Chapter 3.8, Problem 14E

(a)

To determine

To find: The rate at which the rate of change of the area within the circle is increasing after 1 s.

Answer to Problem 14E

The rate at which the rate of change of the area within the circle is increasing after 1 s is, $7200\pi {\text{cm}}^{\text{2}}\text{/s}$.

Explanation of Solution

Given:

A stone which is dropped into a lake that creates a circular ripple whose speed when it travels outward is 60 cm/s.

Calculation:

Area inside the ripple at time t is $A\left(t\right)=\pi {\left(60t\right)}^2$.

That is, $A\left(t\right)=3600\pi t^2$.

The rate of change of the ripple at time t is, $A\text{'}\left(t\right)=7200\pi t$.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 1 s is, $A\left(1\right)=7200\pi {\text{cm}}^{\text{2}}\text{/s}$.

(b)

To determine

To find: The rate at which the rate of change of the area within the circle is increasing after 3 s.

Answer to Problem 14E

The rate at which the rate of change of the area within the circle is increasing after 3 s is, $21,600\pi {\text{cm}}^{\text{2}}\text{/s}$.

Explanation of Solution

From part (a), the rate of change of the ripple at time t is, $A\text{'}\left(t\right)=7200\pi t$.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 3 s is, $A\left(3\right)=21,600\pi {\text{cm}}^{\text{2}}\text{/s}$.

(c)

To determine

To find: The rate at which the rate of change of the area within the circle is increasing after 5 s. What can be concluded from the parts (a), (b) and (c).

Answer to Problem 14E

The rate at which the rate of change of the area within the circle is increasing after 5 s is, $36000\pi {\text{cm}}^{\text{2}}\text{/s}$.

Explanation of Solution

From part (a), the rate of change of the ripple at time t is, $A\text{'}\left(t\right)=7200\pi t$.

Therefore, the rate at which the rate of change of the area within the circle is increasing after 5 s is, $36000\pi {\text{cm}}^{\text{2}}\text{/s}$.

From the parts (a), (b) and (c), it can be concluded that the rate of change increases as time increases.