During a particular day in a Mediterranean city the temperature inside an office building between 10 am and 7.30 pm flunctuates so that t hours after 10 am, the temperature T degrees is give by T=19+6sin(pi*t/6)
When the temperature teachers 24 degrees an air conditioner in the boardroom is switched on and it is switched off when the temperature in the rest of the building falls below 24 degrees. For how long is the air conditioner on in the bathroom?
Help with maths?
2016-08-09 9:11 am
回答 (2)
2016-08-09 9:38 am
A description is given for the boardroom, and the question asks about the bathroom. No solution can be provided on the basis of information given.
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If we assume that we want to know how long building temperature is above 24 degrees, we can solve
.. 19 + 6sin(π/6*t) ≥ 24
.. sin(π/6*t) ≥ (24-19)/6
.. t ≥ (6/π)*arcsin(5/6) and t ≤ (6/π)*(π - arcsin(5/6))
This expression will be true for ...
.. 1.8814 ≤ t ≤ 4.1186
The air conditioner will be on for about 4.1186-1.8814 = 2.2372 hours, about 2 hours 14 minutes 14 seconds.
_____
If we assume that we want to know how long building temperature is above 24 degrees, we can solve
.. 19 + 6sin(π/6*t) ≥ 24
.. sin(π/6*t) ≥ (24-19)/6
.. t ≥ (6/π)*arcsin(5/6) and t ≤ (6/π)*(π - arcsin(5/6))
This expression will be true for ...
.. 1.8814 ≤ t ≤ 4.1186
The air conditioner will be on for about 4.1186-1.8814 = 2.2372 hours, about 2 hours 14 minutes 14 seconds.
2016-08-09 9:44 am
T = 19 + 6sin(πt/6)
so, when T = 24 we have:
24 = 19 + 6sin(πt/6)
i.e. sin(πt/6) = 5/6
=> πt/6 = 0.985 + 2nπ or (π - 0.985) + 2nπ....for n = 0, 1, 2,...
Hence, t = 1.881 + 12n or 4.119 + 12n
Now, 1.881 hours is 1 hour and 53 mins
and 4.119 hours is 4 hours and 7 mins
i.e. the air conditioner switches on at 11:53 and turns off just after 14:07
Hence, duration of operation is 2 hours and 14 minutes.
:)>
so, when T = 24 we have:
24 = 19 + 6sin(πt/6)
i.e. sin(πt/6) = 5/6
=> πt/6 = 0.985 + 2nπ or (π - 0.985) + 2nπ....for n = 0, 1, 2,...
Hence, t = 1.881 + 12n or 4.119 + 12n
Now, 1.881 hours is 1 hour and 53 mins
and 4.119 hours is 4 hours and 7 mins
i.e. the air conditioner switches on at 11:53 and turns off just after 14:07
Hence, duration of operation is 2 hours and 14 minutes.
:)>
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