Chapter 4, Problem 9PS

a.

To determine

Graph the model by using graphing utility.

Answer to Problem 9PS

  

Explanation of Solution

Given information:

The pressure $P$ (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by

  $P=100-20\cos \frac{8\pi t}{3}$

Where $t$ is the time (in seconds).

Use a graphing utility to graph the model.

Calculation:

Here, we will consider the following model:

  $P=100-20\cos \frac{8\pi t}{3}$

Now, we will plot the above graph by using graphing utility:

  $100-20\cos \left(\frac{8\pi }{3}t\right)$

  

Hence, the results are shown in the graph.

b.

To determine

Find the period of the model.

Answer to Problem 9PS

  $\boxed{\frac{3}{4}millimeters}$

Explanation of Solution

Given information:

The pressure $P$ (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by

  $P=100-20\cos \frac{8\pi t}{3}$

Where $t$ is the time (in seconds).

What is the period of the model? What does it represent in the context of the problem?

Calculation:

Here, the period of the model is:

  $\frac{2\pi }{\frac{8\pi }{3}}=\boxed{\frac{3}{4}}$

Hence, the above value tells us that the pressure against the wall completes its one cycle in $\boxed{\frac{3}{4}millimeters}$ of mercury.

c.

To determine

Find the amplitude of the model.

Answer to Problem 9PS

  $\boxed{20millimeters}$

Explanation of Solution

Given information:

The pressure $P$ (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by

  $P=100-20\cos \frac{8\pi t}{3}$

Where $t$ is the time (in seconds).

What is the amplitude of the model? What does it represent in the context of the problem?

Calculation:

Here, the amplitude of the model is:

  $\boxed{20}$

Hence, from the above values we can observe that the strongest pulse is of $\boxed{20millimeters}$ of mercury.

d.

To determine

Find the pulse of the patient.

Answer to Problem 9PS

  $\boxed{80/\min ute}$

Explanation of Solution

Given information:

The pressure $P$ (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by

  $P=100-20\cos \frac{8\pi t}{3}$

Where $t$ is the time (in seconds).

If one cycle of this model is equivalent to one heartbeat, what is the pulse of the patient?

Calculation:

Here, the pulse of the patient is:

  $\frac{60}{\frac{3}{4}}=20\left(4\right)=\boxed{80/\min ute}$

Hence, the pulse of the patient is $\boxed{80/\min ute}$ .

e.

To determine

Find the period and the coefficient of $t$ .

Answer to Problem 9PS

  $\boxed{\frac{15}{16}}$ , $\boxed{\frac{32\pi }{15}}$

Explanation of Solution

Given information:

The pressure $P$ (in millimeters of mercury) against the walls of the blood vessels of a patient is modeled by

  $P=100-20\cos \frac{8\pi t}{3}$

Where $t$ is the time (in seconds).

A physician wants the patient’s pulse rate to be $64$  beats per minute or less. What should the period be? What should the coefficient of $t$ be?

Calculation:

Here, for the person to have a pulse rate of $64$ , the period should be:

  $P=\frac{60}{64}=\boxed{\frac{15}{16}}$

Now, the coefficient of $t$ will be:

  $\frac{2\pi }{\frac{15}{16}}=\boxed{\frac{32\pi }{15}}$

Hence, the period and the coefficient of $t$ is $\boxed{\frac{15}{16}}$ and $\boxed{\frac{32\pi }{15}}$ .