Calculus 2 class Trig question

Question

Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature is 45 degrees at midnight and the high and low temperature during the day are 50 and 40 degrees, respectively. Assuming t is the number of hours since midnight, find an equation for the temperature, D, in terms of t.

D(t)= ?

Answer 1

Assuming that the period is 24 hours, D(t) = ±Asin(Bt) + C, where:

 

      2π/B = period = 24   So, B = π/12

 

      The temperature fluctuates between 40° and 50°.  The equilibrium position is halfway between the high and                                  low, namely 45°.  So, C = 45 (shift the graph of y = Asin(Bt) up 45 units).

 

      A = amplitude = distance between the equilibrium temperature and the high or low temperature.

      So, A = 5.

 

Therefore, D(t) = ±5sin[(π/12)t] + 45.

 

     If the high temperature occurs in the early morning (6 AM), use the + sign.  If the low temperature occurs at 6 AM, use the

     - sign.