Outside temperature over the course of a day can be modeled

Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 64 degrees and 86 degrees during the day and the average daily temperature first occurs at 12AM. How many hours after midnight does the temperature first reach 70 degrees?

Please show your work! I am having trouble understanding this problem.

Solution

Avg Temp = (64 + 86)/2 = 75
Range = 86 - 64 = 22

The Temperature oscillates ±11° about the mean temperature with a period of 24 hours
T(t) = 75 + 11*sin(t/12 + ) : Temperature at time t hours after midnight
T(12) = 75 = 75 + 11*sin(0 + )
sin(0 + ) = 0 = 0

T(t) = 75 + 11*sin(t/12)
T(t) = 75 + 11*sin[t/12]

70 = 75 + 11*sin[t/12]
sin[t/12] = (70 - 75)/11
t/12 = arcsin(-5/11)
t = 12arcsin(-5/11)/ = -1.8

The temperature first reaches 70 degrees at 10.2 hours (10:12 pm) before midnight