Answer:
[tex]x(t) = -1.5cos(\frac{2 \pi t}{12.5}) + 4[/tex]
Step-by-step explanation:
General formula
[tex]x(t) = -Acos(\frac{2 \pi t}{T}) + B[/tex]
First transform data
12:00 am -> t=0 -> x=2.5
6:15 am -> t=6.25 -> x=5.5
12:30 pm -> t=12.5 -> x=2.5
6:45pm -> t=18.75 -> x=5.5
Period (T) is the time between two equal values of x.
t=0 -> x=2.5
t=12.5 -> x=2.5
t=6.25 -> x=5.5
t=18.75 -> x=5.5
T = 12.5 - 0 = 18.75 - 6.25 = 12.5
B is the shift of the cosine function with respect to y-coordinate. It is halfway between maximum and minimum values of the function
B = (5.5 + 2.5)/2 = 4
The amplitude (A) is the distance from the highest point to B
A = 5.5 - 4 = 1.5
Therefore, the correlation is
[tex]x(t) = -1.5cos(\frac{2 \pi t}{12.5}) + 4[/tex]
Model verification
x(0) = -1.5 + 4 = 2.5
x(6.25) = 1.5 + 4 = 5.5
x(12.5) = -1.5 + 4 = 2.5
