Chapter 13.7, Problem 21PPS

a.

To determine

To find: Equation for the motion of boat.

Answer to Problem 21PPS

Equation of given situation is

  $h=4\sin \frac{2}{3}\pi t$

Explanation of Solution

Given information:

The difference between highest point and lowest points of the boat is $8$ inches.

Each cycle of the periodic motion lasts $3$ seconds.

The boat is in equilibrium at $t=0$ .

Given

The difference between highest point and lowest points of the boat is $8$ inches.

It means

Amplitude $=a=\frac{8}{2}=4$

Each cycle of the periodic motion lasts $3$ seconds.

It means Period of the function is $3$

Given

The boat is in equilibrium at $t=0$ .

It means it is Sine Function.

  $\begin{array}{l}\Rightarrow \frac{2\pi }{b}=3\\ \Rightarrow b=\frac{2\pi }{3}\end{array}$

Therefore,

On substituting values

We get

  $\begin{array}{l}y=a\sin b\theta \\ \Rightarrow h=4\sin \frac{2}{3}\pi t\end{array}$

Therefore,

Equation of given situation is

  $h=4\sin \frac{2}{3}\pi t$

b.

To determine

To find: Graph of the function.

Explanation of Solution

Given information:

The difference between highest point and lowest points of the boat is $8$ inches.

Each cycle of the periodic motion lasts $3$ seconds.

The boat is in equilibrium at $t=0$ .

Equation of given situation is

  $h=4\sin \frac{2}{3}\pi t$

Graph of the function is