Problem 58 - Calculus Early Transcendentals: Pearson New International Edition

Question

The total cost of producing and selling \(x\) units of Xbars per month is \(C(x)=100+3.002 x-0.0001 x^{2}\). If the production level is 1600 units per month, find the average cost, \(C(x) / x\), of each unit and the marginal cost.

Step-by-Step Solution

Answer

Average cost at 1600 units is $2.9045; marginal cost is $2.682.

Step 1: Understand the Total Cost Function

The total cost function is given by: \[ C(x) = 100 + 3.002x - 0.0001x^2 \] This function describes the total cost incurred when producing \(x\) units of Xbars. We need to find both the average cost for producing 1600 units and the marginal cost at that production level.

Step 2: Calculate Average Cost at 1600 Units

First, substitute \( x = 1600 \) into the total cost function to find \( C(1600) \).\[ C(1600) = 100 + 3.002(1600) - 0.0001(1600)^2 \]Calculate the above expression to find the total cost:- \( 3.002 \times 1600 = 4803.2 \)- \( 0.0001 imes 1600^2 = 256 \)- Then, \( C(1600) = 100 + 4803.2 - 256 = 4647.2 \)Next, find the average cost by dividing the total cost by the number of units:\[ \text{Average Cost}(1600) = \frac{4647.2}{1600} = 2.9045 \]

Step 3: Determine the Marginal Cost Function

The marginal cost is the derivative of the total cost function \( C(x) \). Differentiate the total cost function with respect to \( x \):\[ C'(x) = \frac{d}{dx}(100 + 3.002x - 0.0001x^2) = 3.002 - 0.0002x \]

Step 4: Calculate Marginal Cost at 1600 Units

Substitute \( x = 1600 \) into the marginal cost function to find \( C'(1600) \):\[ C'(1600) = 3.002 - 0.0002(1600) \]Solve for the marginal cost:\[ C'(1600) = 3.002 - 0.32 = 2.682 \]

Key Concepts

Average Cost Calculation

Calculating the average cost involves determining what the cost per unit is when you produce a certain number of units. Imagine you have made 1600 units of Xbars, and you want to know how much each one costs on average.
Start by finding the total cost using the cost function, which is given as:

• \( C(1600) = 100 + 3.002 \times 1600 - 0.0001 \times 1600^2 \)
• First, calculate each component: \( 3.002 \times 1600 = 4803.2 \) and \( 0.0001 \times 1600^2 = 256 \)
• Therefore, \( C(1600) = 100 + 4803.2 - 256 = 4647.2 \)

Now, to find the average cost per unit, divide the total cost by the number of units, which is 1600:

• \( \text{Average Cost} = \frac{4647.2}{1600} = 2.9045 \)

This means that each Xbar costs approximately \( 2.9045 \) when 1600 are produced. It’s like finding the sum of all expenses and spreading it equally over each unit made. This method is handy for pricing and budgeting.

Marginal Cost Derivative

In economics, the marginal cost is an important concept that measures the cost of producing one more unit of a good. To determine this, you need to find the derivative of the total cost function, commonly symbolized as \( C'(x) \).
The given cost function is:

• \( C(x) = 100 + 3.002x - 0.0001x^2 \)

Taking the derivative with respect to \( x \), we get:

• \( C'(x) = 3.002 - 0.0002x \)

This equation helps you compute how costs change as production increases by one unit. It’s a tool to determine efficiency in production.
Next, plug in the production level of 1600 units to find the specific marginal cost at that level:

• \( C'(1600) = 3.002 - 0.0002 \times 1600 = 2.682 \)

This indicates that producing the 1601st unit costs \( 2.682 \), illustrating how small cost increases might become when more units are made. It helps businesses in planning production adjustments.

Total Cost Function

The total cost function is crucial because it encapsulates fixed, variable, and polynomial costs involved in production. For the Xbar exercise, the function provided is:

• \( C(x) = 100 + 3.002x - 0.0001x^2 \)

Here's how to interpret it:

• The constant 100 is a fixed cost, which does not change with production levels. It could be costs like rent or salaries.
• The linear term \( 3.002x \) reflects variable costs that change with each unit produced, like materials and labor costs.
• The quadratic term \( -0.0001x^2 \) models more complex cost changes as production scales up or down. It may represent economies or diseconomies of scale.

Understanding this function allows you to see how costs behave as production quantities change. It’s essential for making informed business decisions and analyzing cost behaviors efficiently in varying production scenarios.

Calculus in Economics

Calculus plays a pivotal role in economic analysis, especially in understanding cost behaviors and optimizing decision-making. By utilizing calculus, businesses can derive functions like the marginal cost to examine how their costs change with varying levels of production.
Here's how calculus is linked to economic concepts like average and marginal cost:

• It allows businesses to calculate derivatives, providing insights into the rates at which costs rise or fall as production is adjusted.
• Differentiation aids in predicting future cost changes, helping companies allocate resources effectively and maintain competitiveness.
• Integrals can be used for area-under-the-curve problems, which is valuable for calculating total costs over a range of production levels.

Overall, calculus provides the mathematical framework to deeply analyze and interpret various economic phenomena, making it indispensable for understanding production costs, maximizing profits, and efficiently managing resources.