# To express: The radius of the circular ripple as a function of time t . ### 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 1.3, Problem 53E

(a)

To determine

## To express: The radius of the circular ripple as a function of time t.

Expert Solution

The function of the radius in terms of time t is r(t)=60t .

### Explanation of Solution

It is given that the stone travel at a speed of 60 cm/s.

Let the radius of the circular ripple as r.

Note that the radius is same as the distance travelled by a stone.

Recall the formula, Distance = Time × Speed.

Substitute t for time and 60 for speed in a distance formula.

Thus, the function is r(t)=60t where r(t) is measured in cm.

Therefore, the radius of the circular ripple after t seconds is r(t)=60t .

(b)

To determine

### To find: The expression for (A∘r)(t) and interpret where A is the area of the function of radius.

Expert Solution

The value of (Ar)(t)=3600πt2 it represents the area of the circular ripple at a time t.

### Explanation of Solution

Area of the circle is, A(r)=π(r(t))2 (1)

The composite function (Ar)(t) is defined as (Ar)(t)=A(r(t)) .

From part (a), the value of r(t)=60t .

Thus, (Ar)(t)=A(60t) .

Substitute 60t in equation (1),

A(r)=π(60t)2=3600πt2

Thus, (Ar)(t)=3600πt2 , which represents the area of the circular ripple with respect to the radius of time t.

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