Answer:
The cosine function that models this reaction is:
[tex]y=-70cos(\frac{1}{4}\pi x) +90[/tex]
Step-by-step explanation:
The general cosine function has the following form
[tex]y = Acos(bx) + k[/tex]
Where A is the amplitude: half the vertical distance between the highest peak and the lowest peak of the wave.
[tex]\frac{2\pi}{b}[/tex] is the period: time it takes the wave to complete a cycle.
k is the vertical displacement.
The maximum temperature is 160 and the minimum is 20. Then the amplitude A is:
[tex]A =\frac{160-20}{2}\\\\A= 70[/tex]
The reaction completes a cycle in 8 hours
Then the period is 8 hours.
Thus:
[tex]\frac{2\pi}{b}=8\\\\ b=\frac{2\pi}{8}\\\\ b=\frac{1}{4}\pi[/tex]
The function is:
[tex]y = 70cos(\frac{1}{4}\pi x)+k[/tex]
when [tex]t=0[/tex] y is minumum therefore [tex]y=-cos(x)[/tex]
So
[tex]y = -70cos(\frac{1}{4}\pi x)+k[/tex]
Now we substitute [tex]t = 0[/tex] in the function and solve for k
[tex]20 = -70cos(0)+k\\\\k=20+70\\\\k=90[/tex]
Finally
[tex]y=-70cos(\frac{1}{4}\pi x) +90[/tex]
