Question

The temperature of a chemical reaction ranges between –10°C and 50°C. The temperature is at its lowest point when t = 0 and completes one cycle over a six-hour period. What is a sine function that would model this reaction?

Answer 1

y = Asin(Bt + C) + D

A is the amplitude, B is the period, -C/B is the phase shift and D is the vertical shift.

Amplitude A = 1/2 (max y - min y) = 1/2 * (50 - (-10)) = 30.

Period = 2*pi / B. Here period = 6 hours. Therefore, B = 2*pi/6 = pi/3

Find the midline between -10 and 50. It is (-10+50)/2 = 20. This is the vertical shift D.

y = 30sin(pi/3 * t + C) + 20

"The temperature is at its lowest point when t = 0".
sin(x) function attains its minimum when x = 3*pi/2
Put t = 0 in sin(pi/3 * t + C). sin(C) and it will be minimum when C = 3pi/2

y = 30sin(pi/3 * t + 3pi/2) + 20