: \[ C(x) = ext{Fixed Costs} + ext{Variable costs per unit} \times ext{Number of units} \] For this problem: \[ C(x) = 900 + 55x \] where \(x\) is the number of bicycles produced.

Step 3: Interpreting the Linear Function

In the cost function \( C(x) = 900 + 55x \), \(900\) represents the fixed cost, and \(55x\) represents the total variable cost, where \(55 is the cost to produce each bicycle multiplied by the number of bicycles \(x\). This shows that as \(x\) increases, the total cost \(C(x)\) will increase by steps of \)55.

Key Concepts

fixed costs

Fixed costs refer to expenses that a company has to pay regularly, regardless of how many products it produces. They are constant and do not change with the level of output. Examples of fixed costs include rent, salaries of permanent staff, and equipment depreciation. In the context of this bicycle manufacturing problem, the company incurs fixed costs of $900. This means that even if no bicycles are produced, the company will still have to pay $900. By understanding fixed costs, companies can better manage their budgeting since these costs remain constant over different production levels.

variable costs

Variable costs change directly with the production volume. The more units you produce, the higher the total variable costs. They are often associated with manufacturing activities, such as materials and labor costs per unit. For the bicycle company, the variable cost is $55 per bicycle. This means that if the company produces one bicycle, it will incur $55 as a variable cost, two bicycles will be $110, and so on. Variable costs are crucial to grasp because they impact the decision-making process for pricing and production strategies.

linear functions

Linear functions are mathematical expressions that describe a straight-line relationship between two variables. In business, linear functions are used to model relationships such as costs or revenue over time. The cost function for this bicycle company is a linear function, as it takes the form \( C(x) = 900 + 55x \). This equation shows a linear relationship because the cost increases by the same amount for each additional bicycle produced. Linear functions are valuable in helping predict costs and profits based on production changes.

cost function expression

A cost function expression gives a mathematical representation of how total costs change with production levels. It combines fixed and variable costs into a single formula. For this problem, the cost function is \( C(x) = 900 + 55x \), where \( C(x) \) is the total cost when producing \( x \) bicycles. By using this expression, companies can easily calculate the expected cost for producing any number of units. This is particularly important for financial planning, allowing businesses to estimate expenses and decide on production and pricing strategies more effectively. They can determine their break-even point and understand how changes in production will influence their overall financial performance.