Question

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60{cm/s.

(a) Express the radius rof this circle as a function of the time t (in seconds).

(b) If A is the area of this circle as a function of the radius, find\(A \circ r\)and interpret it.

Short answer

(a) \(r\left( t \right) = 60t\) is the function of the radius with respect to time.

(b) \(A\left( {r\left( t \right)} \right) = 3600\pi {t^2}\) is the area of the circle that gives the idea of the covered area at any given time t.

Step-by-step solution

Introduction

Two functions can be expressed as a combination of each other with the addition, subtraction, multiplication, and divide operation. One or more operations can be applied at a time.

If f and g are two functions they can be written as,

\(\left(f+g\right)\left(x\right),\left(f-g\right)\left(x\right),\left(f\right)\left(x\right),\left( {f}/{g}\;\right)\left(x\right)\).

It is giventhat,

Speed is 60cm/s

(a) Radius r as a function of time t.

There are a relation between distance, speed and time, that is,

\(\begin{aligned}{\rm{Distance}} &= {\rm{speed}} \times {\rm{time}}\\r\left( t \right) &= 60t\end{aligned}\)

(b) A is the area of circle and r is the circle of the radius then\(A \circ r\)      

A is the area of circle and r is the circle of the radius and t is the time, so now from the formula of the circle area and the relation between the r and t , the function will be following

\(\begin{array}{l}\left( {A \circ r} \right)t = A\left( {r\left( t \right)} \right)\\A\left( {r\left( t \right)} \right) = \pi {r^2}\\A\left( {r\left( t \right)} \right) = \pi {\left( {60t} \right)^2}\\A\left( {r\left( t \right)} \right) = 3600\pi {t^2}\end{array}\)

Hence, this is the area of the circle that gives the idea of the covered area at any given time t.