Question

The temperature of a chemical reaction ranges between 20 degrees Celsius and 160 degrees Celsius. The temperature is at its lowest point when t = 0, and the reaction completes 1 cycle during an 8-hour period. What is a cosine function that models this reaction?
answer choices

f(t) = -90 cos pi/4t +70

f(t) = -70 cos pi/4t +90

f(t) = 70 cos 8t +90

f(t)= 90 cos 8t +70

Answer 1

Answer:

The answer is teh second option $f(t) = -70cos(\frac{\pi}{4}t) +90$

Step-by-step explanation:

The cosine functions have the following form:

$f(t) = Acos (bt) + k$

Where A is the amplitude of the wave. The amplitude of a wave is equal to half the distance between its largest peak and its smallest peak.

$\frac{2\pi}{b}$ is the period. The period is the time t that the wave takes to complete a cycle.

k is the vertical displacement of the wave.

In this problem f(t) measures the temperature in degrees Celsius and goes from 20 degrees Celsius to 160 degrees. It is never negative, then

Then we can find its amplitude A.

$A = |\frac{(160-20)}{2}|= 70$

The period is 8 hours.

Then we can find b.

$8 = \frac{2\pi}{b}\\\\b = \frac{2\pi}{8}\\\\b =\frac{\pi}{4}$

Then, we know that when t = 0, the wave must be at its minimum value $f(t) = 20$

Then we find k

$f(t = 0) = 20 = -70cos(\frac{\pi}{4}(0)) + k\\\\20 = -70 + k\\\\k = 90$

So

The function is:

$f(t) = -70cos(\frac{\pi}{4}t) +90$