91Ó°ÊÓ

The given sin function$P\left(t\right)=50\sin \left(\mathrm{Ó\neg }t\right)+50$

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

$$\begin{gathered} \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T} \\ =\frac{2\mathrm{Ï€}}{23} \\ \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{23} \end{gathered}$$

$$\begin{gathered} \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T} \\ =\frac{2\mathrm{Ï€}}{28} \\ \mathrm{Ó\neg }=\frac{\mathrm{Ï€}}{14} \end{gathered}$$

For intellectual potential with period of 33 days-:

$$\begin{gathered} \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T} \\ =\frac{2\mathrm{Ï€}}{33} \\ \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{33} \end{gathered}$$

Hence,required graph of 3 functions

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

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

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

To find the emotional potential for the next 30 days with $\mathrm{Ó\neg }=\frac{\mathrm{Ï€}}{14}$

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

Therefore, my physical potential is localid="1646484480886" $P_1\left(t\right)≈-66.168$, emotional potential is localid="1646484515078" $P_2\left(t\right)≈6.3$and the intellectual potential is localid="1646484549362" $P_3≈-7.774$for the next 30 days if 20 years old.