Chapter 4.5, Problem 87E

a.

To determine

Time for one full respiratory cycle.

Answer to Problem 87E

  $\boxed{4}$

Explanation of Solution

Given information:

After exercising for a few minutes, a person has a respiratory cycle for which the velocity of airflow is approximated by

  $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$

where $t$ is the time (in seconds). (Inhalation occurs when $v>0$ , and exhalation occurs when $v<0$ . )

Find the time for one full respiratory cycle.

Calculation:

The velocity of airflow after time $t$ is given by:

  $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$

It is known that the period for a function of the form $y=a\sin bx$ is given by:

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

Then period of $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$ is

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

  $\begin{array}{l}=\begin{array}{c}2\pi \\ \left(\begin{array}{c}\pi \\ 2\end{array}\right)\end{array}\\ =4\end{array}$

Hence, the time for one full cycle is $\boxed{4}$ seconds.

b.

To determine

Find the number of cycles per minute.

Answer to Problem 87E

  $\boxed{15}$

Explanation of Solution

Given information:

After exercising for a few minutes, a person has a respiratory cycle for which the velocity of airflow is approximated by

  $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$

Where $t$ is the time (in seconds). (Inhalation occurs when $v>0$ , and exhalation occurs when $v<0$ . )

Find the number of cycles per minute.

Calculation:

One full cycle requires 4sec., then the number of cycles in $1$ minute will be,

  $\begin{array}{c}60\\ 4\end{array}=15$

Hence, the number of cycles per minute is $\boxed{15}$ .

c.

To determine

Sketch the graph of the velocity function.

Answer to Problem 87E

  

Explanation of Solution

Given information:

After exercising for a few minutes, a person has a respiratory cycle for which the velocity of airflow is approximated by

  $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$

Where $t$ is the time (in seconds). (Inhalation occurs when $v>0$ , and exhalation occurs when $v<0$ . )

Sketch the graph of the velocity function.

Calculation:

The graph of $v=1.75\sin \begin{array}{c}\pi t\\ 2\end{array}$ is given below:

  

Hence, the result.