The measurement of the radius of the end of a log is found to be 5 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log. in?
The measurement of the radius of the end of a log is found to be 5 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log. in?
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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![**Title: Calculating Propagated Error in the Area Measurement of a Log Using Differentials**
### Problem Statement:
The measurement of the radius of the end of a log is found to be 5 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.
### Detailed Problem Description:
- **Radius (r):** 5 inches
- **Possible Error in Radius (Δr):** ±1/2 inch
Using the given information, we are required to use differentials to find the approximate propagated error in the area (A) of the end of the log.
### Formula and Calculation:
1. **Area of a Circle:**
\( A = \pi r^2 \)
2. **Differential of Area (dA):**
\( dA = \frac{dA}{dr} \cdot dr \)
\( dA = 2\pi r \cdot dr \)
3. **Input Values:**
- \( r = 5 \) inches
- \( dr = \Delta r = \pm \frac{1}{2} \) inch
4. **Substituting the values:**
\( dA = 2\pi (5) \cdot \frac{1}{2} \)
\( dA = 5\pi \)
### Result:
The possible propagated error in computing the area of the end of the log is:
\[ \pm 5\pi \, \text{in}^2 \]
This error should be rounded to the nearest hundredth where necessary, ensuring the precision required for the specific application.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F08c083ee-5dc4-4392-bfaa-a5f4117ce6af%2Fb660d4bc-c692-4389-a6ff-fb789cec3e3e%2F5nucl4h.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Propagated Error in the Area Measurement of a Log Using Differentials**
### Problem Statement:
The measurement of the radius of the end of a log is found to be 5 inches, with a possible error of 1/2 inch. Use differentials to approximate the possible propagated error in computing the area of the end of the log.
### Detailed Problem Description:
- **Radius (r):** 5 inches
- **Possible Error in Radius (Δr):** ±1/2 inch
Using the given information, we are required to use differentials to find the approximate propagated error in the area (A) of the end of the log.
### Formula and Calculation:
1. **Area of a Circle:**
\( A = \pi r^2 \)
2. **Differential of Area (dA):**
\( dA = \frac{dA}{dr} \cdot dr \)
\( dA = 2\pi r \cdot dr \)
3. **Input Values:**
- \( r = 5 \) inches
- \( dr = \Delta r = \pm \frac{1}{2} \) inch
4. **Substituting the values:**
\( dA = 2\pi (5) \cdot \frac{1}{2} \)
\( dA = 5\pi \)
### Result:
The possible propagated error in computing the area of the end of the log is:
\[ \pm 5\pi \, \text{in}^2 \]
This error should be rounded to the nearest hundredth where necessary, ensuring the precision required for the specific application.
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