Problem 34 - Single Variable Calculus

Question

The weekly total cost in dollars incurred by the BMC Recording Company in manufacturing \(x\) compact discs is $$C(x)=4000+3 x-0.0001 x^{2} \quad 0 \leq x \leq 10,000$$ a. What is the actual cost incurred by the company in producing the 2001 st disc? The 3001 st disc? b. What is the marginal cost when \(x=2000\) ? When \(x=3000 ?\)

Step-by-Step Solution

Answer

a. The actual cost of producing the 2001st disc is $5.20 and the actual cost of producing the 3001st disc is $4.40. b. The marginal cost when \(x=2000\) is $2.6, and the marginal cost when \(x=3000\) is $2.4.

Step 1: Finding the cost for producing 2000 and 3000 discs

We will first find the cost of producing 2000 discs (\(C(2000)\)) and then the cost of producing 3001 discs (\(C(3001)\)): \[C(2000)=4000+3(2000)-0.0001(2000)^{2}\] \[C(3001)=4000+3(3001)-0.0001(3001)^{2}\] Calculating the values, we get: \[C(2000)=10000\] \[C(3001)=13995.7\]

Step 2: Finding the actual cost for the 2001st and 3001st discs

Now we will find the actual cost of producing the 2001st and 3001st discs by calculating the difference in the costs: Actual cost of 2001st disc: \[C(2001)-C(2000) \Rightarrow (4000+3(2001)-0.0001(2001)^{2})-10000\] Actual cost of 3001st disc: \[C(3001)-C(3000) \Rightarrow (13995.7-(4000+3(3000)-0.0001(3000)^{2}))\] Calculating the values, we get: Actual cost of 2001st disc: \[5.20\] Actual cost of 3001st disc: \[4.40\]

Step 3: Finding the marginal cost function

We need to find the derivative of the cost function, \(C'(x)\), which represents the marginal cost. Differentiate \(C(x)\) with respect to \(x\): \[C'(x)=\frac{d}{dx} (4000+3x-0.0001x^{2})\] The derivative can be found as: \[C'(x)=3-0.0002x\]

Step 4: Calculating the marginal cost when \(x=2000\) and \(x=3000\)

Now, we will calculate the marginal cost for \(x=2000\) and \(x=3000\) using the marginal cost function: Marginal cost when \(x=2000\): \[C'(2000)=3-0.0002(2000)\] Marginal cost when \(x=3000\): \[C'(3000)=3-0.0002(3000)\] Calculating the values, we get: Marginal cost when \(x=2000\): \[2.6\] Marginal cost when \(x=3000\): \[2.4\] #Summary# a. The actual cost of producing the 2001st disc is \(5.20 and the actual cost of producing the 3001st disc is \)4.40. b. The marginal cost when \(x=2000\) is \(2.6, and the marginal cost when \)x=3000\( is \)2.4.

Key Concepts

Marginal Cost

Marginal cost is an essential concept in economics and business. It represents the additional cost incurred when producing one more unit of a good. Understanding marginal cost helps businesses make informed decisions about production levels and pricing.

In the context of the BMC Recording Company, the marginal cost function is derived from their cost equation. For the given exercise, the cost function of producing each compact disc is:

• \(C(x) = 4000 + 3x - 0.0001x^2\)

The calculation of the marginal cost involves finding the derivative of this function. The marginal cost helps the company understand how the cost of producing an additional disc changes as production scales. It is crucial for optimizing pricing strategies and maximizing profitability. Knowing the marginal cost at specific production levels, like at 2000 and 3000 discs, enables the company to determine whether increasing output would be beneficial.

In this exercise, the company finds that when producing the 2001st and 3001st discs, the actual costs differ due to changes in marginal costs. This is indicative of how cost dynamics evolve with increased production levels.

Calculus Derivatives

Calculus derivatives play a vital role in understanding rates of change in various mathematical contexts, including cost functions in economics. By differentiating a cost function, we can derive the marginal cost function, which represents the rate of change of total cost with respect to output.

For the given cost function,

• \(C(x) = 4000 + 3x - 0.0001x^2\)

the derivative is calculated as follows:

• \(C'(x) = \frac{d}{dx} (4000 + 3x - 0.0001x^2) = 3 - 0.0002x\)

This derivative, \(C'(x)\), is the marginal cost function. It provides insights into how the cost of production varies with each additional unit manufactured. Using derivatives in this way can help businesses better plan their production by analyzing how costs change with different levels of output. For instance, when \(x=2000\), the marginal cost decreases compared to when \(x=3000\), indicating that economies of scale might be at play or resource constraints are less pressing at higher outputs.

Manufacturing Costs

Manufacturing costs encompass all the expenses involved in producing goods, such as raw materials, labor, and overhead costs. Understanding these costs is crucial in determining the pricing and operational strategies of a company.

In this exercise, the BMC Recording Company uses a cost function to model their manufacturing costs for compact discs. The function

• \(C(x) = 4000 + 3x - 0.0001x^2\)

represents the total cost of producing \(x\) discs. The components of this function reflect different types of costs:

• The constant 4000 represents fixed costs, which are expenses that do not change with the production level.
• The term \(3x\) suggests variable costs, which increase proportionally with the amount produced.
• The quadratic term \(-0.0001x^2\) captures diminishing returns or cost efficiency that might occur as production scale increases.

These components help businesses analyze how costs fluctuate and develop strategies to control expenses. By understanding these factors, a company can identify cost-saving opportunities and make informed production decisions to enhance profitability.