# Math

Outside temperature over a day can be modelled as a sinusoidal function. Suppose you know the high temperature for the day is 87 degrees and the low temperature of 63 degrees occurs at 5 AM. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

• So I am pretty sure the equation would start y=25sin(x) because 25 is the difference between the highest and lowest temperature
• In a sine equation the point at 3pi/2 is the lowest, so I think we have to move the graph over so that 5 is the lowest point??? Because the lowest temp happens at 5 hours past midnight. That doesn't really make sense but I'm not sure how to go about this problem.

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1. 2 A = 87-63 = 24
so A = 12 degrees
average = (87+63)/2 = 75
so
D = 75 + 12 sin (2 pi t/24 - p)
where p is the phase to give the right temp when t = 5
when t = 5
sin (2 pi 5/24 - p) = -1

or 5 pi/12 - p = -pi/2
p = 6 pi/12 + 5 pi/12 = 11 pi/12

so
D = 75 + 12 sin ( pi t/12 - 11 pi/12)
or
D = 75 + 12 sin [ pi/12(t-11) ]

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