   Chapter 2.7, Problem 13E

Chapter
Section
Textbook Problem

# (a) Find the average rate of change of the area of a circlewith respect to its radius r as r changes from(i) 2 to 3(ii) 2 to 2.5(iii) 2 to 2.1(b) Find the instantaneous rate of change when r = 2.(c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount △ r . How can you approximate the resulting change in area △ A if △ r is small?

To determine

Part (a):

To find: The average rate of change of the area of a circle when r changes from

(i) 2 to 3

(ii) 2 to 2.5

(iii) 2 to 2.1

Explanation

1) Concept:

Use concept of average rate of change

Ar

2) Formula:

I. Area of circle: Ar= πr2

II. The average rate of change of the area of a circle with respect to radius r over [r1, r2]

Ar =Ar2 - Ar1r2 - r1

3) Given:

(i) 2 to 3

(ii) 2 to 2.5

(iii) 2 to 2.1

4) Calculation:

(i)  By using average rate of change formula,

Ar =Ar2 - Ar1r2 - r1

Ar =πr22 - πr12r2 - r1

Substitute r2=3 and r1=2

Ar =π32-π223-2=9π-4π=5π

Thus,

Ar= 5π

(ii)  By using average rate of change formula,

Ar =Ar2 - Ar1r2 - r1

Ar =πr22 - πr12r2 - r1

Substitute r2=2

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Convert the expressions in Exercises 6584 to power form. x3

Finite Mathematics and Applied Calculus (MindTap Course List)

#### let f(x) = x3 + 5, g(x) = x2 2, and h(x)= 2x + 4. Find the rule for each function. 2. f g

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### 0 1 −∞ does not exist

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

#### True or False: converges.

Study Guide for Stewart's Multivariable Calculus, 8th 