SOLUTION: Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 53 degrees occurs at 4 PM and the average temperature for the day i

Question 1149301: Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 53 degrees occurs at 4 PM and the average temperature for the day is 45 degrees. Find the temperature, to the nearest degree, at 8 AM.
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

We have an average temperature of 45 degrees and a high of 53 degrees. That makes the midline 45 and the amplitude 8.

We could spend some time trying to figure out how to get a graph in which x=0 corresponds to midnight. However; that is not necessary. We can use any sinusoidal function we want, as long as we understand which part of the graph corresponds to what hour of the day.

So with midline 45 and amplitude 8, let's choose a simple function:

A graph (showing 1.5 periods, from -pi to 2pi)....

The time at which we are to find the temperature, 8 AM, is 8 hours before 4 PM, which is the time when the temperature is maximum. 8 hours is 1/3 of a day, so we want to find the temperature 1/3 of a cycle before 4 PM.

Since we are using a sine function with no horizontal shift, the time of maximum temperature, 4 PM, corresponds to 1/4 of the way through a cycle, at pi/2.

We want to go back 1/3 of a cycle, or (2/3)pi, from pi/2; that puts us at -pi/6.

That makes it easy to find the temperature at 4 AM without a calculator:

The temperature at 4 AM is 41 degrees.

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