Problem 65 - Applied Calculus

Question

A lumberyard will deliver wood for \(\$ 4\) per board foot plus a delivery charge of \$20. Find a function \(C(x)\) for the cost of having \(x\) board feet of lumber delivered.

Step-by-Step Solution

Answer

The cost function is \( C(x) = 4x + 20 \).

Step 1: Understanding the Problem

We need to find the cost equation, which is composed of two parts: a variable cost per board foot and a fixed delivery charge. The variable cost is $4 per board foot and the fixed delivery charge is $20.

Step 2: Defining the Variables

Let \( x \) be the number of board feet of wood delivered. We are tasked with finding \( C(x) \), the total cost function of delivering \( x \) board feet.

Step 3: Formulating the Function

The total cost \( C(x) \) is composed of the variable cost \( 4x \) plus the fixed delivery charge of \$20. This is expressed as the equation \( C(x) = 4x + 20 \).

Key Concepts

Linear Equation

The total cost of delivering lumber can be represented through a linear equation. A linear equation forms a straight line when graphed and follows the standard format of:

• \(y = mx + b\), where
• \(m\) is the slope of the line, and
• \(b\) is the y-intercept, or the point where the line crosses the y-axis.

In our problem, the total cost function \(C(x) = 4x + 20\) fits the structure of a linear equation.

• The coefficient \(4\) represents the slope, meaning for each additional board foot of lumber, the total cost increases by \\(4.
• The constant term \(20\) represents the fixed delivery charge, showcasing the y-intercept. It tells us that even if no lumber is purchased, the delivery charge of \\)20 applies.

Linear equations are extensively used in real-life contexts, such as predicting profits, determining costs like this example, or calculating distances over time.

Variable Cost

Variable costs are expenses that vary directly with the level of output or activity. They increase or decrease depending on how much is produced or sold.
In our lumber delivery scenario, the variable cost is \\(4 per board foot of lumber. Thus, the cost will change based on the number of board feet ordered.
This direct relationship is expressed as:

• The term \(4x\) in the cost function \(C(x) = 4x + 20\).
• Where \(x\) denotes the number of board feet.
• Each additional foot adds \\)4 to the total cost.

Variable costs allow businesses to determine pricing strategies, manage budgets, and make efficient production decisions. Understanding how these costs affect total expenses is crucial for effective financial planning.

Fixed Cost

Fixed costs are expenses that do not change with the level of goods or services produced. They remain constant, no matter how much or how little is produced.
In this case, the fixed cost is a delivery charge of \$20. This cost will be the same whether one board foot or many hundreds of board feet are ordered.
The expression of this in our equation is:

• The term \(+20\) in the cost equation \(C(x) = 4x + 20\).

Understanding fixed costs helps in calculating the break-even point of a business, predicting profitability, and controlling expenses. Recognizing fixed costs allows businesses to better forecast budgets and manage long-term financial planning.