Problem 57 - Calculus

Question

The cost of planting seed is usually a function of the number of acres sown. The cost of the equipment is a fixed cost because it must be paid regardless of the num- ber of acres planted. The costs of supplies and labor vary with the number of acres planted and are called variable costs. Suppose the fixed costs are \(\$ 10,000\) and the variable costs are \(\$ 200\) per acre. Let \(C\) be the total cost, measured in thousands of dollars, and let \(x\) be the number of acres planted. (a) Find a formula for \(C\) as a function of \(x\) (b) Graph \(C\) against \(x\) (c) Which feature of the graph represents the fixed costs? Which represents the variable costs?

Step-by-Step Solution

Answer

(a) \(C = 10 + 0.2x\). (b) Line graph. (c) Y-intercept represents fixed costs; slope represents variable costs.

Step 1: Define the cost components

To find the total cost, we need to consider both fixed and variable costs. The fixed cost is a one-time charge of \\(10,000, which is constant regardless of the number of acres. The variable cost depends on the number of acres, costing \\)200 per acre.

Step 2: Establish the cost formula

The total cost \(C\) can be expressed as the sum of the fixed cost and the variable cost multiplied by the number of acres \(x\). Hence, the formula is:\[C = 10 + 0.2x\]This formula represents the cost \(C\) in thousands of dollars, where \\(10,000 is the fixed cost and \\)0.2 (which is \$200 in thousands) is the cost per acre multiplied by the number of acres \(x\).

Step 3: Graph the cost function

To graph the cost function \(C = 10 + 0.2x\), use a graphing tool or plot points manually. Start with \(x = 0\), giving \(C = 10\), and plot this on the graph. Gradually increase \(x\) and plot additional points, such as \(x = 10\) giving \(C = 12\), and so forth. Draw a straight line through these points because the relationship between \(C\) and \(x\) is linear, due to the fixed and variable components.

Step 4: Identify graph features

On the graph, the y-intercept at \(C = 10\) represents the fixed costs since it is the cost when zero acres are planted. The slope of the line, which is 0.2, represents the variable cost per acre. This slope indicates how much the total cost \(C\) increases for each additional acre planted.

Key Concepts

Fixed Costs

Fixed costs are expenses that remain constant, regardless of how much you produce or sell. In the context of our problem, the fixed cost is the infrastructural cost associated with owning equipment used for planting seeds. In simpler terms, it's the cost you incur even if no acres are planted.
The fixed cost in our problem is $10,000. This amount does not change whether you plant one acre or a hundred. It's predetermined before any work commences, making it predictable and easy to budget for.

• Example: Leasing of land, purchase of a tractor.
• Fact: Fixed costs are crucial in breaking down total costs.

The key takeaway is that fixed costs do not vary with activity level and form the base of our cost structure.

Variable Costs

Variable costs, unlike fixed costs, change with the level of activity—the more you produce, the higher they get. In the exercise, these costs come from labor and supplies for planting, which fluctuate based on the number of acres addressed.
In our example, the variable cost is $200 per acre. Thus, as more acres are planted, this cost increases incrementally by this specific amount for each additional acre.

• Example: Seed purchases, hourly wages for workers.
• Important fact: Variable costs are directly proportional to the scale of operation.

Variable costs allow businesses to maintain flexibility. During lower demand periods, these costs naturally decrease as production is scaled back.

Graphing Linear Functions

Graphing a linear function helps visualize the relationship between two variables in a linear equation. In our case, the relationship between the total cost \( C \) and the number of acres planted \( x \) is depicted by a straight line on a graph.
To start graphing, plot the y-intercept, which is the fixed cost (point where the line crosses the y-axis at \( C = 10 \)). Then, use the slope (0.2) to determine how the line increases with each additional acre (each rise of 1 in \( x \) results in a 0.2 or \$200 increase in \( C \)).

• Y-intercept represents no planting activity but fixed costs incurred.
• Slope indicates variable cost, showing how rapidly costs increase.

Graphing linear functions allows for easy visualization of the cost structure and quick identification of fixed and variable cost components.

Linear Equation

A linear equation is a mathematical expression that creates a straight line when graphed. It is usually written in the form \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
In our exercise, the linear equation \( C = 10 + 0.2x \) incorporates both fixed and variable costs. Here, \( C \) is our dependent variable (total cost) and \( x \) is the independent variable (acres). The slope 0.2 represents the variable cost per acre, while the y-intercept 10 represents fixed costs, calculated in thousands.

• Constant change: As \( x \) changes, \( C \) changes linearly.
• Real-world efficacy: Simplifies cost estimation processes.

Linear equations are essential in interpreting relationships between quantities, especially in budgeting and financial forecasting.