Chapter 4.5, Problem 79E

a.

To determine

Find the time for one full respiratory cycle.

Answer to Problem 79E

  $=\frac{2\pi }{\left(\frac{\pi }{2}\right)}$

Explanation of Solution

Given information:

For a person exercising, the velocity v (in litters per second) of airflow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is modelled by $v=1.75$ $\sin \left(\pi t/2\right)$ 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:

  $v=1.75\sin \left(\frac{\pi t}{2}\right)$

It is know that the period for a function of the form $y=a\sin b\chi$ is given by:

Period= $\frac{2\pi }{b}$

Then period of $v=1.75\sin \left(\frac{\pi t}{2}\right)is$

Period = $\frac{2\pi }{b}$

  $=\frac{2\pi }{\left(\frac{\pi }{2}\right)}$

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

b.

To determine

Find the number of cycles per minute.

Answer to Problem 79E

  $\boxed{15}$ Cycles per minute.

Explanation of Solution

Given information:

For a person exercising, the velocity v (in litters per second) of airflow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is modelled by $v=1.75$ $\sin \left(\pi t/2\right)$ 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 $4$ sec, then number of cycles in $1$ minute will be

  $\frac{60}{4}=15$

Hence numbers of cycles per minute are $\boxed{15}$ cycles per minute.

c.

To determine

Sketch the graph of the velocity function.

Answer to Problem 79E

  

Explanation of Solution

Given information:

For a person exercising, the velocity v (in litters per second) of airflow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is modelled by $v=1.75$ $\sin \left(\pi t/2\right)$ 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 $v=1.75\sin \left(\frac{\pi t}{2}\right)$ is given below: