Modeling problem

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 81 degrees occurs at 6 PM and the average temperature for the day is 65 degrees. Find the temperature, to the nearest degree, at 11 AM.

Answer 1

If we suppose a standard 24 hour day, and that there would only be one high point of temperature during a day, this implies that our period is length 24. Taking the difference of the maximum temperature with the average temperature gives us the amplitude. The average temperature is the vertical shift of the graph. There is no horizontal shift needed if we use the sin function, since the function reaches a critical value every 6 hours (either a maximum, minimum, or average). Since the temperature start off going down (instead of up like the sin parent function) we need to have a negative out front. Therefore, the resulting function would be y=-16sin((π/12)x)+65 and you could use that to find the temperature at any hour (hours after noon would be in military time). Thus, for 11AM (or 1100 hours) you would simply plug in 11 for x and get out approximately 61^o