Outside temperature over a day can be modeled as a sinusoidal
function. Suppose you know the high temperature of 57 degrees
occurs at 3 PM and the average temperature for the day is 50
degrees. Find the temperature, to the nearest degree, at 7
We have an average temperature of 50 degrees and a high of 57
degrees. That makes the midline 50 and the amplitude
The time at which we are to find the temperature, 7 AM, is 8 hours before 3 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 3 PM.
Since we are using a sine function with no horizontal shift, the time of maximum temperature, 3 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 7 AM
= 50 + 8sin(-π/6)
= 50 - 4
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