Outside temperature over a day can be modeled as a sinusoida

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 87 degrees occurs at 4 PM and the average temperature for the day is 70 degrees. Find the temperature, to the nearest degree, at 5 AM.

Solution

amplitude is (87-70)=17
center line is 70, so start with the assumption that the minimum occurs at midnight (0 hours):

y = -17cos(kx)+70

The function has a period of 24 hours, so 2/k = 24, making k=/12

y = -17cos(/12 x) + 70

Now we know that the max occurs at x=17, not x=12, so we need to shift by 5:

y = -17cos(/12 (x-5)) + 70