Question

people who believe in biorhythms claim there are three cycles that rule our behavior - the physical, emotional, and mental. each is a sine function of a certain period. the function for our emotional fluctuations is e = sin(\frac{\pi}{10}t) (equation may not be based on actual studies.) where t is measured in days starting at birth. emotional fluctuations, e, are measured from - 1 to 1, with 1 representing peak emotional well - being, - 1 representing the low for emotional well - being, and 0 representing feeling neither emotionally high nor low. a. find e corresponding to t = 5, 10, 15, 20, and 25. describe what you observe. b. what is the period of the emotional cycle? a. find e corresponding to t = 5. e =

Explanation:

Step1: Substitute t = 5 into the formula

Substitute \(t = 5\) into \(E=\sin\frac{\pi}{10}t\). So \(E=\sin(\frac{\pi}{10}\times5)=\sin\frac{\pi}{2}\).

Step2: Evaluate the sine - value

We know that \(\sin\frac{\pi}{2}=1\).

Step3: Substitute t = 10 into the formula

Substitute \(t = 10\) into \(E = \sin\frac{\pi}{10}t\). Then \(E=\sin(\frac{\pi}{10}\times10)=\sin\pi\). Since \(\sin\pi = 0\).

Step4: Substitute t = 15 into the formula

Substitute \(t = 15\) into \(E=\sin\frac{\pi}{10}t\). So \(E=\sin(\frac{\pi}{10}\times15)=\sin\frac{3\pi}{2}\). And \(\sin\frac{3\pi}{2}=- 1\).

Step5: Substitute t = 20 into the formula

Substitute \(t = 20\) into \(E=\sin\frac{\pi}{10}t\). Then \(E=\sin(\frac{\pi}{10}\times20)=\sin2\pi\). Since \(\sin2\pi = 0\).

Step6: Substitute t = 25 into the formula

Substitute \(t = 25\) into \(E=\sin\frac{\pi}{10}t\). So \(E=\sin(\frac{\pi}{10}\times25)=\sin\frac{5\pi}{2}\). And \(\sin\frac{5\pi}{2}=1\).

Observation: The emotional - well - being value \(E\) oscillates between 1, 0, and - 1 as \(t\) increases.

Step7: Find the period of the function

For a sine function of the form \(y = A\sin(Bx - C)+D\), the period \(T\) is given by \(T=\frac{2\pi}{B}\). In the function \(E=\sin\frac{\pi}{10}t\), \(B = \frac{\pi}{10}\). Then \(T=\frac{2\pi}{\frac{\pi}{10}}\).

Step8: Simplify the period formula

\(T=\frac{2\pi}{\frac{\pi}{10}}=2\pi\times\frac{10}{\pi}=20\).

Answer:

a. When \(t = 5\), \(E = 1\); when \(t = 10\), \(E = 0\); when \(t = 15\), \(E=-1\); when \(t = 20\), \(E = 0\); when \(t = 25\), \(E = 1\). We observe that the value of \(E\) oscillates between 1, 0, and - 1 as \(t\) increases.
b. The period of the emotional cycle is 20 days.