# Thread: Application of a Cosine Function

1. ## Application of a Cosine Function

The pressure $P$ (in milimeters of mercury against the walls of the blood vessels of a patient is modeled by
$P = 100-\cos ({\frac {8\pi}{3}t})$ where $t$ is time (in seconds).

(a) Use a graphing utility to graph the model.
Done

(b) What is the period of of the model? What does the period tell you about the situation?
$P = \frac {2\pi}{\frac {8\pi}{3}} = \frac {3}{4}$
I got the period $\frac {3}{4}$, but what does is it mean in this application problem?

(c) What is the amplitude of the model? What does it tell you about the situation?
The amplitude would be 20, but what does it mean in this application problem?

(d) If one cycle of this model is equivalent to one heartbeat, what is the pulse of this patient?
I do not get this question at all. Any ideas?

(e) If a physician wants this patient's pulse rate to be 64 beats per minute or less, what should the period be? What should the coefficient of $t$ be?
Would the period be $\frac {1}{64}$? Then the cofficient of $t$ would be $\frac {2\pi}{\frac {1}{64}} = 128 \pi$ right?

I have attached a picture of the original problem for clarification if needed.

2. Originally Posted by chrozer
The pressure $P$ (in milimeters of mercury against the walls of the blood vessels of a patient is modeled by
$P = 100-\cos ({\frac {8\pi}{3}t})$ where $t$ is time (in seconds).

correction ...

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

(a) Use a graphing utility to graph the model.
Done

(b) What is the period of of the model? What does the period tell you about the situation?
$P = \frac {2\pi}{\frac {8\pi}{3}} = \frac {3}{4}$
I got the period $\frac {3}{4}$, but what does is it mean in this application problem?

period is the amount of time for the pressure to complete one cycle of change.

(c) What is the amplitude of the model? What does it tell you about the situation?
The amplitude would be 20, but what does it mean in this application problem?

amplitude = 20 ... the pressure cycles from a min of 80 to a max of 120 mmhg. ever hear of blood pressure reading as 120 over 80 ?

(d) If one cycle of this model is equivalent to one heartbeat, what is the pulse of this patient?
I do not get this question at all. Any ideas?

pulse rate is the number of beats in one minute ... one cycle of pressure (one beat) occurs in a time of 3/4 seconds ... so, how many cycles (beats) in one minute?

(e) If a physician wants this patient's pulse rate to be 64 beats per minute or less, what should the period be? What should the coefficient of $t$ be?
Would the period be $\frac {1}{64}$? Then the cofficient of $t$ would be $\frac {2\pi}{\frac {1}{64}} = 128 \pi$ right?

desired period > 60/64 = 15/16 sec.

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

hope that helps.