Chapter 4.5, Problem 93E

a.

To determine

Answer to Problem 93E

  $\boxed{20}$ sec

Explanation of Solution

Given information:

A Ferris wheel is built such that the height $h$ (in feet) above ground of a seat on the wheel at time $t$ (in seconds) can be modelled by

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

Find the period of the model. What does the period tell you about the ride?

Calculation: The height of the seat from the ground on Ferris wheel ig given by,

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

It is known for a function of the form $y=d+a\sin \left(bx-c\right)$ , the period and amplitude is given by,

  $Amplitude=\left|a\right|$

  $Period=\begin{array}{c}2\pi \\ b\end{array}$

This shift is $d$ units upwards for $d>0$ and $d$ units downwards for $d<0$ .

The period of the model $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$ is,

  $Period=\begin{array}{c}2\pi \\ b\end{array}$

  $\begin{array}{l}Period=\begin{array}{c}2\pi \\ (\begin{array}{c}\pi \\ 10\end{array})\end{array}\\ =20\end{array}$

The period is found to be $\boxed{20}$ sec.

Hence, the Ferris wheel complete one ride in $\boxed{20}$ sec.

b.

To determine

The amplitude of the model.

What does the amplitude tell you about the ride?

Answer to Problem 93E

  $\boxed{50}$ feet.

Explanation of Solution

Given information:

A Ferris wheel is built such that the height $h$ (in feet) above ground of a seat on the wheel at time $t$ (in seconds) can be modelled by

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

Find the amplitude of the model. What does the amplitude tell you about the ride?

Calculation:

The height of the seat from the ground on Ferris wheel ig given by,

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

It is known for a function of the form $y=d+a\sin \left(bx-c\right)$ , the period and amplitude is given by,

  $Amplitude=\left|a\right|$

  $Period=\begin{array}{c}2\pi \\ b\end{array}$

This shift is $d$ units upwards for $d>0$ and $d$ units downwards for $d<0$ .

The amplitude of the model $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$ is,

  $\begin{array}{l}Amplitude=\left|50\right|\\ =50\end{array}$

The amplitude is found to be $50$ feet above the ground.

Hence, it means a Ferris wheel has a ride of height $\boxed{50}$ feet.

c.

To determine

Graph one cycle of the model.

Answer to Problem 93E

  

Explanation of Solution

Given information:

A Ferris wheel is built such that the height $h$ (in feet) above ground of a seat on the wheel at time $t$ (in seconds) can be modelled by

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

Use a graphing utility to graph one cycle of the model.

Calculation:

The height of the seat from the ground on Ferris wheel ig given by,

  $h(t)=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}t-\begin{array}{c}\pi \\ 2\end{array})$

It is known for a function of the form $y=d+a\sin \left(bx-c\right)$ , the period and amplitude is given by,

  $Amplitude=\left|a\right|$

  $Period=\begin{array}{c}2\pi \\ b\end{array}$

This shift is $d$ units upwards for $d>0$ and $d$ units downwards for $d<0$ .

To graph this function in TI-83, proceed as below,

Press the button $\boxed{Y=}$ and type $Y=53+50\sin (\begin{array}{c}\pi \\ 10\end{array}X-\begin{array}{c}\pi \\ 2\end{array})$

  

Press the button $\boxed{WINDOW}$ and adjust the window as shown:

  

Press the button $\boxed{TRACE}$ or $\boxed{GRAPH}$ to obtain the graph.

  

Hence, the result.