A sinusoidal function, such as cosine, can be used to model the pressure oscillation inside a pipe. With a period of 6 minutes, amplitude of 105 pounds per square foot, and a midline of 185 pounds per square foot, the function P(t) = 105cos(20πt) + 185 models the pressure accurately.
The student is asked to model pressure variation inside a pipe as a function of time, which demonstrates characteristics of oscillatory motion. Since the pressure oscillates ten times an hour, the period of oscillation (T) is 6 minutes. We can use a sinusoidal function to model this oscillation, such as a sine or cosine wave, which oscillates between its minimum and maximum values.
The amplitude (A) of the wave will be half of the pressure range, so:
A = (290 - 80)/2
= 105 pounds per square foot.
The vertical shift (D) is the midpoint of the pressure range,
D = (290 + 80)/2
= 185 pounds per square foot.
Because the pressure starts at its low point of 80, which corresponds to the sine function starting at 0, we'll use a cosine function with a horizontal shift (C) of 0 to model this behavior:
P(t) = Acos(Bt) + D
where B is the frequency variable, related to the period by B = 2π/T. For T = 6 minutes, B = 2π/6. However, since the function must operate in minutes, and there are 60 minutes in an hour, we convert this to B = (2π)/(6/60) = 20π radians per hour.
Thus, the model for the pressure inside the pipe over time is:
P(t) = 105cos(20πt) + 185