Problem 4 - Applied Calculus

Question

The cost of producing \(q\) items is \(C(q)=2500+12 q\) dollars. (a) What is the marginal cost of producing the \(100^{\text {th }}\) item? the \(1000^{\text {th }}\) item? (b) What is the average cost of producing 100 items? 1000 items?

Step-by-Step Solution

Answer

(a) The marginal cost for both 100th and 1000th items is \(12\) dollars. (b) The average cost for 100 items is \(37\) dollars, and for 1000 items is \(14.5\) dollars.

Step 1: Understanding the Problem

We are asked to find the marginal cost of producing the 100th and 1000th items and the average cost of producing 100 and 1000 items. The cost function given is \( C(q) = 2500 + 12q \).

Step 2: Finding the Marginal Cost

The marginal cost is the derivative of the cost function with respect to \( q \). Differentiate \( C(q) = 2500 + 12q \), which gives \( C'(q) = 12 \). This implies that the marginal cost remains constant at 12 dollars for each item, regardless of the specific number of items produced.

Step 3: Answering Part (a) - Marginal Cost Calculations

The marginal cost for the 100th item and the 1000th item is \( C'(100) = 12 \) and \( C'(1000) = 12 \) respectively, as the marginal cost is constant across quantities.

Step 4: Finding the Average Cost Formula

The average cost \( AC(q) \) is defined as the total cost divided by the number of items, or \( AC(q) = \frac{C(q)}{q} = \frac{2500 + 12q}{q} \).

Step 5: Calculating Average Cost for 100 Items

Substitute \( q = 100 \) into the average cost formula: \( AC(100) = \frac{2500 + 12 \times 100}{100} = \frac{2500 + 1200}{100} = \frac{3700}{100} = 37 \) dollars per item.

Step 6: Calculating Average Cost for 1000 Items

Substitute \( q = 1000 \) into the average cost formula: \( AC(1000) = \frac{2500 + 12 \times 1000}{1000} = \frac{2500 + 12000}{1000} = \frac{14500}{1000} = 14.5 \) dollars per item.

Key Concepts

Average Cost

When we talk about average cost, we're looking at how much it costs, on average, to produce each item of a product. This is calculated by taking the total cost of production and dividing it by the number of items produced.
In our equation, the average cost is represented by \( AC(q) = \frac{C(q)}{q} \), where \( C(q) \) is the cost function and \( q \) is the quantity of items. For example, if the total cost of making 100 items is 3700 dollars, then the average cost is \( \frac{3700}{100} = 37 \) dollars per item.
Similarly, when producing 1000 items, the total cost is 14500 dollars, and the average cost becomes \( \frac{14500}{1000} = 14.5 \) dollars per item.
The average cost is a helpful metric for businesses as it shows how scale affects production costs. As you can see from the example, producing more items reduces the average cost per item.

Cost Function

The cost function is a mathematical expression that businesses use to understand their production costs relative to the number of units produced. In our example, the cost function is \( C(q) = 2500 + 12q \).
This function reveals a fixed cost of 2500 dollars and a variable cost of 12 dollars for each additional item produced.

• **Fixed Cost**: This is the component of the total cost that remains constant regardless of the quantity produced. In our example, this is the 2500 dollars.
• **Variable Cost**: This varies with the production level, costing an additional 12 dollars per item in our scenario.

Mainly, the cost function helps to predict how much it will cost to produce a certain number of items, which is crucial for setting price points and profit margins. Understanding this function helps businesses manage costs more effectively.

Derivatives

Derivatives come into play in calculus and are a foundational concept when dealing with changing quantities. When linked to cost functions, derivatives help identify the rate at which costs change with respect to the production level.
The derivative of the cost function, known as the marginal cost, tells us how much the cost will increase for each additional unit produced.
In our example, by taking the derivative of the cost function \( C(q) = 2500 + 12q \), we get \( C'(q) = 12 \). This means the marginal cost is constant at 12 dollars per item.
In practical terms, derivatives provide valuable insights for decision making, helping businesses understand the impact on cost when adjusting production levels. They allow companies to optimize production to meet cost efficiency targets, ensuring the cost of producing one more item does not outweigh potential revenues.