3. The area of a circle increases at a rate of 47 cm²/s. How fast is the radius changing when the radius is 2 cm? A=TTY² 21Tr 4π = 2T (2) 4IT=4 TT
3. The area of a circle increases at a rate of 47 cm²/s. How fast is the radius changing when the radius is 2 cm? A=TTY² 21Tr 4π = 2T (2) 4IT=4 TT
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...
Related questions
Question
![**Problem:**
The area of a circle increases at a rate of \(4\pi \, \text{cm}^2/\text{min}\). How fast is the radius changing when the radius is 2 cm?
**Solution:**
1. **Given equation for area of a circle:**
\[
A = \pi r^2
\]
2. **Differentiate with respect to time \(t\):**
\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]
3. **Given \(\frac{dA}{dt} = 4\pi\), and we need to find \(\frac{dr}{dt}\) when \(r = 2\):**
4. **Substitute the values into the differentiated equation:**
\[
4\pi = 2\pi (2) \frac{dr}{dt}
\]
5. **Simplify:**
\[
4\pi = 4\pi \frac{dr}{dt}
\]
6. **Solve for \(\frac{dr}{dt}\):**
\[
\frac{dr}{dt} = 1
\]
Thus, the radius is changing at a rate of 1 cm/min when the radius is 2 cm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39190787-1e8c-4e2c-8802-b30850ac5f44%2Fcbc6b382-1480-410d-a5d4-9e52bfb7ca6b%2Fxffqe1s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
The area of a circle increases at a rate of \(4\pi \, \text{cm}^2/\text{min}\). How fast is the radius changing when the radius is 2 cm?
**Solution:**
1. **Given equation for area of a circle:**
\[
A = \pi r^2
\]
2. **Differentiate with respect to time \(t\):**
\[
\frac{dA}{dt} = 2\pi r \frac{dr}{dt}
\]
3. **Given \(\frac{dA}{dt} = 4\pi\), and we need to find \(\frac{dr}{dt}\) when \(r = 2\):**
4. **Substitute the values into the differentiated equation:**
\[
4\pi = 2\pi (2) \frac{dr}{dt}
\]
5. **Simplify:**
\[
4\pi = 4\pi \frac{dr}{dt}
\]
6. **Solve for \(\frac{dr}{dt}\):**
\[
\frac{dr}{dt} = 1
\]
Thus, the radius is changing at a rate of 1 cm/min when the radius is 2 cm.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning