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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
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.
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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