need help with trig

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 41 and 69 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 48 degrees?

Answer 1

The peak to peak temperature variation is 69 - 41 = 28 degrees.

The peak variation is 28/2 = 14 degrees.

The average temperature is 41 + 28/2 = 55 degrees. We can write temperature (T) as a function oif time (t) in hours (time runs from 0 to 23 hours) as:

T(t) = 55 + 14*sin((2*pi*t/24) + X)

where X is selected to give the average temperature at t = 8 hours.

T(8) = 55 + 14*sin((16*pi/24) + X) = 55 so

sin((2*pi/3) + X) = 0

or

X = -2*pi/3

To find the time at which the temperature reaches 48 degrees consider

55 + 14*sin((2*pi*t/24) - 2*pi/3) = 48

Therefore

sin((2*pi*t/24 - 2*pi/3) = -0.5

The angle whose sine is -0.5 is -0.5236 radians, so:

-0.5236 = (2*pi*t/24) - 2*pi/3

or

t = 6.00 hours (6 AM)