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}{14}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 = 14, 21, 28, 35, and 42. describe what you observe. b. what is the period of the emotional cycle? a. find e corresponding to t = 14. e = 3.115 find e corresponding to t = 21. e = 4.673 find e corresponding to t = 28. e = 6.231 find e corresponding to t = 35. e = 7.788 find e corresponding to t = 42. e = 9.46

Explanation:

Step1: Calculate E for t = 14

Substitute t = 14 into $E=\sin\frac{\pi}{14}t$. So $E = \sin\frac{\pi}{14}\times14=\sin\pi = 0$.

Step2: Calculate E for t = 21

Substitute t = 21 into $E=\sin\frac{\pi}{14}t$. So $E=\sin\frac{\pi}{14}\times21=\sin\frac{3\pi}{2}=- 1$.

Step3: Calculate E for t = 28

Substitute t = 28 into $E=\sin\frac{\pi}{14}t$. So $E=\sin\frac{\pi}{14}\times28=\sin2\pi = 0$.

Step4: Calculate E for t = 35

Substitute t = 35 into $E=\sin\frac{\pi}{14}t$. So $E=\sin\frac{\pi}{14}\times35=\sin\frac{5\pi}{2}=1$.

Step5: Calculate E for t = 42

Substitute t = 42 into $E=\sin\frac{\pi}{14}t$. So $E=\sin\frac{\pi}{14}\times42=\sin3\pi = 0$.
Observation: The values of E cycle between - 1, 0 and 1.

Step6: Find the period of the function

For a sine - function of the form $y = A\sin(Bx + C)+D$, the period is given by $T=\frac{2\pi}{B}$. In the function $E=\sin\frac{\pi}{14}t$, $B = \frac{\pi}{14}$. So $T=\frac{2\pi}{\frac{\pi}{14}}=28$ days.

Answer:

a. When t = 14, E = 0; when t = 21, E=-1; when t = 28, E = 0; when t = 35, E = 1; when t = 42, E = 0. Observation: The values of E cycle between - 1, 0 and 1.
b. The period of the emotional cycle is 28 days.