Problem 53 - Calculus and its applications

Question

Preston's Leatherworks finds that the cost, in dollars, of producing \(x\) belts is given by \(C(x)=750+34 x-0.068 x^{2} .\) Find the rate at which average cost is changing when 175 belts have been produced.

Step-by-Step Solution

Answer

The rate of change of average cost when 175 belts are produced is approximately \(-0.0925\) dollars per belt.

Step 1: Understanding the Problem

We need to find how the average cost of producing 175 belts is changing. The total cost function given is \( C(x) = 750 + 34x - 0.068x^2 \), and the average cost \( AC(x) \) is calculated as the total cost divided by the number of units, \( x \). Thus, \( AC(x) = \frac{C(x)}{x} \).

Step 2: Derive the Average Cost Function

Using the expression \( AC(x) = \frac{C(x)}{x} \), substitute the cost function to get: \[ AC(x) = \frac{750 + 34x - 0.068x^2}{x} = \frac{750}{x} + 34 - 0.068x \].

Step 3: Differentiate the Average Cost Function

To find the rate of change of the average cost, differentiate \( AC(x) \) with respect to \( x \). This yields: \[ AC'(x) = \frac{d}{dx} \left( \frac{750}{x} + 34 - 0.068x \right) = -\frac{750}{x^2} - 0.068 \].

Step 4: Calculate the Rate of Change at 175 Belts

Substitute \( x = 175 \) into the derivative to find the rate of change of the average cost at that point: \[ AC'(175) = -\frac{750}{175^2} - 0.068 \].

Step 5: Compute the Final Answer

Calculate \( AC'(175) \): \[ AC'(175) = -\frac{750}{30625} - 0.068 = -0.0245 - 0.068 = -0.0925 \]. Thus, the rate at which the average cost is changing when 175 belts have been produced is approximately \(-0.0925\) dollars per belt.

Key Concepts

Cost Function

In economics and business, the cost function is vital for understanding expenses linked with producing goods or services. It provides a mathematical model explaining how the total cost changes as the production volume changes. For Preston's Leatherworks, the cost function given is \( C(x) = 750 + 34x - 0.068x^2 \). This particular function has three components:

• **Fixed Costs**: These are expenses that do not change with the production level. In this case, it is $750.
• **Variable Costs**: The component \( 34x \) indicates costs varying linearly with the number of belts produced.
• **Quadratic Costs**: The term \( -0.068x^2 \) shows the non-linear element of the cost, affecting the overall cost as production increases significantly beyond typical output (potentially due to inefficiencies, bulk discounts, or increased labor costs).

Understanding the cost function's elements is crucial for pricing strategies, budgeting, and making investment or production scaling decisions.

Rate of Change

The rate of change in this context refers to how quickly the average cost per belt is changing as more are produced. To discover this rate, we must understand the differentiation process of our average cost function \( AC(x) \). The average cost function is obtained by dividing the total cost \( C(x) \) by the number of belts \( x \), giving us \( AC(x) = \frac{750}{x} + 34 - 0.068x \).

• The rate of change can tell us whether producing additional units becomes cheaper or costlier in terms of average expenditure.
• A negative rate means the average cost decreases with more production, which is favorable in scaling operations.
• A positive rate could be a signal to reassess production strategies.

In our example, a negative rate of change at 175 belts produced indicates that producing belts is becoming cheaper on average.

Differentiation

Differentiation is a powerful mathematical tool used to determine how a particular function changes over time. In this exercise, we differentiate the average cost function \( AC(x) \) with respect to its variables to find \( AC'(x) \). This derivative helps us pinpoint the rate at which the average cost changes:

• By differentiating \( AC(x) = \frac{750}{x} + 34 - 0.068x \), we derive \( AC'(x) = -\frac{750}{x^2} - 0.068 \).
• The derivative consists of two key terms: \( -\frac{750}{x^2} \) and the constant \( -0.068 \).
• \( -\frac{750}{x^2} \) suggests how rapidly fixed costs dilute as production scales, while \( -0.068 \) relates to the diminishing returns on production costs.

This derivative gives a specific rate of change for any level of production, in our case, at 175 belts produced the cost reduces by about 9.25 cents per belt.