Q. 93 - Precalculus Enhanced with Graphing Utilities

Question

Biorhythms 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 of 23 days
Emotional potential: period of 28 days
Intellectual potential: period of 33 days
(a) Find $\omega$ for each characteristic. 
(b) Using a graphing utility, graph all three functions on the same screen.
(c) Is there a time t when all three characteristics have $100\%$ potential? When is it?
(d) Suppose that you are 20 years old today $(t=7305days)$. Describe your physical, emotional, and intellectual potential for the next 30 days.

Step-by-Step Solution

Answer

(a)

For physical potential with period of 23 days-: $\omega =\frac{2\mathrm{\pi }}{23}$

For emotional potential with period of 28 days-:$\omega =\frac{\mathrm{\pi }}{14}$

For intellectual potential with period of 33 days-: $\omega =\frac{2\mathrm{\pi }}{33}$

(b) Required graph of 3 sin function

(c) No, there is no time t when all the 3 characteristics have 100% potential.

(d) Physical potential is $P_1\left(t\right)\approx -66.168$ , emotional potential is $P_2\left(t\right)\approx 6.3$ and the intellectual potential is $P_3\approx -7.774$ for the next 30 days if  years old. 

Step 1.Given information

The given sin function $P\left(t\right)=50\sin \left(\omega t\right)+50$

Step 2.(a) Find ω for each characteristic.

We can find value of $\omega$ for each characteristic by using the formula $\omega =\frac{2\mathrm{\pi }}{T}$.
Therefore, for physical potential with period of 23 days-: 

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ =\frac{2\mathrm{\pi }}{23} \\ \omega =\frac{2\mathrm{\pi }}{23} \end{gathered}$$

For emotional potential with period of 28 days-:

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ =\frac{2\mathrm{\pi }}{28} \\ \omega =\frac{\mathrm{\pi }}{14} \end{gathered}$$

For intellectual potential with period of 33 days-: 

$$\begin{gathered} \omega =\frac{2\mathrm{\pi }}{T} \\ =\frac{2\mathrm{\pi }}{33} \\ \omega =\frac{2\mathrm{\pi }}{33} \end{gathered}$$

Step 3.(b) Using a graphing utility, graph all three functions on the same screen.

For the physical potential, the function will be-:

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

For the emotional potential, the function will be-:

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

For the intellectual potential, the function will be-:

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

Hence, the graph of 3 functions using graphing utility will be-:

Hence,required graph of 3 functions

Step 4.(c) Is there a time t when all three characteristics have 100 % potential? When is it?

No, there is no time t when all the 3 characteristics have $100\%$ potential. This is because the given function $y=50\sin \left(\omega t\right)+50$ to find the value of percentage of person's potential can have maximum value of $50\%$ only as seen in the graph. 

Step 5. (d) Suppose that you are 20 years old today ( t = 7305 d a y s ) . Describe your physical, emotional, and intellectual potential for the next 30 days.

Since initial age is given to be 20 yrs, hence the initial function will be $y=50\sin \left(7305\omega \right)+50$.
Now to find the physical potential for the next 30 days with $\omega =\frac{2\mathrm{\pi }}{23}$

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

To find the emotional potential for the next 30 days with $\omega =\frac{\mathrm{\pi }}{14}$

$$\begin{gathered} P_3\left(t\right)=P\left(7335\right)-P\left(7305\right) \\ =50\sin (7335.\frac{2\mathrm{\pi }}{33})+50-50\sin (7305.\frac{2\mathrm{\pi }}{33})-50 \\ =92.224-99.008 \\ \approx -7.774 \end{gathered}$$

Therefore, my physical potential is $P_1\left(t\right)\approx -66.168$, emotional potential is $P_2\left(t\right)\approx 6.3$ and the intellectual potential is $P_3\approx -7.774$ for the next 30 days if 20 years old.