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 1

Answer:

The cosine function that models this reaction is:

$y=-70cos(\frac{1}{4}\pi x) +90$

Step-by-step explanation:

The general cosine function has the following form

$y = Acos(bx) + k$

Where A is the amplitude: half the vertical distance between the highest peak and the lowest peak of the wave.

$\frac{2\pi}{b}$ 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:

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

The reaction completes a cycle in 8 hours

Then the period is 8 hours.

Thus:

$\frac{2\pi}{b}=8\\\\ b=\frac{2\pi}{8}\\\\ b=\frac{1}{4}\pi$

The function is:

$y = 70cos(\frac{1}{4}\pi x)+k$

when $t=0$ y is minumum therefore $y=-cos(x)$

So

$y = -70cos(\frac{1}{4}\pi x)+k$

Now we substitute $t = 0$ in the function and solve for k

$20 = -70cos(0)+k\\\\k=20+70\\\\k=90$

Finally

$y=-70cos(\frac{1}{4}\pi x) +90$