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

f(t) = −70 cos pi over 4 t + 90

Explanation:

Answer 2

The cosine function that models this reaction is:

`y=-70cos((1)/(4)\pi x) +90`

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.

`(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 =(160-20)/(2)\\\\A= 70`

The reaction completes a cycle in 8 hours

Then the period is 8 hours.

Thus:

`(2\pi)/(b)=8\\\\ b=(2\pi)/(8)\\\\ b=(1)/(4)\pi`

The function is:

`y = 70cos((1)/(4)\pi x)+k`

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

So

`y = -70cos((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((1)/(4)\pi x) +90`