Question

The temperature of a chemical reaction ranges between −10 degrees Celsius and 50 degrees Celsius. The temperature is at its lowest point when t = 0, and the reaction completes 1 cycle during a 6-hour period. What is a cosine function that models this reaction?

f(t) = −30 cos (pi/3) t + 20

f(t) = −60 cos (pi/3) t + 30

f(t) = 30 cos (6t) + 20

f(t) = 60 cos (6t) + 30

Answer 1

maximum amount of cos is 1 and it's minimum is -1 so the answer should be something you put 1 and -1 (without attaining to cos itself) find -10 and 50
so it's the first one or the third one
(period)T=2pi/a
6=2pi/a
a=pi/3
the first one is the answer

Answer 2

Option 1 -
`f(t)=-30 sin((\pi)/(3)t)+20`

Explanation:

Given : The temperature of a chemical reaction ranges between −10 degrees Celsius and 50 degrees Celsius. The temperature is at its lowest point when t = 0, and the reaction completes 1 cycle during a 6-hour period.

To find : What is a cosine function that models this reaction?

Solution :

General form of cosine function is
`f(x)=A cos(Bx)+C`

Where A is the amplitude

`B=\frac{2\pi}{\text{Period}}`

C is the midline

Now, We have given

The temperature of a chemical reaction ranges between −10 degrees Celsius and 50 degrees Celsius.

A is the average of temperature,

i.e,
`A=(-10-50)/(2)=-30`

Period of 1 cycle is 6 hour

So,
`B=(2\pi)/(6)=(\pi)/(3)`

The temperature is at its lowest point when t = 0 and we know lowest point is -10

So,
`f(t)=A\cos t+C`

`-10=-30\cos 0+C`

`C=20`

Substituting the values we get,

The cosine function is
`f(t)=-30 sin((\pi)/(3)t)+20`

Therefore, Option 1 is correct.