The circumference of a circle is increasing at a rate of 10 cm per second. Find the rate at which the radius is increasing. The unit is cm per second in each possible answer.
The circumference of a circle is increasing at a rate of 10 cm per second. Find the rate at which the radius is increasing. The unit is cm per second in each possible answer.
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 Statement:**
The circumference of a circle is increasing at a rate of 10 cm per second. Find the rate at which the radius is increasing. The unit is cm per second in each possible answer.
**Possible Answers:**
- \(5\pi\)
- \(20\pi\)
- \(5\)
- \(\frac{5}{\pi}\)
- \(\frac{20}{\pi}\) (This is the selected answer)
**Explanation for Educators:**
To solve this problem, use the formula for the circumference of a circle: \(C = 2\pi r\).
Given that \(\frac{dC}{dt} = 10\) cm/s, you need to determine \(\frac{dr}{dt}\).
Using the derivative of the circumference with respect to time:
\[
\frac{dC}{dt} = 2\pi \frac{dr}{dt}
\]
Substitute the given rate of change of the circumference:
\[
10 = 2\pi \frac{dr}{dt}
\]
Solving for \(\frac{dr}{dt}\):
\[
\frac{dr}{dt} = \frac{10}{2\pi} = \frac{5}{\pi}
\]
Therefore, the rate at which the radius is increasing is \(\frac{5}{\pi}\) cm per second.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d31ea21-51a6-4893-9fcb-4fc7e3b87aa7%2Feb60fc93-1ad3-4caf-98d7-d715b24ec3ed%2Fj6hnarr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The circumference of a circle is increasing at a rate of 10 cm per second. Find the rate at which the radius is increasing. The unit is cm per second in each possible answer.
**Possible Answers:**
- \(5\pi\)
- \(20\pi\)
- \(5\)
- \(\frac{5}{\pi}\)
- \(\frac{20}{\pi}\) (This is the selected answer)
**Explanation for Educators:**
To solve this problem, use the formula for the circumference of a circle: \(C = 2\pi r\).
Given that \(\frac{dC}{dt} = 10\) cm/s, you need to determine \(\frac{dr}{dt}\).
Using the derivative of the circumference with respect to time:
\[
\frac{dC}{dt} = 2\pi \frac{dr}{dt}
\]
Substitute the given rate of change of the circumference:
\[
10 = 2\pi \frac{dr}{dt}
\]
Solving for \(\frac{dr}{dt}\):
\[
\frac{dr}{dt} = \frac{10}{2\pi} = \frac{5}{\pi}
\]
Therefore, the rate at which the radius is increasing is \(\frac{5}{\pi}\) cm per second.
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