Hello ! Can anyone help me solve for the second bullet? I tried solving for it, but I didn't get any feasible answer. I was able to solve for the first question and I would also appreciate feedback :D

Question

In a city, at any particular time of the day from November 19 to November 21,

the temperature in Celsius is given by f(t) = 18 − 4 cos ( π /12(t + 1)) degrees Celsius, at t hours since the midnight of November 18.

a. What is the temperature at 3:00 am on November 19?

b. What time will the maximum temperature occur on November 20?

"midnight of November 18" refers to the end of November 18 or 0 hours of November 19

Answer 1

As to the first question, 3:00 am is 3 hours after midnight, so t = 3 and the temperature is

18 - 4 cos (π/3) = 18 - 4*(1/2) = 16

f(t) will be at a maximum when cos ((t+1) π/12) is at a minimum. The smallest possible value of cos θ is -1, when θ is an odd multiple of π. So to maximize f(t), we need to have (t+1)/12 equal to an odd integer. On November 20, t lies between 24 to 48. (t+1)/12 is odd when t = 35, which is 11 hours after midnight.

Please let me know if you need clarification.

Answer 2

Let θ = π/12(t+1), the function inside the cosine function.

Therefore, 18 -4cos(π/12(t+1)) becomes 18-4cos(θ)

Now realize that -1 ≤ cos θ ≤ 1.

-4≤ -4cos θ ≤ 4

14 ≤ 18-4cos θ ≤ 22

18-4cos θ is maximized when cos θ = -1, which results in a maximum value of 22.

Therefore, θ = π + 2πn = π(2n + 1) where n is an integer.

Now make a substitution back into terms of t. Remember, θ = π/12(t+1)

Therefore, π(2n + 1) = π/12(t+1).

Divide by π

2n + 1 = 1/12(t+1)

Multiply by 12

24n + 12 = t + 1

24n + 11 = t

Plug in integer values of n to get t. The temperature is maximized at these value of t.