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Magazine Chapter 13: Problem 49
The marginal cost of producing the \(x\) th roll of film is \(5+2 x+\frac{1}{x} .\) The total cost to produce one roll is $$\$ 1,000$$. Find the cost function \(C(x)\). HINT [See Example 5.]
Short Answer
Expert verified
The cost function $C(x)$ for producing x rolls of film is given by \(C(x) = 5x + x^2 + \ln|x| + 994\).
Step by step solution
01
Identify the given information
The exercise has provided the following information:
- Marginal cost function: \(MC(x) = 5 + 2x + \frac{1}{x}\)
- Total Cost of producing one roll: $1000
02
Integrate the marginal cost function
To find the cost function C(x), we need to integrate the marginal cost function MC(x) with respect to x.
03
Step 2.1: Integrate the terms separately
To do this, we can integrate each term of MC(x) separately with respect to x:
1. Integrate \(5\) with respect to x: \[\int 5 \, dx\]
2. Integrate \(2x\) with respect to x: \[\int 2x \, dx\]
3. Integrate \(\frac{1}{x}\) with respect to x: \[\int \frac{1}{x} \, dx\]
04
Step 2.2: Apply integration rules and calculate the integrals
Now, we can apply the integration rules to each integral term:
1. For \(\int 5 \, dx\), the result is \(5x\).
2. For \(\int 2x \, dx\), the result is \(x^2\).
3. For \(\int \frac{1}{x} \, dx\), the result is \(\ln|x|\).
05
Step 2.3: Add the integration constants and combine the integral results
Now that we have the results for each integral term, we combine them and add the integration constant, say \(C_1\), to create the cost function C(x):
\[C(x) = 5x + x^2 + \ln|x| + C_1\]
06
Find the value of integration constant, C1
According to the problem, the total cost to produce one roll (\(x=1\)) is $1000. By plugging this information into our cost function, we can find the value of the integration constant, C1.
\(C(1) = 5(1) + (1)^2 + \ln|1| + C_1 = 1000\)
By solving this equation, we can find the value of C1:
\(6 + 0 + C_1 = 1000\)
\(C_1 = 994\)
07
Define the final cost function C(x)
Finally, we can plug the value of C1 into our cost function:
\[C(x) = 5x + x^2 + \ln|x| + 994\]
This is the cost function for producing x rolls of film.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Marginal Cost
The marginal cost represents the cost of producing an additional unit of a good or service. In the context of the exercise, we have the marginal cost function given by \(MC(x) = 5 + 2x + \frac{1}{x}\). This function tells us how much the cost changes when we produce one more roll of film for any given production level \(x\).
- The term \(5\) is a constant cost that doesn't change with the number of units produced.
- The term \(2x\) shows that the cost increases linearly with more production.
- The term \(\frac{1}{x}\) suggests a decreasing cost impact as more units are produced, indicating an inverse relationship at lower levels of \(x\).
Integration
Integration is a fundamental concept in calculus used to find total values from rates of change, such as converting a marginal cost function into a total cost function. For our exercise, we need to integrate the marginal cost function \(MC(x) = 5 + 2x + \frac{1}{x}\) to find the total cost function \(C(x)\). This involves calculating the integral of each term individually:
- \(\int 5 \, dx = 5x\) demonstrates integrating a constant.
- \(\int 2x \, dx = x^2\) uses the power rule in integration.
- \(\int \frac{1}{x} \, dx = \ln|x|\) employs the logarithmic rule, highlighting the inverse relationship.
Integration Constant
When performing integration on marginal functions, we must include an integration constant, represented by \(C_1\) in this exercise. This constant accounts for initial unknown values when calculating total functions. In the context of the cost function, the constant \(C_1\) reflects the original baseline cost when \(x = 1\).Here's how to find this constant:- Utilizing known conditions, such as the provided total cost for producing the first roll ($1000), allows us to solve for \(C_1\).- Inserting the initial condition into our integrated function: \(C(1) = 5(1) + (1)^2 + \ln|1| + C_1 = 1000\).- Solving gives \(C_1 = 994\), revealing our total starting cost for production.
Calculus Application
Calculus is a powerful tool that allows us to transition from understanding rates of change to determining exact total values. In this scenario, we apply calculus to transform the marginal cost into a practical total cost function for film production.
- First, establish the relationship of how costs increase with production through the marginal cost function.
- Second, integrate to determine cumulative costs as seen with \(C(x) = 5x + x^2 + \ln|x| + C_1\).
- Finally, use given conditions to solve for any unknown constants, ensuring the cost function accurately reflects all base costs and variable changes.
