find a possible formula for the function.

Question

The pressure P (in pounds per square foot), in a pipe varies over time. Two times an hour, the pressure oscillates from a low of 40 to a high of 200 and then back to a low of 40. The pressure at time t=0 is 40. Let the function P=f(t) denote the pressure in pipe at time t minutes.
Find a possible formula for the function P=f(t) described above.

Answer 1

It should have this form:

 

P(t) = c + a sin(bt + d)

 

c = mid point = (200 + 40)/2 = 120

a = amplitude = (200 - 40)/2 = 80

b = angular frequency = 2πf, where f = linear frequency

f = 2 cycles per hour ⇒ b = 2πf = 2π(2) = 4π radians/hour

d = initial phase

 

So far we have

 

P(t) = 120 + 80sin(4πt + d), with t in hours, P(t) in lbs/ft²

 

P(0) = 40 ⇒ 120 + 80sin[4π(0) + d] = 120 + 80sin(d)

40 = 120 + 80sin(d)

sin(d) = (40 - 120)/80 = -80/80 = -1

d = sin^-1(-1) = 3π/2  [in radians]

 

Finally,

 

P(t) = 120 + 80sin(4πt + 3π/2), with t in HOURS, the argument of the sine in radians, and P in lbs/ft²

 

Test it:

 

Is P(0) = 40?

P(0) = 120 + 80sin(3π/2) = 120 + 80(-1) = 40  [good]

 

Is P_max = 200?

P_max occurs when the sine is maximum, i.e. when the sine is 1.

P_max = 120 + 80(1) = 200  [good]

 

Is P_min = 40?

P_min occurs when the sine is at its minimum of -1.

P_min = 120 + 80(-1) = 40  [good]

 

Is the linear frequency 2 cycles per hour?

f = 4π/(2π) cycles per hour = 2 cycles per hour  [note the units are cycles per hour, not cycles per second as in Hz]

 

If you want the time in minutes, make the following change:

 

Replace 4π radians/hour with 4π/60 radians per minute = π/15 radians/minute.  Then

 

P(t) = 120 + 80sin[(π/15)t + 3π/2], with t in MINUTES.