Chapter 12.6, Problem 30PPS

a.

To determine

To find: Independent variable and dependent variable of the periodic function that represents the given situation.

Answer to Problem 30PPS

Period of given situation is $1.2$ seconds

Independent variable : time

Dependent variable : height

Explanation of Solution

Given information:

The difference between the highest and lowest poits if $2$ feet.

He jumps $50$ times per minute.

Here

Independent variable is time $t$ in seconds.

Dependent variable is height $h$ in feet.

Given

He jumps $50$ times per minute.

It means

Period $=\frac{60}{50}$

  $\Rightarrow$ Period $=1.2$ seconds.

Therefore,

Period of given situation is $1.2$ seconds

b.

To determine

To find: Graph that represents the given situation.

Explanation of Solution

Given information:

The difference between the highest and lowest poits if $2$ feet.

He jumps $50$ times per minute.

Here

Independent variable is time $t$ in seconds.

Dependent variable is height $h$ in feet.

Given

He jumps $50$ times per minute.

It means

Period $=\frac{60}{50}$

  $\Rightarrow$ Period $=1.2$ seconds.

  $\begin{array}{l}\frac{2\pi }{b}=1.2\\ \Rightarrow b=\frac{2\pi }{1.2}\\ \Rightarrow b=\frac{5}{3}\pi \end{array}$

We know amplitude is equal to difference between highest and lowest point divided by $2$

Therefore,

Amplitude $=\frac{2}{2}$

  $\Rightarrow$ Amplitude $=1$

Given that

His starting point is halfway between the highest and lowest point.

It means it lies at Origin.

Therefore,

It is a sine function

On substituting values

We get

  $\begin{array}{l}h=a\sin bt\\ \Rightarrow h=1\sin \left(\frac{5}{3}\pi t\right)\\ \Rightarrow h=\sin \frac{5\pi }{3}t\end{array}$

Therefore,

Equation that represents given situation is

  $h=\sin \frac{5\pi }{3}t$

Graph of above equation is