Final answer:
The cosine function that models the given chemical reaction is T(t) = 70cos(πt/4) + 20, where T(t) represents the temperature at time t.
Step-by-step explanation:
The temperature of a chemical reaction can be modeled using a cosine function. We are given that the temperature ranges from 20°C to 160°C and that the reaction completes one cycle during an eight-hour period. The general form of a cosine function is T(t) = A*cos(Bt+C) + D. To find the specific function that models this reaction, we need to determine the values of A, B, C, and D.
Since the temperature is at its lowest when T equals zero, D will represent the lowest temperature, which is 20°C. The range of the cosine function is the difference between the highest and lowest temperatures, so A will be half of that range, which is (160-20)/2 = 70. To find B, we divide the period (eight hours) by 2π, since one complete cycle corresponds to 2π radians. So B = 2π / 8 = π/4. Finally, to find C, we find the phase shift by determining the value of t when the temperature is at its lowest. The phase shift is t at the lowest temperature divided by the period, which is (0) / 8 = 0.
Putting it all together, the cosine function that models this reaction is T(t) = 70cos(πt/4) + 20.