Problem 5 - Applied Calculus

Question

The cost function is \(C(q)=1000+20 q .\) Find the marginal cost to produce the \(200^{\text {th }}\) unit and the average cost of producing 200 units.

Step-by-Step Solution

Answer

Marginal cost is 20; average cost for 200 units is 25.

Step 1: Understanding the Problem

We need to calculate two things: the marginal cost of producing the 200th unit, and the average cost of producing 200 units, given the cost function \(C(q)=1000+20q\).

Step 2: Calculating Marginal Cost

The marginal cost is the derivative of the cost function \(C(q)\). The cost function is \(C(q) = 1000 + 20q\). The derivative of \(C(q)\) with respect to \(q\) is \(C'(q) = 20\). Therefore, the marginal cost of producing any unit, including the 200th, is \(20\).

Step 3: Calculating Average Cost

Average cost is calculated as the total cost \(C(q)\) divided by the number of units \(q\). At \(q = 200\), the total cost is \(C(200) = 1000 + 20 \times 200 = 5000\). Thus, the average cost \(AC\) is \(\frac{C(200)}{200} = \frac{5000}{200} = 25\).

Key Concepts

Cost Function

A cost function represents the total cost a company incurs in producing goods. This function expresses the relationship between the cost (C) and the quantity of products produced (q). In our exercise, the cost function is denoted by \( C(q) = 1000 + 20q \). Here, \( 1000 \) can be understood as a fixed cost—expenses that do not change regardless of the number of items produced, like rent for a factory or salaries of permanent staff.
On the other hand, \( 20q \) represents the variable costs, which fluctuate depending on production levels, such as the cost of raw materials and labor. The multiplicative factor 20 shows the cost increase per additional unit produced.
Understanding this function helps businesses estimate how changes in production levels affect total costs, allowing them to optimize their manufacturing strategies to minimize unnecessary expenses.

Derivative

The derivative is a powerful mathematical tool used to determine how a function changes at any given point. It tells us the rate at which one quantity changes in relation to another. In cost analysis, derivatives help in understanding the marginal cost, which is the additional cost to produce one more unit.
When we have a linear cost function like \( C(q) = 1000 + 20q \), finding its derivative \( C'(q) \) is straightforward. The derivative represents the rate at which costs increase with each additional unit, often referred to as the slope of the cost curve.
In this case, since \( C'(q) = 20 \), the cost increases by a constant amount—20 units—for each additional product manufactured. This simplicity makes linear functions easy to understand and predict, providing valuable insights for production planning.

Average Cost

Average cost is a measure of the overall cost per unit produced, offering insight into the efficiency of production. It is calculated by dividing the total cost \( C(q) \) by the quantity of products \( q \). In our example, for \( 200 \) units, the average cost gives a useful snapshot of how resources are being utilized across different units.
At \( q = 200 \), the total cost comes out to \( 5000 \). Dividing this number by the number of units (200) provides the average cost as \( \frac{5000}{200} = 25 \). This means each unit, on average, incurs a cost of 25 units.
Understanding average cost is crucial for businesses to set competitive pricing strategies. By comparing this figure with the selling price, companies can determine profitability and make informed financial and operational decisions.