Math Problem Statement
Homework:Section 4.2 Enhanced Homework
Question 19, 4.2.32-BE
HW Score: 79.55%, 17.5 of 22 points
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Part 1
A note card company has found that the marginal cost per card of producing x note cards is given by the function below, where
Upper C prime left parenthesis x right parenthesisC′(x)
is the marginal cost, in cents, per card. Find the total cost of producing
700700
cards, disregarding any fixed costs.
Upper C prime left parenthesis x right parenthesisC′(x)equals=negative 0.08 x plus 86−0.08x+86,
for x less than or equals 1000x≤1000
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Part 1
The total cost is
enter your response here
cents
The total cost iscents
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Solution
To find the total cost of producing 700 cards when the marginal cost is given by , we need to integrate the marginal cost function over the interval . This integral gives the total cost:
Step 1: Set up the integral
Step 2: Integrate the function
The integral of is:
The integral of is:
So the total cost function is:
Since we're finding the cost disregarding fixed costs, we omit .
Step 3: Evaluate the definite integral
We calculate from to :
First, compute :
Now calculate:
Add these together:
Final Answer:
The total cost of producing 700 cards is:
Do you have any questions, or would you like more details?
Here are 5 related questions to explore further:
- How does the total cost change if the number of cards increases to 1000?
- What is the average cost per card for 700 cards?
- How does the marginal cost affect the total cost function?
- Can we include fixed costs into the calculation? How would that change the result?
- What happens to the total cost if the marginal cost function changes to ?
Tip: Always check the limits of integration and ensure units (cents, dollars, etc.) are consistent!
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Math Problem Analysis
Mathematical Concepts
Calculus
Integration
Marginal Cost
Formulas
Integral of a linear function: ∫(ax + b) dx = (a/2)x^2 + bx + C
Definite integral: ∫[a, b] f(x) dx = F(b) - F(a)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12, Undergraduate (Introductory Calculus)
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