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?

Answer 1

Step-by-step explanation:

The general cosine wave function is:

y = A cos(ωt + φ) + B

where A is the amplitude, ω is the frequency, φ is the phase offset, and B is the vertical offset.

The temperature ranges from -10 to 50, so the magnitude of the amplitude is:

|A| = (50 − -10) / 2

|A| = 30

It's at its lowest point at t=0, so the sign of the amplitude is -1:

A = -30

The lowest point, -10, is 20 more than the amplitude, so the vertical offset is:

B = 20

The reaction completes 1 cycle in 6 hours, so f = 1/6. ω = 2πf, so:

ω = 2π (1/6)

ω = π/3

So the function is:

y = -30 cos(π/3 t) + 20

Alternatively, instead of a negative sign, we could have added a phase shift of π:

y = 30 cos(π/3 t + π) + 20

Either of these answers is correct.