Problem 46 - Applied Calculus

Question

For the cost function \(C=1000+300 \ln q\) (in dollars), find the cost and the marginal cost at a production level of 500 . Interpret your answers in economic terms.

Step-by-Step Solution

Answer

The cost is $2864.38 and the marginal cost is $0.60, meaning each additional unit costs $0.60 more.

Step 1: Substitute production level into cost function

The cost function is given as \( C = 1000 + 300 \ln q \). We need to substitute \( q = 500 \) into this function to find the cost at this production level. Thus, the cost is calculated as follows: \[ C = 1000 + 300 \ln(500) \].

Step 2: Compute cost

Now, let's compute \( C = 1000 + 300 \ln(500) \). First, calculate \( \ln(500) \) which is approximately 6.2146. Now, substituting this into the cost function gives: \[ C = 1000 + 300 \times 6.2146 = 1000 + 1864.38 = 2864.38 \]. Thus, the cost at a production level of 500 is approximately $2864.38.

Step 3: Find the marginal cost function

The marginal cost is the derivative of the cost function with respect to \( q \). Thus, the marginal cost function is given by \( MC = \frac{dC}{dq} = \frac{d}{dq} [1000 + 300 \ln q] = 300 \cdot \frac{1}{q} \).

Step 4: Calculate marginal cost at production level 500

Now that we have the marginal cost function \( MC = \frac{300}{q} \), we substitute \( q = 500 \) into this function: \[ MC = \frac{300}{500} = 0.6 \]. Thus, the marginal cost at a production level of 500 is $0.60.

Step 5: Interpret the results economically

Economically, a cost of $2864.38 means that the total cost to produce 500 units is approximately $2864.38. The marginal cost of $0.60 indicates that producing one additional unit beyond the 500 units would increase the total cost by approximately $0.60.

Key Concepts

Marginal Cost

When discussing production, marginal cost (MC) is a crucial concept. It represents the cost of producing one additional unit of goods. To find the marginal cost, you need to take the derivative of the cost function with respect to the quantity produced, denoted as \( q \). In our example, the cost function is \( C = 1000 + 300 \ln q \). By differentiating this function, we find the marginal cost function: \( MC = 300 \cdot \frac{1}{q} \). This means that as production increases, the marginal cost will decrease due to the function's inverse relationship with production level.

• Marginal cost helps in decision-making, especially about how many additional units to produce.
• It varies with different levels of production and can indicate the most cost-effective level for production.

At a production level of 500 units, we compute the marginal cost by substituting \( q = 500 \) into the function, resulting in an MC of \( \\(0.60 \). This tells us that producing one more unit will cost an additional \\)0.60.

Production Level

The production level represents the quantity of goods being produced. In this exercise, we focus on a specific production level of 500 units. Understanding how production level interacts with cost functions is important as it helps managers and producers to optimize production output and minimize costs.

• In our exercise, the production level is mainly used to substitute into the cost function to find the total cost at that level.
• Changes in production level directly affect marginal costs, as seen in our marginal cost formula \( MC = \frac{300}{q} \).

Consider a scenario where increasing the production level from 500 to 501 affects both cost and marginal cost. This slight increase will give an insight into how production decisions can impact overall economic efficiency. Higher production levels can sometimes reduce average costs due to economies of scale, although this isn't directly evident with our logarithmic cost function.

Cost Calculation

Cost calculation involves determining the total production cost for a given quantity of goods. In the given exercise, this is done using the cost function \( C = 1000 + 300 \ln q \). To find the cost at a production level of 500 units, we substitute \( q = 500 \) into the function: \( C = 1000 + 300 \ln(500) \).

• First, calculate the natural logarithm of 500, which is approximately 6.2146.
• Then, substitute this value to compute the total cost as \( 1000 + 300 \times 6.2146 = 2864.38 \).

This calculation shows that the total cost to produce 500 units is approximately \$2864.38. Cost calculations like this are vital for businesses to budget, set prices, and decide on the scale of production. By understanding these costs, businesses can make more informed economic decisions.