# To express: The radius of the balloon 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 54E

(a)

To determine

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

Expert Solution

The function of the radius of the balloon in terms of time t is r(t)=2t .

### Explanation of Solution

It is given that the balloon is being inflated and therefore the radius is increasing at a rate of 2 cm/s.

Let the radius of the balloon be r.

Recall the formula, Distance = Time × Speed.

Note that the distance is same as radius.

Substitute t for time and 2 for speed in a distance formula.

Thus, the function is r(t)=2t where r(t) is measured in cm.

Therefore, the radius of the spherical balloon after t seconds is r(t)=2t .

(b)

To determine

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

Expert Solution

The value of (Vr)(t)=323πt3 , which represents the area of the circular ripple at a time t.

### Explanation of Solution

Area of the circle is, V(r)=43π(r(t))3 (1)

The composite function (Vr)(t) is defined as (Vr)(t)=V(r(t)) .

From part (a), the value of r(t)=2t .

Thus, (Vr)(t)=V(2t) .

Substitute 2t in equation (1),

V(r)=43π(2t)3=43π(8t3)=323πt3

Thus, (Vr)(t)=323πt3 , which represents the area of the circular ripple with respect to the radius of time t.

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