Chapter 4.5, Problem 80E

a.

To determine

Find the time for one full respiratory cycle.

Answer to Problem 80E

  $\boxed{t=6\sec .}$

Explanation of Solution

Given information:

For a person at rest, 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 $\text{v }=0.\text{85}$ sin $\left(\pi \text{t/3}\right),$ where t is the time (in seconds).

Find the time for one full respiratory cycle.

Calculation:

We have given $V=0.85$

  $\sin \frac{\pi t}{3}$ , $t=time$ . Function $y=\sin x$ has a period of $2\pi$ so let’s compare these two equation.

  $\frac{\pi t}{3}=2\pi$

So, $\frac{t}{3}=2$

  $t=6\sec .$

Hence, the one respiratory period lasts in $6$ seconds.

b.

To determine

Find the number of cycles per minute.

Answer to Problem 80E

  $\boxed{10cycle}$

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:

We have given $V=0.85$

  $\sin \frac{\pi t}{3}$ , $t=time$ . Function $y=\sin x$ has a period of $2\pi$ so let’s compare these two equation.

  $\frac{\pi t}{3}=2\pi$

So, $\frac{t}{3}=2$

  $t=6\sec .$

If one cycle in 6 second then cycle in one minute is $\frac{60}{6}=10cycle$ .

Hence, the cycle in one minute is $\boxed{10cycle}$

c.

To determine

Sketch the graph of velocity function.

Answer to Problem 80E

  

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. Use the graph to confirm your answer in part (a) by finding two times when new breaths begin. (Inhalation occurs when v > 0, and exhalation occurs when v < 0.)

Calculation:

We have given $V=0.85$

  $\sin \frac{\pi t}{3}$ , $t=time$ . Function $y=\sin x$ has a period of $2\pi$ so let’s compare these two equation.

  $\frac{\pi t}{3}=2\pi$

So, $\frac{t}{3}=2$

  $t=6\sec .$

  $amplitude=0.85$

  $period=6$

  $frequency=\frac{\pi }{3}$

Hence, It has no phase shift and no vertical shift.