Graphs of the Sine and Cosine Functions

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 80 degrees at midnight and the high and low temperature during the day are 93 and 67 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

Answer 1

T_max = 93

T_min =67

D = f(t) =a(sin(k(x-b)) +h

Where:

|a| = amplitude

Period = 2π/k = 24 hours

k = 2π/24 = π/12

Phase Shift = b (left if b < 0, right if b > 0)

Vertical Shift = h (down if h < 0, up if h > 0) (a.k.a. midline)

h = (93 + 67)/2 = 80ºF

and therefore the amplitude is:

a = 93 - 80 =13ºF

Note since the temperature at midnight is 80ºF (where t = 0, and b = 0) then half way through the 24 day the temperature is also 80ºF at 12 AM (noon). The sinusoidal wave equation has a negative value for "a" since the temperature falls from midnight.

D = -13sin((π/12)t) + 80