SOLUTION: Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 75 degrees at midnight and the high and low temperature during the day a

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Question 1173711: Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 75 degrees at midnight and the high and low temperature during the day are 85 and 65 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.
D(t)=
Found 2 solutions by ikleyn, htmentor:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

    D(t) = 75 -   degrees.


The amplitude 10 is half the difference 85-65.


75 degrees is the mean temperature, and it happens precisely at midnight - so the sine function has no time shift.


The period is 24 hours.

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PLEASE  pay  ATTENTION that the sign before the sine function in my post is  " - "  (MINUS),  as it should be,

in opposite to the post by  @htmentor,  where the sign before the sine function is  " + ",  which is  WRONG.


It is of significant  IMPORTANTCY  (!)


In this problem, to miss these signs has the same effect as to miss a day and a night.



Answer by htmentor(1343)   (Show Source): You can put this solution on YOUR website!
The temperature at time t=0 (midnight)is 75 degrees, and the temperature
oscillates over a range of +/- 10 degrees during a 24 hour period.
Thus the amplitude of oscillation is 10, and the midline temperature is 75.
A complete cycle of 2pi takes place over 24 h, so the "angular frequency" is
w = 2pi/24 = pi/12
So the equation is D(t) = 10sin((pi/12)*t) + 75
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