Chapter 6.4, Problem 93AYU

a.

To determine

Find $\omega$ for each characteristic.

Answer to Problem 93AYU

  $\boxed{\omega =\frac{2\pi }{23}}$ , $\boxed{\omega =\frac{\pi }{14}}$ , $\boxed{\omega =\frac{2\pi }{33}}$ .

Explanation of Solution

Given information:

In the theory of biorhythms a sine function of the form $P\left(t\right)=50\sin \left(\omega t\right)+50$ is used to measure the percent $P$ of a person’s potential at time $t$ , where $t$ is measured in days and $t=0$ is the person’s birthday. Three characteristics are commonly measured.

Physical potential: period $23$ days.

Emotional potential: period $28$ days.

Intellectual potential: period $33$ days.

Find $\omega$ for each characteristic.

Calculation:

We have to find $\omega$ by using $\omega =\frac{2\pi }{T}$ , Physical potential of period $23$ days will be,

  $\begin{array}{l}\omega =\frac{2\pi }{T}\\ \omega =\frac{2\pi }{23}\end{array}$

  $\boxed{\omega =\frac{2\pi }{23}}$

Emotional potential for period $28$ days.

  $\begin{array}{l}\omega =\frac{2\pi }{T}\\ \omega =\frac{2\pi }{28}\\ \omega =\frac{\pi }{14}\end{array}$

  $\boxed{\omega =\frac{\pi }{14}}$

Intellectual potential for period $33$ days.

  $\begin{array}{l}\omega =\frac{2\pi }{T}\\ \boxed{\omega =\frac{2\pi }{33}}\end{array}$

Hence, $\omega$ is $\boxed{\omega =\frac{2\pi }{23}}$ , $\boxed{\omega =\frac{\pi }{14}}$ , $\boxed{\omega =\frac{2\pi }{33}}$ .

b.

To determine

Graph all tree functions on same graph.

Answer to Problem 93AYU

  

Explanation of Solution

Given information:

In the theory of biorhythms a sine function of the form $P\left(t\right)=50\sin \left(\omega t\right)+50$ is used to measure the percent $P$ of a person’s potential at time $t$ , where $t$ is measured in days and $t=0$ is the person’s birthday. Three characteristics are commonly measured.

Physical potential: period $23$ days.

Emotional potential: period $28$ days.

Intellectual potential: period $33$ days.

Using graphing utility, graph all tree functions on same graph.

Calculation:

For Physical potential of period $23$ days function will be ,

  $P\left(t\right)=50\sin \left(\frac{2\pi }{23}t\right)+50$

For emotional potential for period $28$ days function will be .

  $P\left(t\right)=50\sin \left(\frac{\pi }{14}t\right)+50$

For Intellectual potential for period $33$ days function will be.

  $P\left(t\right)=50\sin \left(\frac{2\pi }{33}t\right)+50$

Graph of all functions will be,

  

Hence, the graph is drawn.

c.

To determine

Is there a time $t$ when all the three characteristics have $100\%$ potential? When it is?

Answer to Problem 93AYU

  $\boxed{No}$

Explanation of Solution

Given information:

In the theory of biorhythms a sine function of the form $P\left(t\right)=50\sin \left(\omega t\right)+50$ is used to measure the percent $P$ of a person’s potential at time $t$ , where $t$ is measured in days and $t=0$ is the person’s birthday. Three characteristics are commonly measured.

Physical potential: period $23$ days.

Emotional potential: period $28$ days.

Intellectual potential: period $33$ days.

Is there a time $t$ when all the three characteristics have $100\%$ potential? When it is?

Calculation:

No, there is no time $t$ when all the three characteristics have $100\%$ potential, because the given function $P\left(t\right)=50\sin \left(\omega t\right)+50$ have maximum value of $50\%$ only as seen in the graph.

Hence, No, there is no time $t$ when all the three characteristics have $100\%$ potential.

d.

To determine

Describe your physical, emotional and intellectual potential for the next $30days$ .

Answer to Problem 93AYU

  $\boxed{\begin{array}{l}{P}_1\left(t\right)=-66.168\\ P_2\left(t\right)=6.3\\ P_3\left(t\right)=-7.774\end{array}}$

Explanation of Solution

Given information:

In the theory of biorhythms a sine function of the form $P\left(t\right)=50\sin \left(\omega t\right)+50$ is used to measure the percent $P$ of a person’s potential at time $t$ , where $t$ is measured in days and $t=0$ is the person’s birthday. Three characteristics are commonly measured.

Physical potential: period $23$ days.

Emotional potential: period $28$ days.

Intellectual potential: period $33$ days.

Suppose that you are $20$ years old today $\left(t=7305days\right)$ . Describe your physical, emotional and intellectual potential for the next $30days$ .

Calculation:

Age is given as $20$ years so that age function will be.

  $y=50\sin \left(7305t\right)+50$

Now find physical potential for next $30days$ with $\omega =\frac{2\pi }{23}$ will be,

  $\begin{array}{l}{P}_1\left(t\right)=P\left(7335\right)-P\left(7305\right)\\ =50\sin \left(7335.\frac{2\pi }{23}\right)+50-50\sin \left(7305.\frac{2\pi }{23}\right)-50\\ =0.00185-66.17\\ =-66.168\end{array}$

Now find emotional potential for next $30days$ with $\boxed{\omega =\frac{\pi }{14}}$ will be,

  $\begin{array}{l}{P}_2\left(t\right)=P\left(7335\right)-P\left(7305\right)\\ =50\sin \left(7335.\frac{\pi }{14}\right)+50-50\sin \left(7305.\frac{\pi }{14}\right)-50\\ =6.41-0.11\\ =6.3\end{array}$

Now find intellectual potential for next $30days$ with $\omega =\frac{2\pi }{33}$ will be,

  $\begin{array}{l}{P}_3\left(t\right)=P\left(7335\right)-P\left(7305\right)\\ =50\sin \left(7335.\frac{2\pi }{33}\right)+50-50\sin \left(7305.\frac{2\pi }{33}\right)-50\\ =92.224-99.998\\ =-7.774\end{array}$

Hence, physical, emotional and intellectual potential for the next $30days$ is $\boxed{\begin{array}{l}{P}_1\left(t\right)=-66.168\\ P_2\left(t\right)=6.3\\ P_3\left(t\right)=-7.774\end{array}}$ .