Answer :
The cosine function of a chemical reaction that occurs at temperatures between 40 and 180 degrees Celsius is [tex]f (t) = -70\cos \left(\frac{\pi}{6}\right)t+110[/tex]. Option c is the correct expression.
A cosine function's amplitude is equal to half the vertical distance between its highest and lowest points. The distance between the tallest point and the center line is equal to this. This is of the form f (t) = A cos (bt)+k where A is the wave amplitude, t is the time period, and k is the vertical displacement.
In this problem, the temperature is measured by f(t), which ranges from 40 to 180 degrees Celsius. Then the temperature is not negative. The amplitude will be,
[tex]A=\frac{|180-40|}{2}=70[/tex]
Given the time period is 12 hours.
[tex]\begin{aligned}\text{Period}&= \frac{2\pi}{b}\\12&= \frac{2\pi}{b}\\b&= \frac{2\pi}{12}\\b&=\frac{\pi}{6}\end{aligned}[/tex]
When, t = 0, the wave must be at its lowest point. Then, f(t)=40. Find k using this.
[tex]\begin{aligned}f(t=0)&=40\\40&=-70\cos\left(\frac{\pi}{6}\times 0\right)+k\\40&=-70+k\\k&=110\end{aligned}[/tex]
The required cosine function will be of the form, [tex]f (t) = -70\cos \left(\frac{\pi}{6}\right)t+110[/tex]
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