Problem 1 - Applied Calculus

Question

It costs 4800 to produce 1295 items and it costs \(\$ 4830\) to produce 1305 items. What is the approximate marginal cost at a production level of 1300 items?

Step-by-Step Solution

Answer

The approximate marginal cost is $3 per item.

Step 1: Understanding the Concept of Marginal Cost

The marginal cost refers to the additional cost incurred to produce one more unit of a good or service. In this context, we need to determine how much extra cost is associated with producing one additional item when the production level increases from 1295 items to 1305 items.

Step 2: Calculate the Change in Total Cost

Calculate the change in total cost when production increases from 1295 to 1305 items. The formula to find the change in cost is:\[ \Delta C = C_2 - C_1 \]where \(C_2 = 4830\) (cost to produce 1305 items) and \(C_1 = 4800\) (cost to produce 1295 items). Substitute the values to find \(\Delta C\).

Step 3: Calculate the Change in Quantity

Determine the change in the quantity of items produced when production increases from 1295 to 1305. The formula is:\[ \Delta Q = Q_2 - Q_1 \]where \(Q_2 = 1305\) and \(Q_1 = 1295\). Substitute the values to find \(\Delta Q\).

Step 4: Compute the Marginal Cost

The marginal cost is calculated using the following formula:\[ MC = \frac{\Delta C}{\Delta Q} \]where \(\Delta C\) is the change in total cost calculated in Step 2, and \(\Delta Q\) is the change in quantity determined in Step 3. Substitute these values into the formula to find the marginal cost.

Key Concepts

Change in Total Cost

The change in total cost is essential to assess the additional financial burden when increasing production. This refers to the difference in costs required to produce two different quantities of items. In our exercise example, we have two production cost points: \(4800 for 1295 items and \)4830 for 1305 items. By finding the difference between these two costs, we determine how much extra money is needed for increased production.

The change in total cost formula is simply:

• \( \Delta C = C_2 - C_1 \)

Where \( \Delta C \) represents the change in total cost, \( C_2 \) is the cost for manufacturing 1305 items, and \( C_1 \) is the cost for 1295 items.
Using the numbers from the exercise, \( \Delta C = 4830 - 4800 = 30 \). Thus, the change in total cost is \(30, indicating that producing 10 additional items costs an extra \)30.

Change in Quantity

Change in quantity illustrates how the number of items produced varies as production levels shift. Finding this difference helps in understanding the extent of change in the production process. In our example, we are looking at production levels of 1295 items initially and 1305 items after increasing production.

The change in quantity can be calculated using this straightforward formula:

• \( \Delta Q = Q_2 - Q_1 \)

Where \( \Delta Q \) stands for the change in quantity, \( Q_2 \) is the final quantity (1305 items), and \( Q_1 \) is the initial quantity (1295 items).
Plugging these values into the formula, we find \( \Delta Q = 1305 - 1295 = 10 \). So, the quantity has increased by 10 items. Understanding this change is crucial for calculating marginal costs accurately.

Cost Function Analysis

Cost function analysis is essential for understanding how costs behave in relation to changes in production. It involves examining how different cost components interact as the quantity produced adjusts. In this case, cost function analysis helps us determine the most crucial piece of information: the marginal cost, which signifies how the cost of production changes with each additional unit produced.

To figure out the marginal cost (MC), we divide the change in total cost by the change in quantity. Our formula for marginal cost is:

• \( MC = \frac{\Delta C}{\Delta Q} \)

Substitute \( \Delta C = 30 \) (from our change in total cost) and \( \Delta Q = 10 \) (from our change in quantity) into the formula. The calculation becomes:

• \( MC = \frac{30}{10} = 3 \)

Therefore, the approximate marginal cost is \(3 per item. This indicates that on average, producing one more item costs an additional \)3. Understanding these insights into cost behaviors is critical for businesses striving to optimize their production strategies and manage costs effectively.