# The function f that models the radius as a function of time. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.6, Problem 62E

(a)

To determine

## The function f that models the radius as a function of time.

Expert Solution

The value of the function f is f(t)=t .

### Explanation of Solution

Given:

The radius of the balloon is increasing at the rate of 1cm/s .

The function f models the radius as a function of time.

Calculation:

Since the function f is the function that models the radius as a function of time and the radius of the spherical balloon is increasing at the rate of 1cm/s , therefore the function f is a function of multiplication of increasing rate and time t.

The function f is written as,

f(t)=(1)t=t

Thus, the value of the function f is f(t)=t .

(b)

To determine

### The function g that models the volume as a function of the radius.

Expert Solution

The value of the function g is g(r)=43πr3 .

### Explanation of Solution

Given:

The function g is the function that models the volume as a function of the radius.

Calculation:

The volume of sphere with radius r is given below,

Volumeofsphere=43πr3

Since the g is the function that models the volume of sphere as a function of the radius, therefore the function g is equal to the volume of sphere whose radius is r.

The function g is written as,

g(r)=43πr3

Thus, the value of the function g is g(r)=43πr3 .

(c)

To determine

### The value of g∘f .

Expert Solution

The value of gf is (gf)(t)=43πt3 .

### Explanation of Solution

Given:

From part (a), the value of function f is given below,

f(t)=t (1)

From part (b), the value of function g is given below,

g(r)=43πr3 (2)

Calculation:

The composite function gf is expressed as,

(gf)(t)=g(f(t))

From equation (1), substitute t for f(t) in above expression,

(gf)(t)=g(t) (3)

Substitute t for r in equation (2), to find the value of g(t) ,

g(t)=43π(t)3=43πt3

Substitute 43πt3 for g(t) in equation (3), to find the value of gf ,

(gf)(t)=43πt3

The function gf represents the volume of the spherical balloon as a function of time.

Thus, the value of gf is (gf)(t)=43πt3 .

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