Chapter 1.3, Problem 53E

(a)

To determine

To express: The radius of the circular ripple as a function of time t.

Answer to Problem 53E

The function of the radius in terms of time t is $r\left(t\right)=60t$.

Explanation of Solution

It is given that the stone travel at a speed of 60 cm/s.

Let the radius of the circular ripple as r.

Note that the radius is same as the distance travelled by a stone.

Recall the formula, Distance = Time × Speed.

Substitute t for time and 60 for speed in a distance formula.

Thus, the function is $r\left(t\right)=60t$ where r(t) is measured in cm.

Therefore, the radius of the circular ripple after t seconds is $r\left(t\right)=60t$.

(b)

To determine

To find: The expression for $\left(A\circ r\right)\left(t\right)$ and interpret where A is the area of the function of radius.

Answer to Problem 53E

The value of $\left(A\circ r\right)\left(t\right)=3600\pi t^2$ it represents the area of the circular ripple at a time t.

Explanation of Solution

Area of the circle is, $A\left(r\right)=\pi {\left(r\left(t\right)\right)}^2$ (1)

The composite function $\left(A\circ r\right)\left(t\right)$ is defined as $\left(A\circ r\right)\left(t\right)=A\left(r\left(t\right)\right)$.

From part (a), the value of $r\left(t\right)=60t$.

Thus, $\left(A\circ r\right)\left(t\right)=A\left(60t\right)$.

Substitute $60t$ in equation (1),

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

Thus, $\left(A\circ r\right)\left(t\right)=3600\pi t^2$, which represents the area of the circular ripple with respect to the radius of time t.