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 \(r\) of 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) It is obtained that \(r\left( t \right) = 60t\).

b) \(\left( {A \circ r} \right)\left( t \right) = 3600\pi {t^2}\). The formula provides the distribution of the rippled area (in \({{\mathop{\rm cm}\nolimits} ^2}\)) at any time \(t\).

Step-by-step solution

Composite function 

Consider two functions \(f\) and \(g\), the composite function\(f \circ g\)(also called thecomposition of\(f\)and \(g\)) is defined as:

\(\left( {f \circ g} \right)\left( x \right) = f\left( {g\left( x \right)} \right)\)

Express the radius \(r\) of this circle as a function of time t

a)

With the relationship \({\rm{distance}} = \,{\rm{rate}} \cdot {\rm{time}}\), where \(r\) is the distance.

Express the radius \(r\) of the circle as a function of time \(t\) as shown below:

\(r\left( t \right) = 60t\)

Thus, it is obtained that \(r\left( t \right) = 60t\).

Determine \(A \circ r\) and interpret it

b)

The area of the circle is \(A = \pi {r^2}\).

Express the area of the circle as a function of the radius as shown below:

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

The formula provides the distribution of the rippled area (in \({{\mathop{\rm cm}\nolimits} ^2}\)) at any time \(t\).