# The function g 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 61E

(a)

To determine

## To find: The function g that models the radius as a function of time.

Expert Solution

The value of the function g is g(t)=60t.

### Explanation of Solution

Given:

The circular ripple travels outward at a speed of 60cm/s.

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

Calculation:

The radius of the ripple is the distance travelled by the ripple.

So, the distance travelled is evaluated as follows,

Distance=speed×time=60×t=60t

The radius is given by the function g(t).

The required function is g(t)=60t.

Thus, the value of the function g is g(t)=60t.

(b)

To determine

### To find: The function f that models the area of the circle as a function of the radius.

Expert Solution

The value of the function f is f(r)=πr2.

### Explanation of Solution

Given:

The function f is the function that models the area of the circle as a function of the radius.

Calculation:

The area of a circular ripple with radius r is given by,

Areaofcircularripple=πr2

Since the f is the function that models the area of the circle as a function of the radius, therefore the function f is equal to the area of the circle whose radius is r.

The function f is written as,

f(r)=πr2

Thus, the value of the function f is f(r)=πr2.

(c)

To determine

### To find: The value of the function f∘g.

Expert Solution

The value of fg is (fg)(t)=3600πt2.

### Explanation of Solution

Given:

From part (a), the value of function g is , g(t)=60t (1)

From part (b), the value of function f is , f(r)=πr2 (2)

Calculation:

The composite function fg is expressed as,

(fg)(t)=f(g(t))

From equation (1), substitute g(t)=60t in above expression,

(fg)(t)=f(60t) (3)

Substitute r=60t in equation (2), to find the value of f(60t) as follows,

f(60t)=π(60t)2=3600πt2

Substitute f(60t)=3600πt2 in equation (3) to find the value of fg as follows,

(fg)(t)=3600πt2

The function fg represents the area of the circle as a function of time.

Thus, the value of fg is (fg)(t)=3600πt2.

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