Answer
[tex]y = 28 \sin(\frac{\pi}{6} - \frac{2\pi}{3}) + 49[/tex]
Step by Step Explanation
Given
See attachment for table
Required
Determine the sinusoidal function
The sinusoidal function is represented as:
[tex]y = A \sin(Bx + C) + D[/tex]
Where
[tex]|A| = Amplitude[/tex]
And it is calculated as:
[tex]A = \frac{1}{2}[Highest - Least][/tex]
From the table:
Highest = 77 and Least = 21
So:
[tex]A = \frac{1}{2}[77 - 21][/tex]
[tex]A = \frac{1}{2}[56][/tex]
[tex]A = 28[/tex]
B is calculated as:
[tex]Period = \frac{2\pi}{B}[/tex]
The period is 12 i.e. 12 months
So:
[tex]\frac{2\pi}{B} = 12[/tex]
[tex]B = \frac{2\pi}{12}[/tex]
[tex]B = \frac{\pi}{6}[/tex]
C is calculated as:
[tex]\frac{C}{B} = -4[/tex]
[tex]C = -4 * B[/tex]
[tex]C = -4 * \frac{\pi}{6}[/tex]
[tex]C = -2 * \frac{\pi}{3}[/tex]
[tex]C = -\frac{2\pi}{3}[/tex]
Lastly, D is calculated as:
[tex]D = Least + A[/tex]
So:
[tex]D =21 + 28[/tex]
[tex]D=49[/tex]
So, the function is:
[tex]y = A \sin(Bx + C) + D[/tex]
[tex]y = 28 \sin(\frac{\pi}{6} - \frac{2\pi}{3}) + 49[/tex]
