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find a possible formula for the function.

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.

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Arturo O. | Experienced College Instructor for Physics TutoringExperienced College Instructor for Physi...
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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 Pmax = 200?
Pmax occurs when the sine is maximum, i.e. when the sine is 1.
Pmax = 120 + 80(1) = 200  [good]
 
Is Pmin = 40?
Pmin occurs when the sine is at its minimum of -1.
Pmin = 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.
 

Comments

Note you could also have used a formula of the form
 
P(t) = c + a cos(bt + d)
 
The initial phase "d" will be different if you use a cosine.  The other constants will be the same.