Chapter 4, Problem 10PS

a.

To determine

Graph the three models in the same viewing window by using graphing utility.

Answer to Problem 10PS

  

Explanation of Solution

Given information:

A popular theory that attempts to explain the ups and downs of everyday life states that each person has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by the sine functions below, where $t$ is the number of days since birth.

  $\begin{array}{l}Physical\left(23days\right):P=\sin \frac{2\pi t}{23},t\ge 0\\ Emotional\left(28days\right):E=\sin \frac{2\pi t}{28},t\ge 0\\ Intellectual\left(33days\right):I=\sin \frac{2\pi t}{33},t\ge 0\end{array}$

Consider a person who was born on July $20,1995$ .

Use a graphing utility to graph the three models in the same viewing window for $7300\le t\le 7380$ .

Calculation:

Here, we will consider the models: $P=\sin \frac{2\pi t}{23},E=\sin \frac{2\pi t}{28}$ and $I=\sin \frac{2\pi t}{33}$

Now we will draw the graph of the given function on the given rectangular coordinate system in $7300\le t\le 7380$ :

  

Here,

The black graph is: $P$

The blue graph is: $E$

The green graph is: $I$

Hence, the results are shown in the above graph.

b.

To determine

Describe the person’s biorhythms during the month of September $2015$ .

Answer to Problem 10PS

  $\boxed{firsttheyallstartouthigh,thengodown,andthengobackupagain}$

Explanation of Solution

Given information:

A popular theory that attempts to explain the ups and downs of everyday life states that each person has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by the sine functions below, where $t$ is the number of days since birth.

  $\begin{array}{l}Physical\left(23days\right):P=\sin \frac{2\pi t}{23},t\ge 0\\ Emotional\left(28days\right):E=\sin \frac{2\pi t}{28},t\ge 0\\ Intellectual\left(33days\right):I=\sin \frac{2\pi t}{33},t\ge 0\end{array}$

Consider a person who was born on July $20,1995$ .

Describe the person’s biorhythms during the month of September $2015$ .

Calculation:

Here, our objective is to describe the person’s biorhythms during the month of September $2015$ .

Hence, in the above part, first they all start out high, then go down, and then go back up again.

c.

To determine

Calculate the person’s three energy levels on September $22,2015$ .

Answer to Problem 10PS

  $\boxed{P=0.888}$ , $\boxed{E=0}$ , $\boxed{I=0.815}$

Explanation of Solution

Given information:

A popular theory that attempts to explain the ups and downs of everyday life states that each person has three cycles, called biorhythms, which begin at birth. These three cycles can be modeled by the sine functions below, where $t$ is the number of days since birth.

  $\begin{array}{l}Physical\left(23days\right):P=\sin \frac{2\pi t}{23},t\ge 0\\ Emotional\left(28days\right):E=\sin \frac{2\pi t}{28},t\ge 0\\ Intellectual\left(33days\right):I=\sin \frac{2\pi t}{33},t\ge 0\end{array}$

Consider a person who was born on July $20,1995$ .

Calculate the person’s three energy levels on September $22,2015$ .

Calculation:

Here, our objective is to find the person’s three energy levels on September $22,2015$ .

We are given that the person was born on July $20,1995$ and energy level is to be checked on September $22,2015$ . Therefore, the numbers of days are $t=7364$ .

Now, we will substitute $t=7364$ in $P=\sin \frac{2\pi t}{23}$ to find the value of $P$ .

  $P=\sin \frac{2\pi \left(7364\right)}{23}=0.888$

Now, we will substitute $t=7364$ in $E=\frac{2\pi t}{28}$ to find the value of $E$ .

  $E=\sin \frac{2\pi \left(7364\right)}{28}=0$

Now, we will substitute $t=7364$ in $I=\frac{2\pi t}{33}$ to find the value of $I$ .

  $I=\sin \frac{2\pi \left(7364\right)}{33}=0.815$

Hence, the person’s three energy levels on September $22,2015$ are $\boxed{P=0.888}$ , $\boxed{E=0}$ and $\boxed{I=0.815}$ .