Problem 55 - Applied Calculus

Question

The cost to produce \(q\) items is \(C(q)=1000+2 q^{2}\) dollars. Find the marginal cost of producing the \(25^{\text {th }}\) item. Interpret your answer in terms of costs.

Step-by-Step Solution

Answer

The marginal cost of the 25th item is 100 dollars, indicating an increase in cost by this amount when the 25th item is produced.

Step 1: Understand Marginal Cost

Marginal cost is the additional cost of producing one more unit. Mathematically, it is the derivative of the cost function \(C(q)\) with respect to \(q\), which represents the rate of change of cost with respect to quantity.

Step 2: Calculate Derivative of Cost Function

Given the cost function \(C(q) = 1000 + 2q^2\), we need to find \(\frac{dC}{dq}\). Differentiate the function to get: \(\frac{dC}{dq} = 4q\). This derivative represents the marginal cost in terms of \(q\).

Step 3: Evaluate Derivative at q = 25

To find the marginal cost of producing the 25th item, substitute \(q = 25\) into the derivative: \(\frac{dC}{dq}\bigg|_{q = 25} = 4 \times 25 = 100\).

Step 4: Interpret the Result

The marginal cost at \(q = 25\) is 100 dollars. This means that producing the 25th item increases the total production cost by approximately 100 dollars.

Key Concepts

Derivative of Cost Function

To understand the concept of marginal cost, we first need to grasp the idea of the derivative of a cost function. The cost function, usually expressed as \(C(q)\), tells us the total cost of producing \(q\) items. In the given exercise, the cost function is \(C(q) = 1000 + 2q^2\). Here, the coefficient "1000" is a fixed cost that remains constant regardless of the number of items produced, while \(2q^2\) represents the variable cost, which changes as the quantity changes. The derivative of a function provides a way to measure its rate of change, and for a cost function, the derivative with respect to \(q\) (the quantity of items) is crucial. Differentiating \(C(q)\) with respect to \(q\) gives \(\frac{dC}{dq} = 4q\). This derivative tells us how the cost \(C\) changes as the quantity \(q\) varies.

• The derivative is essentially the slope of the cost function at any point \(q\).
• By understanding this slope, we can determine how much additional cost is incurred by producing one more unit.

By calculating this derivative, we gain valuable insight into the cost behavior.

Rate of Change

The rate of change is a fundamental concept in calculus, and in the context of a cost function, it helps us understand how costs vary with changes in production levels. Specifically, the derivative \(\frac{dC}{dq}\) indicates the rate at which production costs increase as more items are produced.In simpler terms, imagine the cost function as a road map through the cost landscape:

• The rate of change tells us the steepness of the road at any point \(q\).
• A steeper road (higher derivative value) suggests a faster increase in costs as production ramps up.

For the given function \(C(q) = 1000 + 2q^2\), when the derivative \(\frac{dC}{dq} = 4q\) is evaluated at \(q = 25\), it means that at this stage of production, every additional unit produced results in an additional cost of 100 dollars. This rate of change is invaluable for businesses as they plan production strategies, ensuring they manage costs effectively at different production levels.

Production Cost Analysis

Production cost analysis involves examining how different factors influence the total cost of production. Businesses use this analysis to make strategic decisions about pricing, production, and profitability. In the given scenario, comprehending the marginal cost is crucial for understanding how costs behave as production scales. Marginal cost, particularly at specific levels of production such as the 25th unit in our exercise, provides a snapshot of the cost implications of scaling up production by one more unit. Doing this type of analysis:

• Helps businesses determine the most cost-efficient points in production.
• Guides pricing strategies by understanding the minimum price needed to cover additional costs.

For the exercise in question, the marginal cost of producing the 25th item is 100 dollars. This insight is important because:

• It alerts the company to the cost burden of each additional unit at higher levels of production.
• It assists in deciding whether increasing production will be profitable based on forecasted revenue from sales.

Thus, analyzing production costs through marginal cost and cost function derivatives empowers businesses to make informed financial decisions.