Problem 65 - Brief Applied Calculus

Question

BUSINESS: Cost Functions 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: Understand the Problem

We need to derive a function that represents the total cost of purchasing and delivering a certain number of board feet of lumber, defined as \(x\). The cost involves both a variable cost and a fixed cost.

Step 2: Identify the Fixed Cost

The fixed cost is the delivery charge which is a constant amount of \(\$20\). This fee is incurred regardless of how much wood is delivered.

Step 3: Identify the Variable Cost

The variable cost is associated with the quantity of wood ordered. It costs \(\$4\) per board foot of lumber. Thus, if \(x\) board feet of lumber are ordered, the cost will be \(4x\) dollars.

Step 4: Formulate the Cost Function

The total cost function \(C(x)\) includes both the fixed cost and the variable cost. Therefore, we can express it as: \[ C(x) = 4x + 20 \] where \(4x\) is the cost dependent on the number of board feet and \(20\) is the delivery charge.

Key Concepts

Variable Cost

When we think of variable costs, we're talking about expenses that change depending on how much of a product or service is produced. In the context of our lumberyard example, this cost varies based on the number of board feet of lumber you order.
For every board foot, you will pay \(4, so if you order more wood, your cost increases. Mathematically, we represent this by multiplying the cost per unit—here, \)4 per board foot—by the number of units, which is denoted by 'x'.
That's why the variable cost is expressed as \(4x\). The key thing to remember is this cost is directly proportional to the amount of product purchased.

Fixed Cost

Now, onto fixed costs. These are costs that remain constant no matter how much service or product is consumed. In our scenario, the fixed cost is the delivery charge of $20.
Notice how, unlike variable costs, the fixed cost doesn't change if you order one board foot or thousands. Every order, regardless of size, incurs the same flat fee.
This is crucial to understanding cost estimates, as it allows businesses to predict a base level of expenses regardless of sales volume.

• Example: Whether you order 1 board foot or 100 board feet, the delivery remains $20.

Mathematical Modeling

Mathematical modeling is essentially about creating a mathematical representation of a real-world situation. For the lumberyard scenario, we create a model to calculate total costs based on different quantities of wood ordered. This is done using a cost function, which combines both fixed and variable costs.
In our example, the function \(C(x) = 4x + 20\) models the total cost \(C(x)\) as a function of the number of board feet \(x\).
The formula shows that total cost consists of two parts: the cost for the wood (\(4x\)), and the flat delivery charge (20). This model allows us to easily calculate the total cost for any given number of board feet, making it a powerful tool for budgeting and planning.
This is why mathematical modeling is so powerful—it takes complex real-world concepts and simplifies them into easy-to-use equations.