Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 52 degrees occurs at 6 PM and the average temperature for the day is 45 degrees. Find the temperature, to the nearest degree, at 8 AM.

Answer 1

24 hour period
t =0 is midnight
t = 8 is 8 am
t = 18 = 6 pm
temp = y
y = 45 + (52 - 45) cos ( (2 pi /24)(t-18) )
y = 45 + 7 cos ( pi t/12 - 3/2 pi)
y = 45 + 7 cos (pi *8/12 - 18 pi/12) = 45 + 7 cos(-10 pi/12)
= 45 + 7 cos (- 5 pi/6) = 45 + 7 cos (-150 deg)
= 45 + 7 (-0.866) = 45 - 6.06 = 38.9

Answer 2

the period is 24 hours
since cosine has a maximum at t=0, shift that to 6 pm.
y = 45 + 7cos(π/12 (t-18))
now find y(8)