13.) The order and transportation cost C of bottles of Pepsi® is approximated by the function: X C(x) = 10,000+ where x is the order size of bottles of Pepsi® in hundreds. X x+3 According to Rolle's Theorem, the rate of change of cost must be zero interval [3,6]. Find the order size. for some order size in th
13.) The order and transportation cost C of bottles of Pepsi® is approximated by the function: X C(x) = 10,000+ where x is the order size of bottles of Pepsi® in hundreds. X x+3 According to Rolle's Theorem, the rate of change of cost must be zero interval [3,6]. Find the order size. for some order size in th
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem 13: Analyzing Order and Transportation Costs**
The cost of ordering and transporting bottles of Pepsi is approximated by the function:
\[ C(x) = 10,000 \left( \frac{1}{x} + \frac{x}{x+3} \right) \]
where \( x \) is the order size of bottles of Pepsi (in hundreds).
**Objective**
According to Rolle's Theorem, you need to find the order size where the rate of change of cost is zero within the interval \([3, 6]\).
**Solution Steps**
1. **Understanding the Function:**
- The function \( C(x) \) represents the total cost based on the order size \( x \).
- The expression involves both a reciprocal term and a rational term.
2. **Application of Rolle's Theorem:**
- Rolle’s Theorem states that if a function is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in the interval \((a, b)\) such that the derivative \( f'(c) = 0 \).
- First, verify that the conditions for Rolle’s Theorem are met: continuity, differentiability, and equal values at endpoints.
3. **Find the Derivative \( C'(x) \):**
- Differentiate \( C(x) \) with respect to \( x \).
4. **Solve \( C'(x) = 0 \):**
- Solve the equation for \( x \) to determine the critical points.
- Ensure the solution lies within the interval \([3, 6]\).
5. **Verify the Conditions of Rolle’s Theorem:**
- Check if \( C(3) = C(6) \) to confirm applicability of the theorem.
6. **Interpret the Result:**
- The resulting value of \( x \) (in hundreds) provides the optimal order size where the rate of change of cost is zero.
**Conclusion**
By following the above steps, you can determine the order size that minimizes the rate of change in cost, ensuring efficient logistics and cost management for ordering Pepsi bottles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66d30f65-22c6-47d5-b371-c8d8c2d2b3f2%2Feaeaa291-397a-4351-b097-e489314fbb2b%2F1ehkyy8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 13: Analyzing Order and Transportation Costs**
The cost of ordering and transporting bottles of Pepsi is approximated by the function:
\[ C(x) = 10,000 \left( \frac{1}{x} + \frac{x}{x+3} \right) \]
where \( x \) is the order size of bottles of Pepsi (in hundreds).
**Objective**
According to Rolle's Theorem, you need to find the order size where the rate of change of cost is zero within the interval \([3, 6]\).
**Solution Steps**
1. **Understanding the Function:**
- The function \( C(x) \) represents the total cost based on the order size \( x \).
- The expression involves both a reciprocal term and a rational term.
2. **Application of Rolle's Theorem:**
- Rolle’s Theorem states that if a function is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in the interval \((a, b)\) such that the derivative \( f'(c) = 0 \).
- First, verify that the conditions for Rolle’s Theorem are met: continuity, differentiability, and equal values at endpoints.
3. **Find the Derivative \( C'(x) \):**
- Differentiate \( C(x) \) with respect to \( x \).
4. **Solve \( C'(x) = 0 \):**
- Solve the equation for \( x \) to determine the critical points.
- Ensure the solution lies within the interval \([3, 6]\).
5. **Verify the Conditions of Rolle’s Theorem:**
- Check if \( C(3) = C(6) \) to confirm applicability of the theorem.
6. **Interpret the Result:**
- The resulting value of \( x \) (in hundreds) provides the optimal order size where the rate of change of cost is zero.
**Conclusion**
By following the above steps, you can determine the order size that minimizes the rate of change in cost, ensuring efficient logistics and cost management for ordering Pepsi bottles.
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