Question 1179528
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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?
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<pre>
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  {{{sin(alpha)}}} = 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  <U>ANSWER</U>  is THIS:  first time after midnight, the temperature will reach 65 degs at 6 am.
</pre>

Solved.