Outside temperature over a day can be modeled as a sinusoidal function.

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 85 degrees occurs at 6 PM and the average temperature for the day is 70 degrees. Find the temperature, to the nearest degree, at 9 AM.

Answer 1

Since the average temperature of the day is 70°, that must be the vertical shift of the sine function we use to model the temperature. Also, the maximum temperature, 85°, minus that average gives the amplitude of the sine function as 15. The period of the sine function is 24 hours, so equating that to 2π radians gives the frequency multiplier (the C in A + B sin (Cx +D)) as π/12. Finally, the phase shift is found by noting that the maximum temperature occurs at 6 P.M., which means that (Cx + D) must equal π/2 when x = 18. This means D = -π. Putting it all together, we have T = 70 + 15 sin (πx/12 - π), and therefore the temperature at 9 A.M., when x = 9, is equal to 70 - (15√2)/2.