# The rate at which the rate of change of the area within the circle is increasing after 1 s.

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 3.8, Problem 14E

(a)

To determine

## To find: The rate at which the rate of change of the area within the circle is increasing after 1 s.

Expert Solution

The rate at which the rate of change of the area within the circle is increasing after 1 s is, 7200πcm2/s.

### Explanation of Solution

Given:

A stone which is dropped into a lake that creates a circular ripple whose speed when it travels outward is 60 cm/s.

Calculation:

Area inside the ripple at time t is A(t)=π(60t)2.

That is, A(t)=3600πt2.

The rate of change of the ripple at time t is, A'(t)=7200πt.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 1 s is, A(1)=7200πcm2/s.

(b)

To determine

### To find: The rate at which the rate of change of the area within the circle is increasing after 3 s.

Expert Solution

The rate at which the rate of change of the area within the circle is increasing after 3 s is, 21,600πcm2/s.

### Explanation of Solution

From part (a), the rate of change of the ripple at time t is, A'(t)=7200πt.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 3 s is, A(3)=21,600πcm2/s.

(c)

To determine

### To find: The rate at which the rate of change of the area within the circle is increasing after 5 s. What can be concluded from the parts (a), (b) and (c).

Expert Solution

The rate at which the rate of change of the area within the circle is increasing after 5 s is, 36000πcm2/s.

### Explanation of Solution

From part (a), the rate of change of the ripple at time t is, A'(t)=7200πt.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 5 s is, 36000πcm2/s.

From the parts (a), (b) and (c), it can be concluded that the rate of change increases as time increases.

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