SOLUTION: I tried to do with examples and couldn't make it!!! Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 60 an

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Question 1179528: I tried to do with examples and couldn't make it!!!
Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 60 and 80 degrees during the day and the average daily temperature first occurs at 8 AM. How many hours after midnight, to two decimal places, does the temperature first reach 65 degrees?
? hours after midnight.
Found 2 solutions by ikleyn, htmentor:
Answer by ikleyn(52814)   (Show Source): You can put this solution on YOUR website!
.
Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies
between 60 and 80 degrees during the day and the average daily temperature first occurs at 8 AM.
How many hours after midnight, to two decimal places, does the temperature first reach 65 degrees?
~~~~~~~~~~~~~~~ 


Temperature varies between 60 and 80 degrees - - - it means that the average daily temperature is 70 degrees.


So, the temperature is 70 degrees at 8 am (given).


Assuming that the temperature is sinusoidal during the 24 hours, we must take it as a given that 
the temperature is 70 degrees 12 hours after 8 am, i.e. at 8 pm.


Then we can construct this table of temperatures


    average, 70 degs     8 am           = 8 am

    maximum, 80 degs     8 am + 6 hours = 2 pm

    average, 70 degs     2 pm + 6 hours = 8 pm

    minimum, 60 degs     8 pm + 6 hours = 2 am   next day

    average, 70 degs     8 am           = 8 am   next day




           From this table, you see that you should examine
        two time intervals: one from  8 pm today to 2 am next day
          and the second from 2 am next dat to 8 am next day.



So, after 8 pm, the temperature continues decreasing from 70 degs to the lowest temperature of 60 degs.

In this way, the temperature passes 65 degrees  when   = 1/2, i.e.  with the shift of 1/12  of the 24-hour period.



        1/12 of the 24-hours period is 2 hours.



So we conclude that for the first time interval, the temperature will reach 65 degrees at  8 pm + 2 hours, which is 10 pm.
It is still BEFORE midnight, so we do not count this time moment.


Next time moment with the 65 degs will be 2 hours BEFORE 8 am next day, which is 6 am next day.


So, your  ANSWER  is THIS:  first time after midnight, the temperature will reach 65 degs at 6 am.

Solved.



Answer by htmentor(1343)   (Show Source): You can put this solution on YOUR website!
The sinusoidal variation with temperature will have the general form:
T(t) = T0 + Asin(wt - phi), where T0 is the average temperature, A is
the amplitude of variation, w is the angular frequency, and phi is a phase shift
Since the day starts at midnight, i.e. t = 0, and the day does not
necessarily start out at the average temperature, we need to introduce
a phase shift. We are told that the temperature varies between 60 and 80,
thus the average temperature = 70, and the amplitude of variation about the
average is 10. Since the temperature completes one cycle in 24 h, the
angular frequency w = 2*pi/24 = pi/12
Since the average temperature is first reached at 8:00 am, which is 8 hours
after midnight, sin(pi*8/12 - phi) = 0 -> phi = 8/12*pi = (2/3)pi
Thus, the equation is T(t) = 70 + 10sin((pi/12)t - (2pi/3))
T(t) = 65 = 70 + 10sin((pi/12)t - (2pi/3)) -> -1/2 = sin((pi/12)t - (2pi/3))
Since asin(-1/2) = -pi/6 -> (pi/12)t - 2pi/3 = -pi/6
Solving for t gives t = 6. Thus the temperature is 65 at 6:00 am
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